A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

Abstract

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate.

1. Introduction

Focused on fractional-order integrals, derivatives and applications over real and complex domains, fractional calculus is a branch of mathematics. The use of classical analysis arithmetic is necessary to produce findings from fractional analysis that are more realistic. The use of fractional differential equations and integral equations allows for the solving of a wide range of mathematical models. Due to the fact that classical mathematical models are specific examples of fractional-order mathematical models, the findings drawn from fractional-order mathematical models are more comprehensive and correct. Fractional theory allows for the processing of any number of orders, real or integer, as opposed to just integer orders, making it a more suitable approach. In the modern day, fractional calculus techniques and tools have an impact on almost every nonlinear subject or field of research. Electrical engineering, control theory, mechanical engineering, viscoelasticity, rheology, optics and physics are only a few of the many domains of engineering that have numerous and fruitful applications; for further information, see References [1,2,3,4,5,6,7,8].

Rounding and measuring mistakes are reduced in mathematical computations using a mathematical approach known as interval analysis. The most influential study on interval analysis was written by the Japanese scientist Teruo Sunag, as referenced in [9]. In this article, the laws governing fundamental operations with intervals are not only thoroughly investigated but also mathematically analyzed. The range of rational functions can be calculated by using only the terminal points of rational functions over intervals. As multidimensional intervals, interval vectors are also discussed along with the appropriate operations. By enclosing the residual term and confining the value of a definite integral, interval arithmetic tools are used to calculate a pointwise enclosure for the solution of an initial value problem. A compact interval has developed over the past three decades into an autonomous object in numerical analysis for confirming or containing solutions to a variety of mathematical problems or demonstrating that such problems cannot be solved in a specific domain. See References [10,11] for additional information on some real-world uses of interval analysis in various linear and nonlinear disciplines. The H-H-inequality, as we are all aware, is the first geometric interpretation of a convex function and frequently employed in fields involving convex optimization. We also know that inequalities must be incorporated in order to confirm their existence, uniqueness and stability when evaluating the accuracies of various mathematical models based on actual events. In order to address uncertainty and assess the stability of differential models, writers have devised fractional forms of inequality since they frequently utilize mathematical models based on natural phenomena to portray results with greater accuracy. For more information, see [12,13] and the references therein.

But, both in practical and pure analysis, generalized convexity mapping can handle a wide range of issues. There are a number of well-known inequalities that have been created utilizing related classes of convexity, including the Simpson, Ostrowski, Opial, Hardy and well-known H-H models, that have been extended to interval-valued functions. Different concepts of convex classes and integral operators, including the traditional Riemann integral, Caputo Fabrizo, Riemann–Liouville and k-fractional operators, are used by different authors to construct these inequalities. Hadamard fractions and Riemann–Liouville fractions were used by Wang et al. [14] to study various identities for differentiable functions, and they also used these identities to show some inequalities based on s-, r-, m-, (s, m)- and other convex functions. Additionally, Iscan [15] used fractional operators to generate a number of new H-H forms using the preinvexity concept. For the product of two convex maps, Pachpette constructed the refined form of H-H-inequality using a fractional integral in [16]. Chan developed a number of H-H-inequalities for the products of convex functions based on Riemann–Liouville fractional integrals (see Reference [17]). For harmonically convex mappings in the context of fuzzy interval-valued setting, Khan et al. [18] proposed inequalities in an enlarged version using the Riemann–Liouville fractional integral operator. Shi et al.’s [19] work was then expanded to include coordinated convex interval-valued mappings (${\rm I}VM$s) by fractional integrals. They started by creating H-H-inequality results utilizing h-convex and harmonically h-convex functions. Dragomir created Riemann–Liouville fractional integrals with the concept of h-convexity in order to create H-H-inequalities, as seen in Ref. [20]. By means of Riemann–Liouville integral operators, Khan and his coworkers produced H-H-inequalities for left–right set-valued functions [21]. Inequalities of the H-H-Fejer type for harmonically convex functions were created by Kunt et al. [22]. H-H-inequalities with bounds were constructed by Moshin et al. [23] using q-calculus in a variety of ways. Refer to references [24,25,26,27,28,29] for some recent advancements regarding developed inequities. In 2014, Bhunia used a variety of interval measures to investigate the optimality of situations with multiple objectives. Using the radius and interval midpoints, he also created the idea of the center-radius order [30]. In a recent paper by the following authors, Bhunia’s idea was used to more precisely create several kinds of inequalities based on various integral operators [31,32,33,34]. The majority of these inequities, according to our evaluation of the literature, are brought about by partial-order or inclusion relations. The fundamental benefit of fuzzy-order relations for coordinated h-convex functions over the fuzzy codomain is that the inequality term may be predicted more precisely. This claim can be supported by useful illustrations of suggested theorems. It is crucial to comprehend the application of fuzzy total order relations to the analysis of various coordinated convex mapping classes. For more information related to coordinated interval calculus and coordinated fuzzy calculus, see [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58] and the references therein.

Motivated by the recent research, we introduce convexity for the fuzzy-number-valued mappings of two variables in order to create a new set of H-H-type inequalities for the $FNVM$s that are coordinated. The main advantage of these inequalities is their ability to be transformed into new fuzzy Riemann–Liouville fractional H-H inequalities for coordinated convex $FNVM$s. For more information related to classical H-H inequalities for coordinated convex interval-valued mappings (${\rm I}VM$s), see [59].

For our research, we consulted Refs. [60,61,62,63,64] for examples of high-caliber books and papers that serve as inspiration. We created a few different H-H-inequalities on fuzzy-number-valued mappings using newly established class-coordinated $\mathrm{ỽ}$-convex mappings over fuzzy codomain and fractional integrals. For the purpose of illustrating their accuracy, we also made several samples. The essay concludes in the manner described below. Section 2 provides a succinct historical context. Section 3 talks about the main results, and in Section 4, a succinct conclusion is provided.

2. Preliminaries

We will go over the fundamental phrases and ideas that aid in understanding the ideas behind our recent discoveries in this section. In upcoming definitions, remarks, examples and findings, we will use the notions ${\mathbb{R}}_I$, ${\mathbb{R}}_I^{+}$, ${\mathbb{F}}_0$ and ${\mathbb{F}}_0^{+}$ and ${\mathbb{R}}_I$ for a set of fuzzy numbers, set of positive fuzzy numbers, set of intervals and collection of positive intervals, respectively.

Definition 1

([38,39]). Given $\widetilde{\mathrm{Ỿ}}\in {\mathbb{F}}_0$, the level sets or cut sets are provided by

$${\left[\widetilde{\mathrm{Ỿ}}\right]}^{\gamma }=\left\{x\in \mathfrak{R}|\widetilde{\mathrm{Ỿ}}\left(x\right)>\gamma \right\},$$

(1)

$\forall$ $\gamma \in [0, 1]$ and by

$${\left[\widetilde{\mathrm{Ỿ}}\right]}^0=cl\left(\left\{x\in \mathfrak{R}|\widetilde{\mathrm{Ỿ}}\left(x\right)>0\right\}\right).$$

(2)

These sets are known as $\gamma$-level sets or $\gamma$-cut sets of $\widetilde{\mathrm{Ỿ}}$.

Proposition 1

([62]). Let $\widetilde{\mathrm{Ỿ}},\widetilde{₼}\in {\mathbb{F}}_0$. Then, relation $“{\le }_{\mathbb{F}}”$ is provided on ${\mathbb{F}}_0$ by $\widetilde{\mathrm{Ỿ}}{\le }_{\mathbb{F}}\widetilde{₼}$ when and only when ${{\left[\widetilde{\mathrm{Ỿ}}\right]}^{\gamma }\le }_I{\left[\widetilde{₼}\right]}^{\gamma }$, for every $\gamma \in [0, 1],$ which are left- and right-order relations.

Proposition 2

([61]). Let $\widetilde{\mathrm{Ỿ}},\widetilde{₼}\in {\mathbb{F}}_0$. Then, relation $“{\supseteq }_{\mathbb{F}}”$ is provided on ${\mathbb{F}}_0$ by $\widetilde{\mathrm{Ỿ}}{\supseteq }_{\mathbb{F}}\widetilde{₼}$ when and only when ${\left[\widetilde{\mathrm{Ỿ}}\right]}^{\gamma }{\supseteq }_I{\left[\widetilde{₼}\right]}^{\gamma }$, for every $\gamma \in [0, 1],$ which is the $UD-$order relation on ${\mathbb{F}}_0$.

Recall the approaching notions, which are offered in the literature. If $\widetilde{\mathrm{Ỿ}},\widetilde{₼}\in {\mathbb{F}}_0$ and $\mathcal{t}\in \mathfrak{R}$, then, for every $\gamma \in \left[0, 1\right],$ the arithmetic operations addition “$\oplus ”$, multiplication “$\otimes ”$ and scaler multiplication “$\odot ”$ are established by

$${\left[\widetilde{\mathrm{Ỿ}}\oplus \widetilde{₼}\right]}^{\gamma }={\left[\widetilde{\mathrm{Ỿ}}\right]}^{\gamma }+{\left[\widetilde{₼}\right]}^{\gamma },$$

(3)

$${\left[\widetilde{\mathrm{Ỿ}}\otimes \widetilde{₼}\right]}^{\gamma }={\left[\widetilde{\mathrm{Ỿ}}\right]}^{\gamma }\times {\left[ \widetilde{₼}\right]}^{\gamma },$$

(4)

$${\left[\mathcal{t}\odot \widetilde{\mathrm{Ỿ}}\right]}^{\gamma }=\mathcal{t}{\left[\widetilde{\mathrm{Ỿ}}\right]}^{\gamma },$$

(5)

For these results, Equations (4)–(6) have immediate implications.

