Some New Fractional Inequalities for Coordinated Convexity over Convex Set Pertaining to Fuzzy-Number-Valued Settings Governed by Fractional Integrals

Abstract

In this study, we first propose some new concepts of coordinated up and down convex mappings with fuzzy-number values. Then, Hermite–Hadamard-type inequalities via coordinated up and down convex fuzzy-number-valued mapping (coordinated $UD$-convex $FNVMs$) are introduced. By taking the products of two coordinated $UD$-convex $FNVMs$, Pachpatte-type inequalities are also obtained. Some new conclusions are also derived by making particular decisions with the newly defined inequalities, and it is demonstrated that the recently discovered inequalities are expansions of comparable findings in the literature. It is important to note that the main outcomes are validated using nontrivial examples.

1. Introduction

One of the most well-known concepts in the field of function theory is the Hermite–Hadamard inequality, which was found by C. Hermite and J. Hadamard (and described in sources such as [1] and [2] (p. 137)). This inequality has several real-world applications in addition to its geometric interpretation.

The Hermite–Hadamard inequalities have been established by numerous mathematicians. It is important to note that the Hermite–Hadamard inequality, which naturally follows on from Jensen’s inequality, can be seen as a development of the idea of convexity. Recently, there has been renewed interest in the Hermite–Hadamard inequality for convex functions, leading to a wide range of improvements and expansions that have been thoroughly investigated (see, for example, publications like [3,4,5,6,7,8,9,10,11,12]).

Interval analysis is a crucial topic since it is used in math and computer models as one method of addressing interval uncertainty. Even though this theory has a long history, going back to Archimedes’ calculation of a circle’s circumference, significant research on the subject was not published until the 1950s. The first book [13] on interval analysis was published in 1966 by Ramon E. Moore, who is credited with developing interval calculus. After that, other academics studied the theory and uses of interval analysis.

Furthermore, well-known inequality types, such as Ostrowski, Minkowski, and Beckenbach, as well as some of their applications were supplied by taking into account the interval-valued functions (see [14,15,16,17,18]). Additionally, Budak et al. [19] developed a few inequalities utilizing the interval-valued Riemann–Liouville fractional integrals. The definition of interval-valued harmonically convex functions was provided by Liu et al. [20], and, as a result, they were able to derive several Hermite–Hadamard-type inequalities, including interval fractional integrals. The authors provided a fuzzy integral-based variation of Jensen’s inequality for interval-valued functions [21,22] and demonstrated several integral inequalities [23,24,25,26]. In their proofs of Hermite–Hadamard-type inequalities for set-valued functions [27,28,29], Mitroi et al. made use of the general forms of interval-valued convex functions. Rom’an Flores et al. found a few Gronwal-type inequalities for interval-valued functions [30]. Zhao et al. showed many kinds of integral inequalities for interval-valued functions [31,32,33,34,35,36,37,38,39,40,41,42].

In [43], Jleli and Samet discovered a brand-new Hermite–Hadamard-type inequality involving fractional integrals with regard to a different function. The fractional integrals of a function with respect to another function were first introduced by Tunc in [44]. The Riemann–Liouville and Hadamard fractional integrals were generalized into a single form by Katugompala’s novel fractional integration. Budak and Agarwal used generalized fractional integrals, which generalize some significant fractional integrals like the Riemann–Liouville fractional integrals, the Hadamard fractional integrals, and the Katugampola fractional integrals in [45] to establish the Hermite–Hadamard-type inequalities for coordinated convex functions. The interval-valued left- and right-sided generalized fractional double integrals were defined by Kara et al. [46]. Numerous authors have concentrated on interval-valued functions in recent years. The authors of [47] introduced the idea of interval-valued general convex functions and used it to demonstrate a number of novel Hermite–Hadamard-type inequalities. A fractional version of Hermite–Hadamard-type inequalities for interval-valued harmonically convex functions was also provided by the authors in [48]. Researchers recently expanded the idea of interval-valued convexity and described various types of $UD$-convexity for interval-valued functions in [49,50]. For $UD$-fuzzy-number-valued convex functions, they also discovered a large number of Hermite–Hadmard type inequalities.