Lemma 1

([40]). Let $\tilde{\mathcal{G}}:[\mathcal{p},\mathcal{q}]\subset \mathfrak{R}\to {\mathbb{F}}_0$ be an $FNVM$, and its ${\rm I}VM$s are classified according to their $\gamma$-levels ${\mathcal{G}}_{\gamma }:[\mathcal{p},\mathcal{q}]\subset \mathfrak{R}\to {\mathbb{R}}_I$, which are provided by ${\mathcal{G}}_{\gamma }\left(x\right)=\left[{\mathcal{G}}_{*}\left(x,\gamma \right), {\mathcal{G}}^{*}\left(x,\gamma \right)\right]$ $\forall$ $x\in [\mathcal{p},\mathcal{q}]$ and $\forall$ $\gamma \in (0, 1].$ Then, $\tilde{\mathcal{G}}$ is $FA$-integrable over $[\mathcal{p},\mathcal{q}]$ if and only if ${\mathcal{G}}_{*}\left(x,\gamma \right)$ and ${\mathcal{G}}^{*}\left(x,\gamma \right)$ are both $A$-integrable over $[\mathcal{p},\mathcal{q}]$. Moreover, if $\tilde{\mathcal{G}}$ is $FA$-integrable over $\left[\mathcal{p},\mathcal{q}\right],$ then

$$\begin{gathered} {\left[\left(FA\right){\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x\right)dx\right]}^{\gamma }=\left[\left(A\right){\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{G}}_{*}\left(x,\gamma \right)dx, \left(A\right){\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{G}}^{*}\left(x,\gamma \right)dx\right] \\ =\left(IA\right){\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{G}}_{\gamma }\left(x\right)dx, \end{gathered}$$

(6)

$\forall$ $\gamma \in (0, 1].$ For all $\gamma \in \left(0, 1\right],$ ${\mathcal{F}\mathcal{A}}_{\left(\left[\mathcal{p},\mathcal{q}\right], \gamma \right)}$ denotes the collection of all $FA$-integrable $FNVM$s over $[\mathcal{p},\mathcal{q}]$.

Definition 2

([42]). Let $\mathcal{G}:\left[ⱱ, ⱳ\right]\to {\mathbb{R}}_I^{+}$ be $IVM$ and $\mathcal{G}\in {\mathcal{I}\mathcal{R}}_{\left[ⱱ, ⱳ\right]}$. Then, interval Riemann–Liouville-type integrals of $\mathcal{G}$ are established as

$${\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\mathcal{G}\left(y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{ⱱ}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}\mathcal{G}\left(\mathtt{t}\right)d\mathtt{t}\left(y>ⱱ\right),$$

(7)

$${\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\mathcal{G}\left(y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_y^{ⱳ}{\left(\mathtt{t}-y\right)}^{\alpha -1}\mathcal{G}\left(\mathtt{t}\right)d\mathtt{t}(y<ⱳ),$$

(8)

where $\alpha >0$ and $\Gamma$ is the gamma function.

Recently, Allahviranloo et al. [43] introduced the fuzzy version of established fractional integral integrals for the following definition:

Definition 3.

Let $\alpha >0$  and $L\left(\left[ⱱ, ⱳ\right],{\mathbb{F}}_0\right)$  be the collection of all Lebesgue measurable $FNVM$s on $[ⱱ,ⱳ]$. Then, the fuzzy left and right Riemann–Liouville fractional integrals of $\tilde{\mathcal{G}}\in$ $L\left(\left[ⱱ, ⱳ\right],{\mathbb{F}}_0\right)$  with order $\alpha >0$  are defined by

$${\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{ⱱ}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}\tilde{\mathcal{G}}\left(\mathtt{t}\right)d\mathtt{t},\left(y>ⱱ\right),$$

(9)

and

$${\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_y^{ⱳ}{\left(\mathtt{t}-y\right)}^{\alpha -1}\tilde{\mathcal{G}}\left(\mathtt{t}\right)d\mathtt{t},(y<ⱳ),$$

(10)

respectively, where $\Gamma \left(y\right)={\int }_0^{\infty }{\mathtt{t}}^{y-1}{e}^{-\mathtt{t}}d\mathtt{t}$ represents the Euler gamma function.

The fuzzy left and right Riemann–Liouville fractional integral $y$ based on left and right end point functions can then be established, that is

$$\begin{array}{l}{\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(y\right)\right]}^{\gamma }={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{ⱱ}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\right)d\mathtt{t}\\ ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{ⱱ}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}\left[{\mathcal{G}}_{*}\left(\mathtt{t},\gamma \right),{\mathcal{G}}^{*}\left(\mathtt{t},\gamma \right)\right]d\mathtt{t},\left(y>ⱱ\right),\end{array}$$

(11)

where

$${\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{*}\left(y, \gamma \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{ⱱ}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}{\mathcal{G}}_{*}\left(\mathtt{t},\gamma \right)d\mathtt{t},\left(y>ⱱ\right),$$

and

$${\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}^{*}\left(y, \gamma \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{ⱱ}^y{\left(y-\mathtt{t}\right)}^{\alpha -1}{\mathcal{G}}^{*}\left(\mathtt{t},\gamma \right)d\mathtt{t},\left(y>ⱱ\right),$$

(12)

Similarly,

$$\begin{array}{l}{\left[{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(y\right)\right]}^{\gamma }={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_y^{ⱳ}{\left(\mathtt{t}-y\right)}^{\alpha -1}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\right)d\mathtt{t}\\ ={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_y^{ⱳ}{\left(\mathtt{t}-y\right)}^{\alpha -1}\left[{\mathcal{G}}_{*}\left(\mathtt{t},\gamma \right),{\mathcal{G}}^{*}\left(\mathtt{t},\gamma \right)\right]d\mathtt{t},\left(y<ⱳ\right),\end{array}$$

(13)

where

$${\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{*}\left(y, \gamma \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_y^{ⱳ}{\left(\mathtt{t}-y\right)}^{\alpha -1}{\mathcal{G}}_{*}\left(\mathtt{t},\gamma \right)d\mathtt{t},\left(y<ⱳ\right),$$

and

$${\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}^{*}\left(y, \gamma \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_y^{ⱳ}{\left(\mathtt{t}-y\right)}^{\alpha -1}{\mathcal{G}}^{*}\left(\mathtt{t},\gamma \right)d\mathtt{t},\left(y<ⱳ\right).$$

(14)

Lemma 2

([44]). Let $\mathrm{ỽ} :\left[0, 1\right]\to {\mathbb{R}}^{+}$ and $\tilde{\mathcal{G}}:\left[\mathcal{p},\mathcal{q}\right]\to {\mathbb{F}}_0^{+}$ be convex $FNVMs$ on $\left[\mathcal{p},\mathcal{q}\right],$ whose $\gamma$-cuts set up the sequence of ${\rm I}VM$s ${\mathcal{G}}_{\gamma }:\left[\mathcal{p},\mathcal{q}\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ provided by ${\mathcal{G}}_{\gamma }\left(y\right)=\left[{\mathcal{G}}_{*}\left(y,\gamma \right), {\mathcal{G}}^{*}\left(y,\gamma \right)\right]$ for all $y\in \left[\mathcal{p},\mathcal{q}\right]$ and for all $\gamma \in [0, 1]$. If $\tilde{\mathcal{G}}\in L\left(\left[\mathcal{p},\mathcal{q}\right],{\mathbb{F}}_0\right)$, then

$${\displaystyle \frac{1}{\alpha \mathrm{ỽ}\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\alpha }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(\mathcal{p}\right)\right]{\le }_{\mathbb{F}}\left[\tilde{\mathcal{G}}\left(\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(\mathcal{q}\right)\right]{\int }_0^1{\mathtt{t}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}.$$

(15)

Interval- and fuzzy Aumann-type integrals are established as follows for the coordinated $IVM$ $\mathcal{G}\left(x,y\right)$ and coordinated $FNVM$ $\tilde{\mathcal{G}}\left(x,y\right)$:

Lemma 3

([46]). Let $\tilde{\mathcal{G}}:∆\left[ⱱ, ⱳ\right]\times \left[\mathcal{p},\mathcal{q}\right]\subset {\mathbb{R}}^2\to {\mathbb{F}}_0$ be an $FNVM$ on coordinates, whose $\gamma$-cuts set up the sequence of ${\rm I}VMs$ ${\mathcal{G}}_{\gamma }:∆\subset {\mathbb{R}}^2\to {\mathbb{R}}_I$ provided by ${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆=\left[ⱱ, ⱳ\right]\times \left[\mathcal{p},\mathcal{q}\right]$ and for all $\gamma \in \left[0, 1\right].$ Then, $\tilde{\mathcal{G}}$ is fuzzy double-integrable ($FD$-integrable) over $∆$ if and only if ${\mathcal{G}}_{*}\left(x,\gamma \right)$ and ${\mathcal{G}}^{*}\left(x,\gamma \right)$ are both $D$-integrable over $∆.$ Moreover, if $\tilde{\mathcal{G}}$ is $FD$-integrable over $∆,$ then

$$\begin{gathered} {\left[\left(FD\right){\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)dydx\right]}^{\begin{array}{c} \\ \gamma \end{array}}=\left[\left(D\right){\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right)dydx, \left(D\right){\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)dydx\right] \\ =\left(ID\right){\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\mathcal{G}}_{\gamma }\left(x,y\right)dydx, \end{gathered}$$

(16)

for all $\gamma \in \left[0, 1\right].$

The family of all $FD$-integrable $FNVM$s over coordinates and $D$-integrable functions over coordinates are denoted by ${\mathcal{F}\mathfrak{O}}_{∆}$ and ${\mathfrak{O}}_{\left(∆, \gamma \right)}$ for all $\gamma \in \left[0, 1\right].$