To express the collection of all the positive fuzzy numbers over real numbers, we introduce the notation ${\mathbb{F}}_0$ in the context of this article. The terms ${\mathcal{A}}_{[ʋ, ȡ]}$, ${\mathcal{I}\mathcal{A}}_{[ʋ, ȡ]},$ and ${\mathcal{F}\mathcal{A}}_{\left(\left[ʋ, ȡ\right]\right)}$ refer to the set of all $FNVMFNVMs$ that are Riemann integrable real-valued functions, Aumann’s integrable IV-Fs, and fuzzy Aumann’s integrable on the interval $[ʋ, ȡ]$. The following theorem draws a link between the functions that are integrable in terms of Riemann ($\mathcal{A}$-integrable) and functions that are integrable in terms of ${\mathcal{F}\mathcal{A}}_{ }$. Additionally, the sign “${\supseteq }_{\mathbb{F}}$” is used to denote the up and down ($UD$) fuzzy inclusion relationship for $\widetilde{\mathsf{Ɖ}}$ and $\widetilde{₼}$ belonging to ${\mathbb{F}}_0$, where $\widetilde{₼}$ is thought of as a fuzzy subset of $\widetilde{\mathsf{Ɖ}}$. If and only if for $ƺ$-levels, the condition ${\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}{\supseteq }_I{\left[\widetilde{₼}\right]}^{ƺ}$ is met, this $UD$-inclusion is true. Integral fuzzy inequalities generated from $FNVM$s have recently attracted the attention of several academics.

Theorem 1 

([51]). Assume that the $UD$-convex $FNVM$ $\widetilde{Ɣ}:\left[ʋ, ȡ\right]\to {\mathbb{F}}_0$ is an ${\rm I}VM$ with ${Ɣ}_{ƺ}\left(⍬\right)=\left[{Ɣ}_{*}\left(⍬,ƺ\right), {Ɣ}^{*}\left(⍬,ƺ\right)\right]$ for all $⍬\in \left[ʋ, ȡ\right]$ and for all $ƺ\in \left[0, 1\right]$. Then, there are disparities:

$$\widetilde{Ɣ}\left(\frac{ʋ+ȡ}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{ȡ-ʋ}\odot \left(FA\right){\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬\right)d⍬{\supseteq }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ȡ\right)\oplus \widetilde{Ɣ}\left(ʋ\right)}{2}.$$

(1)

We provide the ideas of generalized fractional integrals for two-variable $FNVMs$ in order to demonstrate Hermite–Hadmard-type inequalities for the convex and coordinated convex functions, which are inspired by ongoing investigations. The main benefit of the newly established inequalities is that they can be converted into classical Hermite–Hadamard integral inequalities for coordinated $UD$-convex $FNVM$s and fuzzy Riemann–Liouville fractional Hermite–Hadamard, Hadamard, and Katugampola fractional Hermite–Hadamard inequalities without having to prove each one separately; for more information, see [52,53,54,55,56,57,58,59,60,61,62,63,64] and the references therein.

The format of this paper is as follows: A brief summary of the foundations of fuzzy-number-valued calculus and other relevant works in this area are presented in Section 2. In Section 3, we provide some generalized fractional integrals for $UD$-convex $FNVMFNVMs$ with two variables. For $UD$-convex $FNVMFNVMs$, we create a novel Hermite–Hadamard-type inequality. Several Hermite–Hadamard-type inequalities for coordinated $UD$-convex $FNVMFNVMs$ are presented in Section 3. It is also taken into consideration how these findings compare with the findings of a similar nature in the literature. Finally, Section 4 makes some suggestions for additional studies.

2. Preliminaries

We will go through the fundamental terminologies and findings in this section, which helps to comprehend the ideas behind our fresh findings.

Definition 1 

([58,59]). Given $\widetilde{\mathsf{Ɖ}}\in {\mathbb{F}}_0$, the level or cut sets are given by ${\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}=\left\{⍬\in \mathfrak{R}|\widetilde{\mathsf{Ɖ}}\left(⍬\right)>ƺ\right\}$ $\forall$ $ƺ\in [0, 1]$ and by

$${\left[\widetilde{\mathsf{Ɖ}}\right]}^0=\left\{\mathfrak{⍬}\in \mathfrak{R}|\widetilde{\mathsf{Ɖ}}\left(⍬\right)>0\right\}.$$

(2)

These sets are known as the $ƺ$-level or $ƺ$-cut sets of $\widetilde{\mathsf{Ɖ}}$.

Proposition 1 

([22]). Let $\widetilde{\mathsf{Ɖ}},\widetilde{₼}\in {\mathbb{F}}_0$. Then, the relation “${\le }_{\mathbb{F}}$” is given on ${\mathbb{F}}_0$ by $\widetilde{\mathsf{Ɖ}}{\le }_{\mathbb{F}}\widetilde{₼}$ when and only when ${{\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}\le }_I{\left[\widetilde{₼}\right]}^{ƺ}$, for every $ƺ\in [0, 1],$ which are left- and right-order relations.

Proposition 2 

([57]). Let $\widetilde{\mathsf{Ɖ}},\widetilde{₼}\in {\mathbb{F}}_0$. Then, the relation $“{\supseteq }_{\mathbb{F}}”$ is given on ${\mathbb{F}}_0$ by $\widetilde{\mathsf{Ɖ}}{\supseteq }_{\mathbb{F}}\widetilde{₼}$ when and only when ${\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}{\supseteq }_I{\left[\widetilde{₼}\right]}^{ƺ}$ for every $ƺ\in [0, 1],$ which is $the UD—$order relation on ${\mathbb{F}}_0$.

Remember the approaching notions, which are offered in the literature. If $\widetilde{\mathsf{Ɖ}},\widetilde{₼}\in {\mathbb{F}}_0$ and $\mathcal{t}\in \mathfrak{R}$, then, for every $ƺ\in \left[0, 1\right],$ the arithmetic operation addition “ $\oplus$”, multiplication “$\otimes$”, and scalar multiplication “$\odot$” are defined by

$${\left[\widetilde{\mathsf{Ɖ}}\oplus \widetilde{₼}\right]}^{ƺ}={\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}+{\left[\widetilde{₼}\right]}^{ƺ},$$

(3)

$${\left[\widetilde{\mathsf{Ɖ}}\otimes \widetilde{₼}\right]}^{ƺ}={\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}\times {\left[ \widetilde{₼}\right]}^{ƺ},$$

(4)

$${\left[\mathcal{t}\odot \widetilde{\mathsf{Ɖ}}\right]}^{ƺ}=\mathcal{t}{\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}.$$

(5)

Equations (4) to (6) have immediate consequences for these outcomes.

Theorem 2 

([22]). The space ${\mathbb{F}}_0$ dealing with a supremum metric, i.e., for $\widetilde{\mathsf{Ɖ}}, \widetilde{₼}\in {\mathbb{F}}_0$

$$d_{\infty }\left(\widetilde{\mathsf{Ɖ}}, \widetilde{₼}\right)=\underset{0\le ƺ\le 1}{\mathit{sup}}{d}_H\left({\left[\widetilde{\mathsf{Ɖ}}\right]}^{ƺ}, {\left[\widetilde{₼}\right]}^{ƺ}\right),$$

(6)

is a complete metric space, where $H$ indicates the well-known Hausdorff metric on the space of the intervals.

Theorem 3. 

Let $\widetilde{Ɣ}:[ʋ, ȡ]\subset \mathfrak{R}\to {\mathbb{F}}_0$ be an $FNVM$, and the ${\rm I}VM$s classified according to their $ƺ$-levels ${Ɣ}_{ƺ}:[ʋ, ȡ]\subset \mathfrak{R}\to {\mathbb{R}}_I^{ }$ are given by ${Ɣ}_{ƺ}\left(⍬\right)=\left[{Ɣ}_{*}\left(⍬,ƺ\right), {Ɣ}^{*}\left(⍬,ƺ\right)\right]$ $\forall$ $⍬\in [ʋ, ȡ]$ and $\forall$ $ƺ\in (0, 1].$ Then, $\widetilde{Ɣ}$ is $FA$-integrable over $[ʋ, ȡ]$ if and only if, ${Ɣ}_{*}\left(⍬,ƺ\right)$ and ${Ɣ}^{*}\left(⍬,ƺ\right)$ are both $A$-integrable over $[ʋ, ȡ]$. Moreover, if $\widetilde{Ɣ}$ is $FA$-integrable over $\left[ʋ, ȡ\right],$ then

$$\begin{gathered} {\left[\left(FA\right){\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬\right)d⍬\right]}^{ƺ}=\left[\left(A\right){\int }_{ʋ}^{ȡ}{Ɣ}_{*}\left(⍬,ƺ\right)d⍬, \left(A\right){\int }_{ʋ}^{ȡ}{Ɣ}^{*}\left(⍬,ƺ\right)d⍬\right] \\ =\left(IA\right){\int }_{ʋ}^{ȡ}{Ɣ}_{ƺ}\left(⍬\right)d⍬, \end{gathered}$$

(7)

$\forall$ $ƺ\in \left(0, 1\right],$ where ${\mathcal{F}\mathcal{A}}_{\left(\left[ʋ, ȡ\right], ƺ\right)}$ denotes the collection of all $FA$-integrable $FNVM$s over $[ʋ, ȡ]$.