Here, the main definition of the fuzzy Riemann–Liouville fractional integral on the coordinates of the function $\tilde{\mathcal{G}}\left(x,y\right)$ is given by the following definition:

Definition 4

([45]). Let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ and $\tilde{\mathcal{G}}\in {\mathcal{F}\mathfrak{O}}_{∆}$. The double fuzzy interval Riemann–Liouville-type integrals ${\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+} }^{\alpha , \beta }, {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-} }^{\alpha , \beta },{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+} }^{\alpha , \beta }, and {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-} }^{\alpha , \beta }$ of $\mathcal{G}$ order $\alpha , \beta >0$ are established by the following equations:

$${\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(x,y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_{ⱱ}^x{\int }_{\mathcal{p}}^y{{\left(x-\mathtt{t}\right)}^{\alpha -1}\left(y-\mathtt{s}\right)}^{\beta -1}\tilde{\mathcal{G}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x>ⱱ, y>\mathcal{p}\right),$$

(17)

$${\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(x,y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_{ⱱ}^x{\int }_y^{\mathcal{q}}{{\left(x-\mathtt{t}\right)}^{\alpha -1}\left(\mathtt{s}-y\right)}^{\beta -1}\tilde{\mathcal{G}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (x>ⱱ, y<\mathcal{q}),$$

(18)

$${\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(x,y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_x^{ⱳ}{\int }_{\mathcal{p}}^y{{\left(\mathtt{t}-x\right)}^{\alpha -1}\left(y-\mathtt{s}\right)}^{\beta -1}\tilde{\mathcal{G}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(x<ⱳ, y>\mathcal{p}\right),$$

(19)

$${\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(x,y\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}{\int }_x^{ⱳ}{\int }_y^{\mathcal{q}}{{\left(\mathtt{t}-x\right)}^{\alpha -1}\left(\mathtt{s}-y\right)}^{\beta -1}\tilde{\mathcal{G}}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (x<ⱳ, y<\mathcal{q}).$$

(20)

Here, the newly established concept of coordinated $\mathrm{ỽ}$-convexity over fuzzy number space in the codomain via fuzzy relation is provided by the following definition:

Definition 5.

The $FNVM$ $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ is referred to as a coordinated $\mathrm{ỽ}$-convex $FNVM$ on$∆$ if

$$\begin{array}{l}\tilde{\mathcal{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right),\end{array}$$

(21)

for all$\left(ⱱ, ⱳ\right), \left(\mathcal{p},\mathcal{q}\right)\in ∆$ and $\mathtt{t},\mathtt{ԟ}\in \left[0, 1\right],$ where $\tilde{\mathcal{G}}\left(x\right){\ge }_{\mathbb{F}}\tilde{0}.$ If inequality (21) is reversed, then $\tilde{\mathcal{G}}$ is referred to as a coordinated $\mathrm{ỽ}$-concave $FNVM$ on $∆$.

Lemma 4.

Let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ be a coordinated $FNVM$ on $∆$. Then, $\tilde{\mathcal{G}}$ is a coordinated $\mathrm{ỽ}$-convex $FNVM$ on $∆$ if and only if there are two coordinated $\mathrm{ỽ}$-convex $FNVM$s: ${\tilde{\mathcal{G}}}_x:\left[\mathcal{p},\mathcal{q}\right]\to {\mathbb{F}}_0$, ${\tilde{\mathcal{G}}}_x\left(w\right)=\tilde{\mathcal{G}}\left(x,w\right)$ and ${\tilde{\mathcal{G}}}_y:\left[ⱱ,ⱳ\right]\to {\mathbb{F}}_0$, ${\tilde{\mathcal{G}}}_y\left(z\right)=\tilde{\mathcal{G}}\left(z,y\right)$.

Theorem 1.

Let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ be an $FNVM$ on $∆$. Then, from $\gamma$-levels, we identify the collection of ${\rm I}VM$s ${\mathcal{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ provided by

$${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right],$$

(22)

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. Then, $\tilde{\mathcal{G}}$ is coordinated $\mathrm{ỽ}$-convex $FNVM$ on $∆$ if and only if, for all $\gamma \in \left[0, 1\right],$ ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)$ and ${\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ are both coordinated $\mathrm{ỽ}$-convex.

Proof. 

We assume that for each $\gamma \in \left[0, 1\right],$ ${\mathcal{G}}_{*}\left(\mathcal{x},\gamma \right)$ and ${\mathcal{G}}^{*}\left(\mathcal{x},\gamma \right)$ are coordinated $\mathrm{ỽ}$-convex and $\mathrm{ỽ}$-concave on $∆$, respectively. Then, for Equation (21), for all $\left(ⱱ, ⱳ\right), \left(\mathcal{p},\mathcal{q}\right)\in ∆, \mathtt{t}$ and $\mathrm{ԟ}\in \left[0, 1\right]$, we have

$$\begin{array}{l}{\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right)\\ \le \mathrm{ỽ}\left(\mathtt{t}\right){\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right)+{\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right),\end{array}$$

and

$$\begin{array}{l}{\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right)\\ \le \mathrm{ỽ}\left(\mathtt{t}\right){\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right)+\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathrm{ỽ}\left(1-\mathtt{t}\right){\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right),\end{array}$$

Then, by Equations (22), (3) and (5), we obtain

$$\begin{array}{l}{\mathcal{G}}_{\gamma }\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\right)\\ =\left[{\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right)\right]\\ {\le }_I\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)\right]+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right), {\mathcal{G}}^{\begin{array}{c} \\ \\ \begin{array}{c} \\ *\end{array}\end{array}}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right)\right]\\ +\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathrm{ỽ}\left(1-\mathtt{t}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)\right]+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right)\right]\end{array}$$

that is

$$\begin{array}{l}\tilde{\mathcal{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right),\end{array}$$

so $\tilde{\mathcal{G}}$ is coordinated $\mathrm{ỽ}$-convex $FNVM$ on $∆.$

Conversely, let $\tilde{\mathcal{G}}$ be coordinated $\mathrm{ỽ}$-convex $FNVM$ on $∆.$ Then, for all $\left(ⱱ, ⱳ\right), \left(\mathcal{p},\mathcal{q}\right)\in ∆, \mathtt{t}$ and $\mathrm{ԟ}\in \left[0, 1\right],$ we have

$$\begin{array}{l}\tilde{\mathcal{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right).\end{array}$$

Therefore, again from Equation (22), for each $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{l}{\mathcal{G}}_{\gamma }\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\right)\\ =\left[{\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right)\right].\end{array}$$

Again, using Equations (3) and (5), we obtain

$$\begin{array}{l}\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right){\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right){\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\\ =\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)\right]+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right)\right]\\ +\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathrm{ỽ}\left(1-\mathtt{t}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{p}\right), \gamma \right)\right]+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\left[{\mathcal{G}}_{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right), {\mathcal{G}}^{*}\left(\left(ⱱ,\mathcal{q}\right), \gamma \right)\right],\end{array}$$

for all $\mathcal{x},\omega \in ∆$ and $\mathtt{t}\in \left[0, 1\right].$ Then, by the coordinated $\mathrm{ỽ}$-convexity of $\tilde{\mathcal{G}}$, we have, for all $\mathcal{x},\omega \in ∆$ and $\mathtt{t}\in \left[0, 1\right]$, such that

$$\begin{array}{l}{\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right)\\ \le \mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(ⱱ,\mathcal{p}\right)+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(ⱱ,\mathcal{q}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(ⱳ,\mathcal{p}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(ⱳ,\mathcal{q}\right),\end{array}$$

and

$$\begin{array}{l}{\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right), \gamma \right)\\ \le \mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right){\mathcal{G}}^{*}\left(ⱱ,\mathcal{p}\right)+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}^{*}\left(ⱱ,\mathcal{q}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(\mathrm{ԟ}\right){\mathcal{G}}^{*}\left(ⱳ,\mathcal{p}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}^{*}\left(ⱳ,\mathcal{q}\right),\end{array}$$

for each $\gamma \in \left[0, 1\right].$ Hence, the result is as follows. □

Example 1.

We consider the $FNVM$ $\tilde{\mathcal{G}}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0$ established by

$$\mathcal{G}\left(x,y\right)\left(\sigma \right)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\sigma -xy}{5-xy}},& \left[xy, 5\right]\hfill \\ \hfill {\displaystyle \frac{\left(6+e^x\right)\left(6+e^y\right)-\sigma }{\left(6+e^x\right)\left(6+e^y\right)-5}},& \sigma \in \left(5, \left(6+e^x\right)\left(6+e^y\right)\right]\hfill \\ \hfill 0,& otherwise,\hfill \end{array}\right.$$

(23)

Then, for each $\gamma \in \left[0, 1\right],$ we have ${\mathcal{G}}_{\gamma }\left(x\right)=\left[\left(1-\gamma \right)xy+5\gamma ,\left(1-\gamma \right)\left(6+e^x\right)\left(6+e^y\right)+5\gamma \right]$. The endpoint functions ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right),$ ${\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ are coordinated $\mathrm{ỽ}$-concave functions for each $\gamma \in \left[0, 1\right]$. Hence, $\tilde{\mathcal{G}}\left(x,y\right)$ is a coordinated $\mathrm{ỽ}$-convex $FNVM$.

From Lemma 4 and Example 1, we can easily note that each $\mathrm{ỽ}$-convex $FNVM$ is a coordinated $\mathrm{ỽ}$-convex $FNVM$. But, the opposite is not true.

Remark 1.