Fuzzy Aumann’s and Fractional Calculus on Coordinates

Definition 2. 

([19,38]). Let $Ɣ:\left[ծ, ռ\right]\to {\mathbb{R}}_I^{+}$ be an $IVM$ and $Ɣ\in {\mathcal{I}\mathcal{R}}_{\left[ծ, ռ\right]}$. Then, the interval Riemann–Liouville type integrals of $Ɣ$ are defined as

$${\mathcal{I}}_{{ծ}^{+}}^{ɤ}Ɣ\left(\psi \right)=\frac{1}{\Gamma \left(ɤ\right)}{\int }_{ծ}^{\psi }{\left(\psi -\mathtt{t}\right)}^{ɤ-1}Ɣ\left(\mathtt{t}\right)d\mathtt{t}\left(\psi >ծ\right),$$

(8)

$${\mathcal{I}}_{{ռ}^{-}}^{ɤ}Ɣ\left(\psi \right)=\frac{1}{\Gamma \left(ɤ\right)}{\int }_{\psi }^{ռ}{\left(\mathtt{t}-\psi \right)}^{ɤ-1}Ɣ\left(\mathtt{t}\right)d\mathtt{t}(\psi <ռ),$$

(9)

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

Recently, Allahviranloo et al. [39] introduced the fuzzy version of the defined fractional integral integrals such that:

Definition 3. 

Let $ɤ>0$ and $L\left(\left[ծ, ռ\right],{\mathbb{F}}_0\right)$ be the collection of all the Lebesgue measurable $FNVM$s on$[ծ,ռ]$. Then, the fuzzy left and right Riemann–Liouville fractional integrals of $\widetilde{Ɣ}\in$ $L\left(\left[ծ, ռ\right],{\mathbb{F}}_0\right)$ with the order $ɤ>0$ are defined by

$${\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(\psi \right)=\frac{1}{\Gamma (ɤ)}{\int }_{ծ}^{\psi }{\left(\psi -\mathtt{t}\right)}^{ɤ-1}\widetilde{Ɣ}\left(\mathtt{t}\right)d\mathtt{t},\left(\psi >ծ\right),$$

(10)

and

$${\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(\psi \right)=\frac{1}{\Gamma (ɤ)}{\int }_{\psi }^{ռ}{\left(\mathtt{t}-\psi \right)}^{ɤ-1}\widetilde{Ɣ}\left(\mathtt{t}\right)d\mathtt{t},(\psi <ռ),$$

(11)

respectively, where $\Gamma \left(\psi \right)={\int }_0^{\infty }{\mathtt{t}}^{\psi -1}{e}^{-\mathtt{t}}d\mathtt{t}$ is the Euler gamma function. The fuzzy left and right Riemann–Liouville fractional integral $\psi$ based on the left and right end point functions can be defined, that is

$$\begin{array}{cc}{\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(\psi \right)\right]}^{ƺ}& =\frac{1}{\Gamma \left(ɤ\right)}{\int }_{ծ}^{\psi }{\left(\psi -\mathtt{t}\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(\mathtt{t}\right)d\mathtt{t}\hfill \\ & =\frac{1}{\Gamma (ɤ)}{\int }_{ծ}^{\psi }{\left(\psi -\mathtt{t}\right)}^{ɤ-1}\left[{Ɣ}_{*}\left(\mathtt{t},ƺ\right),{Ɣ}^{*}\left(\mathtt{t},ƺ\right)\right]d\mathtt{t},\left(\psi >ծ\right),\hfill \end{array}$$

(12)

where

$${\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{*}\left(\psi , ƺ\right)=\frac{1}{\Gamma (ɤ)}{\int }_{ծ}^{\psi }{\left(\psi -\mathtt{t}\right)}^{ɤ-1}{Ɣ}_{*}\left(\mathtt{t},ƺ\right)d\mathtt{t},\left(\psi >ծ\right),$$

(13)

and

$${\mathcal{I}}_{{ծ}^{+}}^{ɤ} {Ɣ}^{*}\left(\psi , ƺ\right)=\frac{1}{\Gamma (ɤ)}{\int }_{ծ}^{\psi }{\left(\psi -\mathtt{t}\right)}^{ɤ-1}{Ɣ}^{*}\left(\mathtt{t},ƺ\right)d\mathtt{t},\left(\psi >ծ\right).$$

(14)

The right Riemann–Liouville fractional integral, denoted by${\left[{\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(\psi \right)\right]}^{ƺ}$, can also be defined using the left and right end point functions.