If one assumes that ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right)={\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ with $\gamma =1$, then $\mathcal{G}$ is referred to as a classical coordinated $\mathrm{ỽ}$-convex function if $\mathcal{G}$ meets the inequality stated here:

$$\begin{array}{l}\mathcal{G}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ \le \mathrm{ỽ}\left(\mathtt{t}\right){\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathcal{G}}_{*}\left(ⱱ,\mathcal{p}\right)+\mathrm{ỽ}\left(\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(ⱱ,\mathcal{q}\right)+{\mathrm{ỽ}\left(\mathrm{ԟ}\right)\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathcal{G}}_{*}\left(ⱱ,\mathcal{p}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\mathrm{ỽ}\left(1-\mathrm{ԟ}\right){\mathcal{G}}_{*}\left(ⱱ,\mathcal{q}\right).\end{array}$$

(24)

If one assumes that $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right)={\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ with $\gamma =1$, then $\mathcal{G}$ is referred to as a classical coordinated convex function if $\mathcal{G}$ meets the inequality stated here:

$$\begin{array}{l}\mathcal{G}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ \le \mathtt{t}\mathtt{ԟ}\mathcal{G}\left(ⱱ,\mathcal{p}\right)+\mathtt{t}\left(1-\mathrm{ԟ}\right)\mathcal{G}\left(ⱱ,\mathcal{q}\right)+\left(1-\mathtt{t}\right)\mathrm{ԟ}\mathcal{G}\left(ⱳ,\mathcal{p}\right)+\left(1-\mathtt{t}\right)\left(1-\mathrm{ԟ}\right)\mathcal{G}\left(ⱳ,\mathcal{q}\right).\end{array}$$

(25)

Let us assume that $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ with $\gamma =1$, as well as that ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right)$ is an affine function and ${\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ is a concave function. If it meets the inequality stated here, see the following equation [63]:

$$\begin{array}{l}\mathcal{G}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ \supseteq \mathtt{t}\mathtt{ԟ}\mathcal{G}\left(ⱱ,\mathcal{p}\right)+\mathtt{t}\left(1-\mathrm{ԟ}\right)\mathcal{G}\left(ⱱ,\mathcal{q}\right)+\left(1-\mathtt{t}\right)\mathrm{ԟ}\mathcal{G}\left(ⱳ,\mathcal{p}\right)+\left(1-\mathtt{t}\right)\left(1-\mathrm{ԟ}\right)\mathcal{G}\left(ⱳ,\mathcal{q}\right),\end{array}$$

(26)

is true.

Definition 6.

Let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ be an $FNVM$ on $∆$, provided using

$${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right],$$

(27)

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)$ and ${\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{ỽ}$-convex (concave) and affine functions on $∆$ for all $\gamma \in \left[0, 1\right],$ respectively, then $\tilde{\mathcal{G}}$ is a coordinated left-$\mathrm{ỽ}$-convex (concave) $FNVM$ on $∆$.

Definition 7.

Let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ be an $FNVM$ on $∆$, established using

$${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right],$$

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)$ and ${\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{ỽ}$-affine and $\mathrm{ỽ}$- (concave) functions on $∆$, respectively, then $\tilde{\mathcal{G}}$ is a coordinated right-$\mathrm{ỽ}$-convex (concave) $FNVM$ on $∆$.

Theorem 2.

Let $∆$ be a coordinated convex set, and let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0$ be an $FNVM$. Then, from $\gamma$-levels, we obtain the collection of IVMs ${\mathcal{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ provided by

$${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right],$$

for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. Then, $\tilde{\mathcal{G}}$ is a coordinated $\mathrm{ỽ}$-concave $FNVM$ on $∆$ if and only if, for all $\gamma \in \left[0, 1\right],$ ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)$ and ${\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ are coordinated $\mathrm{ỽ}$-concave and $\mathrm{ỽ}$-convex functions, respectively.

Proof. 

The demonstration of proof of Theorem 2 is similar to the demonstration proof of Theorem 1. □

In the next section, to avoid confusion, we will not include the symbols $\left(R\right)$, $\left(IR\right)$, $\left(FR\right)$, $\left(ID\right)$, and $\left(FD\right)$ before the integral sign.

The main goal of this article is to develop a number of original fractional coordinated integral inequalities for the H-H types using a coordinated $\mathrm{ỽ}$-concave $FNVM$. We acquired the most recent estimates for mappings, whose products were coordinated $\mathrm{ỽ}$-concave $FNVM$s using the fuzzy fractional operators.

3. Main Results

Here is first result of the coordinated integral inequalities for the H-H type using the fuzzy fractional operators via the coordinated $\mathrm{ỽ}$-concave $FNVM$s.

Theorem 3.

Let $\tilde{\mathcal{G}}:∆\to {\mathbb{F}}_0^{+}$ be a coordinated $\mathrm{ỽ}$-convex $FNVM$ on $∆$ and let $\mathrm{ỽ} :\left[0, 1\right]\to {\mathbb{R}}^{+}$. Then, from $\gamma$-cuts, we set up the sequence of ${\rm I}VMs$: ${\mathcal{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ are provided by ${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\tilde{\mathcal{G}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{l}{\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}\\ \oplus {\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{ \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)d\mathtt{t}\\ {\le }_{\mathbb{F}}\alpha \beta \left[\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\left[\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}.\end{array}$$

(28)

If $\tilde{\mathcal{G}}\left(x,y\right)$ is a coordinated $\mathrm{ỽ}$-concave $FNVM$, then

$$\begin{array}{l}{\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\ge }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]\\ {\ge }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\ge }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}\\ \oplus {\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{ \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\left[\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}\\ {\ge }_{\mathbb{F}}\alpha \beta \left[\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\left[\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)\right]d\mathtt{t}.\end{array}$$

(29)

Proof. 

Let $\tilde{\mathcal{G}}:\left[ⱱ, ⱳ\right]\to {\mathbb{F}}_0$ be a coordinated $\mathrm{ỽ}$-convex $FNVM$. Then, based on the hypothesis, we have

$${\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_{\mathbb{F}}\tilde{\mathcal{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)\mathtt{ⱳ},\mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right).$$

By using Theorem 1, for every $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{c}{\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}{\mathcal{G}}_{*}\left(\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right), \gamma \right) \\ \le {\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)\mathtt{ⱳ},\mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right), \gamma \right)+{\mathcal{G}}_{*}\left(\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right), \gamma \right), \\ {\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}{\mathcal{G}}^{*}\left(\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right), \gamma \right) \\ \le {\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)\mathtt{ⱳ},\mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right), \gamma \right)+{\mathcal{G}}^{*}\left(\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right),\gamma \right).\end{array}$$

By using Lemma 4, we have

$$\begin{array}{c}{\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}_{*}\left(\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right), \gamma \right)\le {\mathcal{G}}_{*}\left(\left(x, \mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right), \gamma \right)+{\mathcal{G}}_{*}\left(\left(x, \left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right), \gamma \right), \\ {\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}^{*}\left(\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right), \gamma \right)\le {\mathcal{G}}^{*}\left(\left(x, \mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right), \gamma \right)+{\mathcal{G}}^{*}\left(\left(x, \left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right),\gamma \right),\end{array}$$

(30)

and

$$\begin{array}{c}{\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}_{*}\left(\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right), \gamma \right)\le {\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right), \gamma \right)+{\mathcal{G}}_{*}\left(\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},y\right), \gamma \right), \\ {\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}^{*}\left(\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right), \gamma \right)\le {\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right), \gamma \right)+{\mathcal{G}}^{*}\left(\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},y\right),\gamma \right).\end{array}$$

(31)

From (30) and (31), we have

$$\begin{array}{l}{\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}\left[{\mathcal{G}}_{*}\left(\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right), \gamma \right),{\mathcal{G}}^{*}\left(\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right), \gamma \right)\right]\\ {\le }_I\left[{\mathcal{G}}_{*}\left(\left(x, \mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right), \gamma \right),{\mathcal{G}}^{*}\left(\left(x, \mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right), \gamma \right)\right]\\ +\left[{\mathcal{G}}_{*}\left(\left(x, \left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right), \gamma \right),{\mathcal{G}}^{*}\left(\left(x, \left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right),\gamma \right)\right],\end{array}$$

and

$$\begin{array}{l}{\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}\left[{\mathcal{G}}_{*}\left(\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right), \gamma \right),{\mathcal{G}}^{*}\left(\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right), \gamma \right)\right]\\ {\le }_I\left[{\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right), \gamma \right),{\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right), \gamma \right)\right]\\ +\left[{\mathcal{G}}_{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right), \gamma \right),{\mathcal{G}}^{*}\left(\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right),\gamma \right)\right],\end{array}$$

It follows that

$${\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\mathcal{G}}_{\gamma }\left(x, \mathtt{t}\mathcal{p}+\left(1-\mathtt{t}\right)\mathcal{q}\right)+{\mathcal{G}}_{\gamma }\left(x, \left(1-\mathtt{t}\right)\mathcal{p}+\mathtt{t}\mathcal{q}\right),$$

(32)

and

$${\displaystyle \frac{1}{\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right){\le }_I{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right)+{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, y\right).$$

(33)

For ${\mathcal{G}}_{\gamma }\left(x,.\right)$ and ${\mathcal{G}}_{\gamma }\left(.,y\right)$, both are coordinated $\mathrm{ỽ}$-convex-${\rm I}VM$s; then, using inequality (15), for every $\gamma \in \left[0, 1\right]$, for inequalities (32) and (43), we have

$${\displaystyle \frac{1}{\beta \mathrm{ỽ}\left(\frac{1}{2}\right)}}{{\mathcal{G}}_{\gamma }}_x\left({\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\beta \right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{{\mathcal{G}}_{\gamma }}_x\left(\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta } {{\mathcal{G}}_{\gamma }}_x\left(\mathcal{p}\right)\right]{\le }_I\left[{{\mathcal{G}}_{\gamma }}_x\left(\mathcal{p}\right)+{{\mathcal{G}}_{\gamma }}_x\left(\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}$$

(34)

and

$${\displaystyle \frac{1}{\alpha \mathrm{ỽ}\left(\frac{1}{2}\right)}}{{\mathcal{G}}_{\gamma }}_y\left({\displaystyle \frac{ⱱ+ⱳ}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{{\mathcal{G}}_{\gamma }}_y\left(ⱳ\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{{\mathcal{G}}_{\gamma }}_y\left(ⱱ\right)\right]{\le }_I\left[{{\mathcal{G}}_{\gamma }}_y\left(ⱱ\right)+{{\mathcal{G}}_{\gamma }}_y\left(ⱳ\right)\right]{\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)d\mathtt{t}$$

(35)

Since ${{\mathcal{G}}_{\gamma }}_x\left(w\right)={\mathcal{G}}_{\gamma }\left(x,w\right)$, then (34) can be written as

$${\displaystyle \frac{1}{\beta \mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\beta \right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(x,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(x,\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{G}}_{\gamma }\left(x,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(x,\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.$$

(36)

that is

$$\begin{gathered} {\displaystyle \frac{1}{\beta \mathrm{ỽ}\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\displaystyle \frac{1}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[ {\int }_{\mathcal{p}}^{\mathcal{q}}{\left(\mathcal{q}-ԟ\right)}^{\beta -1}{\mathcal{G}}_{\gamma }\left(x,\mathrm{ԟ}\right)d\mathrm{ԟ}+{\int }_{\mathcal{p}}^{\mathcal{q}}{\left(ԟ-\mathcal{p}\right)}^{\beta -1}{\mathcal{G}}_{\gamma }\left(x,\mathrm{ԟ}\right)d\mathrm{ԟ}\right] \\ {\le }_I\left[{\mathcal{G}}_{\gamma }\left(x,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(x,\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}. \end{gathered}$$

By multiplying double inequality (36) by ${\displaystyle \frac{{\left(ⱳ-x\right)}^{\alpha -1}}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}$ and integrating with respect to $x$ over $\left[ⱱ, ⱳ\right],$ we have

$$\begin{array}{l}{\displaystyle \frac{1}{{\beta \left(ⱳ-ⱱ\right)}^{\alpha }\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\int }_{ⱱ}^{ⱳ}{\mathcal{G}}_{\gamma }\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\left(ⱳ-x\right)}^{\alpha -1}dx\\ {\le }_I{\displaystyle \frac{1}{{{\left(ⱳ-ⱱ\right)}^{\alpha }\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\left(ⱳ-x\right)}^{\alpha -1}{\left(\mathcal{q}-ԟ\right)}^{\beta -1}{\mathcal{G}}_{\gamma }\left(x,\mathrm{ԟ}\right)d\mathrm{ԟ}dx+{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\left(ⱳ-x\right)}^{\alpha -1}{\left(ԟ-\mathcal{p}\right)}^{\beta -1}{\mathcal{G}}_{\gamma }\left(x,\mathrm{ԟ}\right)d\mathrm{ԟ}dx\\ {\le }_I{\displaystyle \frac{1}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\int }_{ⱱ}^{ⱳ}{\left(ⱳ-x\right)}^{\alpha -1}{\mathcal{G}}_{\gamma }\left(x,\mathcal{p}\right)dx+{\int }_{ⱱ}^{ⱳ}{\left(ⱳ-x\right)}^{\alpha -1}{\mathcal{G}}_{\gamma }\left(x,\mathcal{q}\right)dx\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.\end{array}$$

(37)

Again, by multiplying double inequality (36) by ${\displaystyle \frac{{\left(x-ⱱ\right)}^{\alpha -1}}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}$ and integrating with respect to $x$ over $\left[ⱱ, ⱳ\right],$ we have

$$\begin{array}{l}{\displaystyle \frac{1}{{\beta \left(ⱳ-ⱱ\right)}^{\alpha }\mathrm{ỽ}\left(\frac{1}{2}\right)}}{\int }_{ⱱ}^{ⱳ}{\mathcal{G}}_{\gamma }\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\left(x-ⱱ\right)}^{\alpha -1}dx\\ {\le }_I{\displaystyle \frac{1}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\left(x-ⱱ\right)}^{\alpha -1}{\left(\mathcal{q}-ԟ\right)}^{\beta -1}{\mathcal{G}}_{\gamma }\left(x,\mathrm{ԟ}\right)d\mathrm{ԟ}dx\\ +{\displaystyle \frac{1}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}{\left(x-ⱱ\right)}^{\alpha -1}{\left(ԟ-\mathcal{p}\right)}^{\beta -1}{\mathcal{G}}_{\gamma }\left(x,\mathrm{ԟ}\right)d\mathrm{ԟ}dx\\ {\le }_I{\displaystyle \frac{1}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\int }_{ⱱ}^{ⱳ}{\left(x-ⱱ\right)}^{\alpha -1}{\mathcal{G}}_{\gamma }\left(x,\mathcal{p}\right)dx+{\int }_{ⱱ}^{ⱳ}{\left(x-ⱱ\right)}^{\alpha -1}{\mathcal{G}}_{\gamma }\left(x,\mathcal{q}\right)dx\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.\end{array}$$

(38)

From (37), we have

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\\ {\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right]\\ {\le }_I{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.\end{array}$$

(39)

From (38), we have

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\\ {\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)\right]\\ {\le }_I{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.\end{array}$$

(40)

Since, from $\gamma$-cuts, we obtain the collection of ${\rm I}VM$s ${\mathcal{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}$, we then have

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.\end{array}$$

(41)

And

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\right]{\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}.\end{array}$$

(42)

Similarly, since ${\tilde{\mathcal{G}}}_y\left(z\right)=\tilde{\mathcal{G}}\left(z,y\right)$, from (35), (41) and (42), we then have

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+} }^{\beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta } \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\right].\end{array}$$

(43)

and

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{2\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\alpha }}}\left[{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\right].\end{array}$$

(44)

The second, third and fourth inequalities of (28) will be the consequence of adding the inequalities of (41)–(44).

Now, for any $\gamma \in \left[0, 1\right]$, we have inequality (15)’s left portion:

$${\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]$$

(45)

and

$${\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right].$$

(46)

The following inequality is created by adding inequalities (45) and (46):

$$\begin{array}{l}{\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right){\le }_I{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\\ +{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right].\end{array}$$

(47)

Similarly, since we obtain the set of ${\rm I}VM$s ${\mathcal{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ for $\gamma \in \left[0, 1\right]$, the inequality can be expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{1}{{\mathrm{ỽ}}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{\mathrm{ỽ}\left(\frac{1}{2}\right)\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{\mathrm{ỽ}\left(\frac{1}{2}\right)\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right].\end{array}$$

(48)

The first inequality of (28) is this one.

Now, for any $\gamma \in \left[0, 1\right]$, we have inequality (15)’s right portion:

$${\displaystyle \frac{\Gamma \left(\beta \right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta } {\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}$$

(49)

$${\displaystyle \frac{\Gamma \left(\beta \right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta } {\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}$$

(50)

$${\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha } {\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)\right]{\le }_I\left[{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)d\mathtt{t}$$

(51)

$${\displaystyle \frac{\Gamma \left(\alpha \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha } {\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)\right]{\le }_I\left[{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)d\mathtt{t}$$

(52)

By summing inequalities (49)–(52) and then taking the multiplication of the resultant with $\alpha \beta$, we have

$$\begin{array}{l}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)\right]\\ +{\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right]\\ {\le }_I\left[{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)+{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)d\mathtt{t}.\end{array}$$

Since we receive the collection of ${\rm I}VM$s ${\mathcal{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ from $\gamma$-cuts, we have

$$\begin{array}{l}{\displaystyle \frac{\beta \Gamma \left(\alpha +1\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\right]\\ \oplus {\displaystyle \frac{\alpha \Gamma \left(\beta +1\right)}{{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}\left[\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\right]\times {\int }_0^1{\mathrm{ԟ}}^{\beta -1}\left[\mathrm{ỽ}\left(\mathrm{ԟ}\right)+\mathrm{ỽ}\left(1-\mathrm{ԟ}\right)\right]d\mathrm{ԟ}{\int }_0^1{\mathtt{t}}^{\alpha -1}\mathrm{ỽ}\left(\mathtt{t}\right)+\mathrm{ỽ}\left(1-\mathtt{t}\right)d\mathtt{t}.\end{array}$$

(53)

This is the final inequality of (28), and the conclusion has been established. □

Now, we acquire the exceptional cases that can be obtained from Theorem 3 by applying some restrictions over the parameters.

Remark 2.

If one assumes that $\alpha =1,$ $\beta =1$ and $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then using (28), as a result, there will be inequity [55]:

$$\begin{array}{l}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\supseteq }_{\mathbb{F}}{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{2}}\left[{\displaystyle \frac{1}{ⱳ-ⱱ}}{\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{G}}\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)dx\oplus {\displaystyle \frac{1}{\mathcal{q}-\mathcal{p}}}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right)dy\right]{\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)dydx\\ {\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{4\left(ⱳ-ⱱ\right)}}\left[{\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{G}}\left(x,\mathcal{p}\right)dx\oplus {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{G}}\left(x,\mathcal{q}\right)dx\right]\oplus {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathcal{q}-\mathcal{p}\right)}}\left[{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(ⱱ,y\right)dy\oplus {\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(ⱳ,y\right)dy\right]\\ {\supseteq }_{\mathbb{F}}{\displaystyle \frac{\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)}{4}}.\end{array}$$

(54)

If one assumes that $\alpha =1$ and $\beta =1$, while $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and $\tilde{\mathcal{G}}$ is the coordinated left-$UD$-$\mathrm{ỽ}$-convex, then using (28), as a result, there will be inequity [46]:

$$\begin{array}{l}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{2}}\left[{\displaystyle \frac{1}{ⱳ-ⱱ}}{\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{G}}\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)dx\oplus {\displaystyle \frac{1}{\mathcal{q}-\mathcal{p}}}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right)dy\right]{\le }_{\mathbb{F}}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)dydx\\ {\le }_{\mathbb{F}}{\displaystyle \frac{1}{4\left(ⱳ-ⱱ\right)}}\left[{\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{G}}\left(x,\mathcal{p}\right)dx\oplus {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{G}}\left(x,\mathcal{q}\right)dx\right]\oplus {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathcal{q}-\mathcal{p}\right)}}\left[{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(ⱱ,y\right)dy\oplus {\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(ⱳ,y\right)dy\right]\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)}{4}}.\end{array}$$

(55)

If $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then using (28), we succeed in bringing about the upcoming inequity [35]:

$$\begin{array}{l}\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ \supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\mathcal{G}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\mathcal{G}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]+{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{4\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]\\ \supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}} \left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{p}\right)\right]\\ \supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{8\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+} }^{\alpha }\mathcal{G}\left(ⱳ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\mathcal{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\mathcal{G}\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\mathcal{G}\left(ⱱ,\mathcal{q}\right)\right].\\ +{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{8\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+} }^{ \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\mathcal{G}\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\mathcal{G}\left(ⱳ,\mathcal{p}\right)\right]\\ \supseteq {\displaystyle \frac{\mathcal{G}\left(ⱱ,\mathcal{p}\right)+\mathcal{G}\left(ⱳ,\mathcal{p}\right)+\mathcal{G}\left(ⱱ,\mathcal{q}\right)+\mathcal{G}\left(ⱳ,\mathcal{q}\right)}{4}}.\end{array}$$

(56)

If $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then using (28), we succeed in bringing about the upcoming inequity [63]:

$$\begin{array}{l}\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ \supseteq {\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{2}}\left[{\displaystyle \frac{1}{ⱳ-ⱱ}}{\int }_{ⱱ}^{ⱳ}\mathcal{G}\left(x,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)dx+{\displaystyle \frac{1}{\mathcal{q}-\mathcal{p}}}{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},y\right)dy\right]\supseteq {\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}} {\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{G}\left(x,y\right)dydx\\ \supseteq {\displaystyle \frac{1}{4\left(ⱳ-ⱱ\right)}}\left[{\int }_{ⱱ}^{ⱳ}\mathcal{G}\left(x,\mathcal{p}\right)dx+{\int }_{ⱱ}^{ⱳ}\mathcal{G}\left(x,\mathcal{q}\right)dx\right]+{\displaystyle \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\mathcal{q}-\mathcal{p}\right)}}\left[{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{G}\left(ⱱ,y\right)dy+{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{G}\left(ⱳ,y\right)dy\right]\\ \supseteq {\displaystyle \frac{\mathcal{G}\left(ⱱ,\mathcal{p}\right)+\mathcal{G}\left(ⱳ,\mathcal{p}\right)+\mathcal{G}\left(ⱱ,\mathcal{q}\right)+\mathcal{G}\left(ⱳ,\mathcal{q}\right)}{4}}.\end{array}$$

(57)

If $\tilde{\mathcal{G}}$ is the coordinated right-$UD$-$\mathrm{ỽ}$-convex with $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)={\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then using (28), we succeed in bringing about the upcoming inequity [48]:

$$\begin{array}{l}\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha } \mathcal{G}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\mathcal{G}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]+{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{4\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta } \mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta } \mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]\\ \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}} \left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+} }^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta } \mathcal{G}\left(ⱳ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+} }^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta } \mathcal{G}\left(ⱱ,\mathcal{p}\right)\right]\\ \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{8\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+} }^{\alpha }\mathcal{G}\left(ⱳ,\mathcal{p}\right)\mathcal{G}{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha } \mathcal{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-} }^{\alpha }\mathcal{G}\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{ⱳ}^{-}}^{\alpha } \mathcal{G}\left(ⱱ,\mathcal{q}\right)\right]\\ +{\displaystyle \frac{\Gamma \left(\beta +1\right)}{{8\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+} }^{ \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)\widetilde{+}{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta } \mathcal{G}\left(ⱱ,\mathcal{p}\right)+{\mathcal{I}}_{{\mathcal{p}}^{+} }^{\beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta } \mathcal{G}\left(ⱳ,\mathcal{p}\right)\right]\\ \le {\displaystyle \frac{\mathcal{G}\left(ⱱ,\mathcal{p}\right)+\mathcal{G}\left(ⱳ,\mathcal{p}\right)+\mathcal{G}\left(ⱱ,\mathcal{q}\right)+\mathcal{G}\left(ⱳ,\mathcal{q}\right)}{4}}.\end{array}$$

(58)

In the next outcomes, we are going to find very interesting outcomes that will be obtained over the product of two coordinated $\mathrm{ỽ}$ -convex  $FNVM$s. These inequalities are known as Pachpatte inequalities.

Theorem 4.

Let $\tilde{\mathcal{G}}, \tilde{\mathfrak{G}}:∆\to {\mathbb{F}}_0^{+}$ be two coordinated $\mathrm{ỽ}$-convex $FNVM$s on $∆$ and let ${\mathrm{ỽ}}_1,{\mathrm{ỽ}}_2 :\left[0, 1\right]\to {\mathbb{R}}^{+}$. Then, from $\gamma$-cuts, we set up the sequence of ${\rm I}VM$s ${\mathcal{G}}_{\gamma },{\mathfrak{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ provided by ${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right]$ and ${\mathfrak{G}}_{\gamma }\left(x,y\right)=\left[{\mathfrak{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathfrak{G}}^{*}\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\tilde{\mathcal{G}}\otimes \tilde{\mathfrak{G}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}.\end{array}$$

(59)

If $\tilde{\mathcal{G}}$ and $\tilde{\mathfrak{G}}$ are both coordinated $\mathrm{ỽ}$-concave $FNVM$s on $∆$, then the inequality above can be expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\ge }_{\mathbb{F}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}.\end{array}$$

(60)

where

$$\begin{array}{l}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right),\\ \widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right),\\ \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right),\\ \widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus \tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus \tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right),\end{array}$$

and for each $\gamma \in \left[0, 1\right],$ $\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$ and $\widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$ are established as follows:

$$\begin{array}{l}{\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\left[{\mathcal{E}}_{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right), {\mathcal{E}}^{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right)\right],\\ {\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\left[{\mathrm{Ѵ}}_{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right), {\mathrm{Ѵ}}^{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right)\right],\\ {\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\left[{\mathrm{Ѡ}}_{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right), {\mathrm{Ѡ}}^{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right)\right],\\ {\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)=\left[{\mathrm{Ϗ}}_{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right), {\mathrm{Ϗ}}^{*}\left(\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right), \gamma \right)\right].\end{array}$$

Proof. 

Let $\tilde{\mathcal{G}}$ and $\tilde{\mathfrak{G}}$ be two coordinated ${\mathrm{ỽ}}_1$ and ${\mathrm{ỽ}}_2$-convex $FNVM$s on $\left[ⱱ, ⱳ\right]\times \left[\mathcal{p},\mathcal{q}\right]$, respectively. Then,

$$\begin{array}{l}\tilde{\mathcal{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right),\\ \tilde{\mathcal{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right),\\ \tilde{\mathcal{G}}\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right),\\ \tilde{\mathcal{G}}\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right),\end{array}$$

and

$$\begin{array}{l}\tilde{\mathfrak{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right),\\ \tilde{\mathfrak{G}}\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right),\\ \tilde{\mathfrak{G}}\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right),\\ \tilde{\mathfrak{G}}\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ {\le }_{\mathbb{F}}{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right),\end{array}$$

Since $\tilde{\mathcal{G}}$ and $\tilde{\mathfrak{G}}$ both are coordinated ${\mathrm{ỽ}}_1$ and ${\mathrm{ỽ}}_2$-convex $FNVM$s on $\left[ⱱ, ⱳ\right]\times \left[\mathcal{p},\mathcal{q}\right]$, respectively, then, for any $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{l}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ {\le }_I{\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right].\end{array}$$

By taking the multiplication of above fuzzy inclusion with ${\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}$ and then taking the double integration of the result over $\left[0, 1\right]\times \left[0, 1\right]$ with respect to ($\mathtt{t},\mathtt{ԟ}$), we find that

$$\begin{array}{l}{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ {\le }_I{\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\end{array}$$

(61)

From the right hand side of (61), we have

$$\begin{array}{l}{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ +{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)d\mathtt{t}d\mathrm{ԟ}\\ ={\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right].\end{array}$$

(62)

Combining (61) and (62), for each $\gamma \in \left[0, 1\right]$, we have

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right]\\ {\le }_I{\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ +{\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}.\end{array}$$

Moreover, we have

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ {\le }_{\mathbb{F}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]d\mathtt{t}d\mathrm{ԟ}.\end{array}$$

Hence, the required result is found. □

In Remark 3, some of the classical results are discussed as special cases of the Theorem 4.

Remark 3.

If one assumes that$\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, $\alpha =1$ and $\beta =1$, then using (59), as a result, there will be inequity [55]:

$$\begin{array}{l}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)\otimes \tilde{\mathfrak{G}}\left(x,y\right)dydx\\ {\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{9}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\oplus {\displaystyle \frac{1}{18}}\left[\widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{1}{36}}\widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(63)

If $\tilde{\mathcal{G}}$ is a coordinated left-$UD$-$\mathrm{ỽ}$-convex with $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and one assumes that $\alpha =1$ and $\beta =1$, then using (59), as a result, there will be inequity [46]:

$$\begin{array}{l}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)\otimes \tilde{\mathfrak{G}}\left(x,y\right)dydx\\ {\le }_{\mathbb{F}}{\displaystyle \frac{1}{9}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\oplus {\displaystyle \frac{1}{18}}\left[\widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{1}{36}}\widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(64)

If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ then, using (59), we succeed in bringing about the upcoming inequity [64]:

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}[\begin{array}{c}{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{p}\right)\end{array}\hspace{1em}]\\ +{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{p}\right)\end{array}\right]\\ \supseteq \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)\mathcal{E}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\right)\mathrm{Ѵ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\right){\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\mathrm{Ѡ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\mathrm{Ϗ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(65)

If $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$, then using (59), we succeed in bringing about the upcoming inequity [63]:

$$\begin{array}{l}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{G}\left(x,y\right)\times \mathfrak{G}\left(x,y\right)dydx\\ \supseteq {\displaystyle \frac{1}{9}}\mathcal{E}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{1}{18}}\left[\mathrm{Ѵ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+\mathrm{Ѡ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\right]+{\displaystyle \frac{1}{36}}\mathrm{Ϗ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(66)

If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)={\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ and ${\mathfrak{G}}_{*}\left(\left(x,y\right), \gamma \right)={\mathfrak{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then using (59), we succeed in bringing about the upcoming inequity [47]:

$$\begin{array}{l}{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}[\begin{array}{c}{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{p}\right)\end{array}\hspace{1em}]\\ +{\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}} \left[\begin{array}{c}+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{p}\right)\end{array}\right]\\ \le \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)\mathcal{E}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\right)\mathrm{Ѵ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\right){\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}\mathrm{Ѡ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+{\displaystyle \frac{\beta }{(\beta +1)(\beta +2)}}{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}\mathrm{Ϗ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(67)

Theorem 5.

Let $\tilde{\mathcal{G}}, \tilde{\mathfrak{G}}:∆\to {\mathbb{F}}_0^{+}$ be a coordinated $\mathrm{ỽ}$-convex $FNVM$ on $∆$ and let $\mathrm{ỽ} :\left[0, 1\right]\to {\mathbb{R}}^{+}$. Then, from $\gamma$-cuts, we set up the sequence of ${\rm I}VM$s ${\mathcal{G}}_{\gamma },{\mathfrak{G}}_{\gamma }:∆\to {\mathbb{R}}_I^{+}$ provided by ${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[{\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)\right]$ and ${\mathfrak{G}}_{\gamma }\left(x,y\right)=\left[{\mathfrak{G}}_{*}\left(\left(x,y\right),\gamma \right), {\mathfrak{G}}^{*}\left(\left(x,y\right),\gamma \right)\right]$ for all $\left(x,y\right)\in ∆$ and for all $\gamma \in \left[0, 1\right]$. If ${\tilde{\mathcal{G}}\otimes \tilde{\mathfrak{G}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{l}{\displaystyle \frac{1}{2\alpha \beta {{\mathrm{ỽ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ỽ}}_2}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ \oplus \tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}.\end{array}$$

(68)

If $\tilde{\mathcal{G}}$ and $\tilde{\mathfrak{G}}$ both are coordinated $\mathrm{ỽ}$-concave $FNVM$s on $∆$, then the inequality above can be expressed as follows:

$$\begin{array}{l}{\displaystyle \frac{1}{2\alpha \beta {{\mathrm{ỽ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ỽ}}_2}^2\left(\frac{1}{2}\right)}}\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\ge }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{2\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ \oplus {\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{2{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\otimes \tilde{\mathfrak{G}}\left(ⱱ,\mathcal{p}\right)\right]\\ \oplus \tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ \oplus \widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}.\end{array}$$

(69)

where$\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$, $\widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$ and $\widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)$ are provided in Theorem 5.

Proof. 

Since $\tilde{\mathcal{G}},\tilde{\mathfrak{G}} :∆\to {\mathbb{F}}_0$ for two $\mathrm{ỽ}$-convex $FNVM$s, then using inequality (17) and for each $\gamma \in \left[0, 1\right],$ we have

$$\begin{array}{l}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times {\mathfrak{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ ={\mathcal{G}}_{\gamma }\left({\displaystyle \frac{\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ}{2}}+{\displaystyle \frac{\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ}}{2}},{\displaystyle \frac{ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}}{2}}+{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times {\mathfrak{G}}_{\gamma }\left({\displaystyle \frac{\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ}{2}}+{\displaystyle \frac{\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ}}{2}},{\displaystyle \frac{ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}}{2}}+{\displaystyle \frac{\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}}{2}}\right)\\ {\le }_I{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \left[\begin{array}{c}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)+{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)+{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\end{array}\right]\\ \times \left[\begin{array}{c}{\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)+{\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)+{\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\end{array}\right]\\ {\le }_I{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \left[\begin{array}{c}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\end{array}\right]\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times \left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]{\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\\ \times \left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\right]{\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\\ \times \left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\right]{\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\\ \times \left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]{\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

By taking the multiplication of above fuzzy inclusion with ${\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}$ and then taking the double integration of the result over $\left[0, 1\right]\times \left[0, 1\right]$ with respect to ($\mathtt{t},\mathtt{ԟ}$), we have

$$\begin{array}{l}{\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times {\mathfrak{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)d\mathtt{t}d\mathrm{ԟ}\\ {\le }_I{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right)\times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ,ԟ\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\mathtt{ԟ}\mathcal{p}+\left(1-\mathrm{ԟ}\right)\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\mathtt{t}\mathtt{ⱱ}+\left(1-\mathtt{t}\right)ⱳ, \left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\\ +{\mathcal{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(\left(1-\mathtt{t}\right)\mathtt{ⱱ}+\mathtt{t}\mathtt{ⱳ},\left(1-\mathrm{ԟ}\right)\mathcal{p}+ԟ\mathcal{q}\right)\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ \times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ \times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\end{array}\right]d\mathtt{t}d\mathrm{ԟ}\\ +{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ \times {\int }_0^1{\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[\begin{array}{c}{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\\ \begin{array}{c}+{\mathrm{ỽ}}_1\left(1-\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\\ +{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\end{array}\end{array}\right]d\mathtt{t}d\mathrm{ԟ},\end{array}$$

which implies that

$$\begin{array}{l}{\displaystyle \frac{1}{\alpha \beta }}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times {\mathfrak{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\le }_{\mathbb{F}}{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\mathrm{ỽ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ỽ}}_2}^2\left(\frac{1}{2}\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\right]\\ +{\displaystyle \frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right){{\mathrm{ỽ}}_1}^2\left(\frac{1}{2}\right){{\mathrm{ỽ}}_2}^2\left(\frac{1}{2}\right)}{{\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)\times {\mathfrak{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)\right]\\ +2{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathcal{E}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ +2{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѵ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ +2{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ѡ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ}\\ +2{{\mathrm{ỽ}}_1}^2\left({\displaystyle \frac{1}{2}}\right){{\mathrm{ỽ}}_2}^2\left({\displaystyle \frac{1}{2}}\right){\mathrm{Ϗ}}_{\gamma }\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right){\int }_0^1{\mathtt{t}}^{\alpha -1}{\mathrm{ԟ}}^{\beta -1}\left[{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)\right]+{\mathrm{ỽ}}_1\left(\mathtt{t}\right){\mathrm{ỽ}}_1\left(1-\mathrm{ԟ}\right)\left[{\mathrm{ỽ}}_2\left(1-\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(\mathrm{ԟ}\right)+{\mathrm{ỽ}}_2\left(\mathtt{t}\right){\mathrm{ỽ}}_2\left(1-\mathrm{ԟ}\right)\right]\right]d\mathtt{t}d\mathrm{ԟ},\end{array}$$

Since $\gamma \in \left[0, 1\right]$, after simplification, we reached the required conclusion. □

In Remark 4, some classical results have been obtained. It is clear that the results can be obtained using Theorem 5 instead of using other classical results.

Remark 4.

If one assumes that $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, $\alpha =1$ and $\beta =1$, then using (68), as a result, there will be inequity [55]:

$$\begin{array}{l}4\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\supseteq }_{\mathbb{F}}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)\otimes \tilde{\mathfrak{G}}\left(x,y\right)dydx\oplus {\displaystyle \frac{5}{36}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ \oplus {\displaystyle \frac{7}{36}}\left[\widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\widetilde{+}\widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{2}{9}}\widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(70)

If $\tilde{\mathcal{G}}$ is coordinated left-$UD$-$\mathrm{ỽ}$-convex with $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and one assumes that $\alpha =1$ and $\beta =1$, then using (68), as a result, there will be inequity [46]:

$$\begin{array}{l}4\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\otimes \tilde{\mathfrak{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ {\le }_{\mathbb{F}}{\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}}{\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\tilde{\mathcal{G}}\left(x,y\right)\otimes \tilde{\mathfrak{G}}\left(x,y\right)dydx\oplus {\displaystyle \frac{5}{36}}\tilde{\mathcal{E}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ \oplus {\displaystyle \frac{7}{36}}\left[\widetilde{\mathrm{Ѵ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\oplus \widetilde{\mathrm{Ѡ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\right]\oplus {\displaystyle \frac{2}{9}}\widetilde{\mathrm{Ϗ}}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(71)

If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$ and $\gamma =1$, then using (68), we succeed in bringing about the upcoming inequity [63]:

$$\begin{array}{l}4 \mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times \mathfrak{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ \supseteq {\displaystyle \frac{1}{\left(ⱳ-ⱱ\right)\left(\mathcal{q}-\mathcal{p}\right)}} {\int }_{ⱱ}^{ⱳ}{\int }_{\mathcal{p}}^{\mathcal{q}}\mathcal{G}\left(x,y\right)\times \mathfrak{G}\left(x,y\right)dydx+{\displaystyle \frac{5}{36}}\mathcal{E}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +{\displaystyle \frac{7}{36}}\left[\mathrm{Ѵ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)+\mathrm{Ѡ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\right]+{\displaystyle \frac{2}{9}}\mathrm{Ϗ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(72)

If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)\ne {\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then using (68), we succeed in bringing about the upcoming inequity [64]:

$$\begin{array}{l}4\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times \mathfrak{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ \supseteq {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+} }^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta } \mathcal{G}\left(ⱳ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{p}\right)\\ +{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+} }^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta } \mathcal{G}\left(ⱱ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{p}\right)\end{array}\right]\\ +\left[{\displaystyle \frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}}+{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\right]\mathcal{E}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѵ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѡ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{4}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ϗ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(73)

If ${\mathcal{G}}_{*}\left(\left(x,y\right), \gamma \right)={\mathcal{G}}^{*}\left(\left(x,y\right), \gamma \right)$ and ${\mathfrak{G}}_{*}\left(\left(x,y\right), \gamma \right)={\mathfrak{G}}^{*}\left(\left(x,y\right), \gamma \right)$ with $\gamma =1$ and $\mathrm{ỽ}\left(\mathtt{t}\right)=\mathtt{t},\mathtt{ỽ}\left(\mathrm{ԟ}\right)=\mathrm{ԟ}$, then using (68), we succeed in bringing about the upcoming inequity [47]:

$$\begin{array}{l}4\mathcal{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\times \mathfrak{G}\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\\ \le {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[\begin{array}{c}{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱳ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱳ,\mathcal{p}\right)\\ +{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{q}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{q}\right)+{\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }\mathcal{G}\left(ⱱ,\mathcal{p}\right)\times \mathfrak{G}\left(ⱱ,\mathcal{p}\right)\end{array}\right]\\ +\left[{\displaystyle \frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}}+{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)\right]\mathcal{E}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѵ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right)+{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ѡ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right)\\ +\left[{\displaystyle \frac{1}{4}}-{\displaystyle \frac{\alpha }{(\alpha +1)(\alpha +2)}}{\displaystyle \frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}}\right]\mathrm{Ϗ}\left(ⱱ,ⱳ,\mathcal{p},\mathcal{q}\right).\end{array}$$

(74)

4. Examples

In Example 2, we discuss $\mathrm{ỽ}$-concave $FNVMs$ over coordinates.

Example 2.

We consider the $FNVM$ $\tilde{\mathcal{G}}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0$ established by

$$\tilde{\mathcal{G}}\left(x,y\right)\left(\sigma \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sigma -\left(6-e^x\right)\left(6-e^y\right)}{\left(6-e^x\right)\left(6-e^y\right)-25}}, \sigma \in \left[\left(6-e^x\right)\left(6-e^y\right), 25\right]\\ {\displaystyle \frac{35xy-\sigma }{35xy-25}}, \sigma \in \left(25, 35xy\right]\\ 0, otherwise.\end{array}\right.$$

(75)

Then, for each $\gamma \in \left[0, 1\right],$ we have ${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[\left(1-\gamma \right)\left(6-e^x\right)\left(6-e^y\right)+25\gamma ,35\left(1-\gamma \right)xy+25\gamma \right]$. The endpoint functions ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right),$ ${\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ are coordinated $\mathrm{ỽ}$-concave and $\mathrm{ỽ}$-convex functions for each $\gamma \in \left[0, 1\right]$. Hence, $\tilde{\mathcal{G}}\left(x,y\right)$ is a coordinated $\mathrm{ỽ}$-concave $FNVM$.

Example 3.

We assume that the $FNVM$s $\tilde{\mathcal{G}}:\left[0, 2\right]\times \left[0, 2\right]\to {\mathbb{F}}_0$ is established by

$$\mathcal{G}\left(x,y\right)\left(\sigma \right)=\left\{\begin{array}{c}{\displaystyle \frac{\sigma }{\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)}}, \sigma \in \left[0, \left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)\right]\\ {\displaystyle \frac{2\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)-\sigma }{\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)}}, \sigma \in \left(\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right), 2\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)\right]\\ 0, otherwise,\end{array}\right.$$

(76)

Then, for each $\gamma \in \left[0, 1\right],$ we have ${\mathcal{G}}_{\gamma }\left(x,y\right)=\left[\gamma \left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right),\left(2-\gamma \right)\left(2-\sqrt{x}\right)\left(2-\sqrt{y}\right)\right]$. The end point functions ${\mathcal{G}}_{*}\left(\left(x,y\right),\gamma \right) ,$ ${\mathcal{G}}^{*}\left(\left(x,y\right),\gamma \right)$ are coordinated $\mathrm{ỽ}$-convex functions for each $\gamma \in \left[0, 1\right]$, respectively. Hence, $\tilde{\mathcal{G}}\left(x,y\right)$ is a $\mathrm{ỽ}$-coordinated convex $FNVM$.

$$\begin{array}{l}{\mathcal{G}}_{\gamma }\left({\displaystyle \frac{ⱱ+ⱳ}{2}}, {\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)=\left[\gamma ,\left(2-\gamma \right)\right]\\ {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,{\displaystyle \frac{\mathcal{p}+\mathcal{q}}{2}}\right)\right]\oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{4\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left({\displaystyle \frac{ⱱ+ⱳ}{2}},\mathcal{p}\right)\right]\\ =\left[\gamma \left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\pi \right),\left(2-\gamma \right)\left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\pi \right)\right]\\ {\displaystyle \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{{4\left(ⱳ-ⱱ\right)}^{\alpha }{\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{ⱱ}^{+},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{p}}^{+}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-},{\mathcal{q}}^{-}}^{\alpha , \beta }{\mathcal{G}}_{\gamma }\left(ⱱ,\mathcal{p}\right)\right]\\ =\left[\gamma \left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\pi +{\displaystyle \frac{\pi }{8}}+{\displaystyle \frac{{\pi }^2}{32}}\right),\left(2-\gamma \right)\left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\pi +{\displaystyle \frac{\pi }{8}}+{\displaystyle \frac{{\pi }^2}{32}}\right)\right]\\ {\displaystyle \frac{\Gamma \left(\alpha +1\right)}{{8\left(ⱳ-ⱱ\right)}^{\alpha }}}\left[{\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱱ}^{+}}^{\alpha }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{ⱳ}^{-}}^{\alpha }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\right]\\ \oplus {\displaystyle \frac{\Gamma \left(\beta +1\right)}{{8\left(\mathcal{q}-\mathcal{p}\right)}^{\beta }}}\left[{\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{p}}^{+}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{q}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱱ,\mathcal{p}\right)\oplus {\mathcal{I}}_{{\mathcal{q}}^{-}}^{\beta }\tilde{\mathcal{G}}\left(ⱳ,\mathcal{p}\right)\right]\\ =\left[{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}}\gamma ,{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}}\left(2-\gamma \right)\right]\\ {\displaystyle \frac{{\mathcal{G}}_{\gamma }\left(\mathcal{p},ⱳ\right)+{\mathcal{G}}_{\gamma }\left(\sigma ,ⱳ\right)+{\mathcal{G}}_{\gamma }\left(\mathcal{p},\mathcal{q}\right)+{\mathcal{G}}_{\gamma }\left(\sigma ,\mathcal{q}\right)}{4}}=\left[\gamma \left({\displaystyle \frac{9}{2}}-2\sqrt{2}\right),\left(2-\gamma \right)\left({\displaystyle \frac{9}{2}}-2\sqrt{2}\right)\right].\end{array}$$

That is

$$\begin{array}{l}\left[\left(1-\gamma \right)+4\gamma ,9\left(1-\gamma \right)+4\gamma \right]{\le }_I\left[\left(1-\gamma \right)\left(2-{\displaystyle \frac{\sqrt{2}}{4}}-{\displaystyle \frac{\sqrt{2}}{8}}\pi \right)+4\gamma ,\left(1-\gamma \right)\left(2+{\displaystyle \frac{\sqrt{2}}{4}}+{\displaystyle \frac{\sqrt{2}}{8}}\pi \right)+4\gamma \right]\\ {\le }_I\left[\left(1-\gamma \right)\left({\displaystyle \frac{33}{8}}-\sqrt{2}-{\displaystyle \frac{\sqrt{2}}{2}}\pi +{\displaystyle \frac{\pi }{8}}+{\displaystyle \frac{{\pi }^2}{32}}\right)+4\gamma ,\left(1-\gamma \right)\left({\displaystyle \frac{33}{8}}+\sqrt{2}+{\displaystyle \frac{\sqrt{2}}{2}}\pi +{\displaystyle \frac{\pi }{8}}+{\displaystyle \frac{{\pi }^2}{32}}\right)+4\gamma \right]\\ {\le }_I\left[{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}}\left(1-\gamma \right)+4\gamma ,{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}+4\right)\pi +24}{8\sqrt{2}}}\left(1-\gamma \right)+4\gamma \right]\\ {\le }_I{\displaystyle \frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}}\left(1-\gamma \right)+4\gamma .\end{array}$$

Hence, Theorem 4 has been verified.

5. Conclusions and Future Plans

Different fields examine the theory of inequality from different angles. The establishment of novel, cohesive and improved versions of already established outcomes is one of the rigorous viewpoints. We present a new expanded notion of convex mappings in the current proceedings and analyze some implications of the proposed notion; the new class is known as coordinated $\mathrm{ỽ}$-convex $FNVMs$. Additionally, we investigate the containments of the HH and Pachpatte types, effectively putting the recently developed concept of coordinated $\mathrm{ỽ}$-convex convexity into practice along with fractional integral operators. The findings’ dependability is confirmed by simulations and instances. This class has the benefit of combining multiple other recognized convexity classes into one. As a result, the results obtained from this class likewise coincide with classical results. In future, this class will be extended to non-convex classes, and their applications will be found via various fractional integral operators by using the well-known “Condition C”.