Theorem 4 

([57]). Let $\widetilde{Ɣ}:\left[ʋ, ȡ\right]\to {\mathbb{F}}_0$ be a $UD$-convex $FNVM$ on $\left[ʋ, ȡ\right],$ whose $ƺ$-cuts set up the sequence of ${\rm I}VM$s ${Ɣ}_{ƺ}:\left[ʋ, ȡ\right]\subset \mathbb{R}\to {{\mathbb{R}}_C}^{+}$ are given by ${Ɣ}_{ƺ}\left(\psi \right)=\left[{Ɣ}_{*}\left(\psi ,ƺ\right), {Ɣ}^{*}\left(\psi ,ƺ\right)\right]$ for all $\psi \in \left[ʋ, ȡ\right]$ and for all $ƺ\in \left[0, 1\right]$. If $\widetilde{Ɣ}\in L\left(\left[ʋ, ȡ\right],{\mathbb{F}}_0\right)$, then

$$\widetilde{Ɣ}\left(\frac{ʋ+ȡ}{2}\right){\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{2\left(ȡ-ʋ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ʋ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ʋ\right)\right]{\supseteq }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ʋ\right)\oplus \widetilde{Ɣ}\left(ȡ\right)}{2}.$$

(15)

Theorem 5 

([57]). Let $\widetilde{Ɣ},\tilde{\mathcal{J}} :\left[ʋ, ȡ\right]\to {\mathbb{F}}_0$ be two $UD$-convex $FNVM$s$.$ Then, from the $ƺ$-cuts, we set up the sequence of ${\rm I}VM$s ${Ɣ}_{ƺ}, {\mathcal{J}}_{ƺ}:\left[ʋ, ȡ\right]\subset \mathbb{R}\to {\mathbb{R}}_I^{+}$ are given by ${Ɣ}_{ƺ}\left(⍬\right)=\left[{Ɣ}_{*}\left(⍬,ƺ\right), {Ɣ}^{*}\left(⍬,ƺ\right)\right]$ and ${\mathcal{J}}_{ƺ}\left(⍬\right)=\left[{\mathcal{J}}_{*}\left(⍬,ƺ\right), {\mathcal{J}}^{*}\left(⍬,ƺ\right)\right]$ for all $⍬\in \left[ʋ, ȡ\right]$ and for all $ƺ\in \left[0, 1\right]$. If$\widetilde{Ɣ}\otimes \tilde{\mathcal{J}}\in L\left(\left[ʋ, ȡ\right],{\mathbb{F}}_0\right)$ is the fuzzy Riemann integrable, then

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ȡ-ʋ\right)}^{ɤ}}[{\mathcal{I}}_{{ʋ}^{+}}^{ɤ}\widetilde{\mathrm{Ɣ}}\left(ȡ\right)& \otimes \tilde{\mathcal{J}}\left(ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{ɤ}\widetilde{\mathrm{Ɣ}}\left(ʋ\right)\otimes \tilde{\mathcal{J}}\left(ʋ\right)]\hfill \\ & {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\tilde{\mathcal{M}}\left(ʋ,ȡ\right)\oplus \left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\tilde{\mathcal{N}}\left(ʋ,ȡ\right),\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\hfill \widetilde{Ɣ}\left(\frac{ʋ+ȡ}{2}\right)& \otimes \tilde{\mathcal{J}}\left(\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{4\left(ȡ-ʋ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ʋ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ȡ\right)\otimes \tilde{\mathcal{J}}\left(ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ʋ\right)\otimes \tilde{\mathcal{J}}\left(ʋ\right)\right]\hfill \\ & \hspace{1em}\hspace{1em}+\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\tilde{\mathcal{M}}\left(ʋ,ȡ\right)\oplus \frac{1}{2}\left(\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\tilde{\mathcal{N}}\left(ʋ,ȡ\right),\hfill \end{array}$$

(17)

where$\tilde{\mathcal{M}}\left(ʋ,ȡ\right)=\widetilde{Ɣ}\left(ʋ\right)\otimes \tilde{\mathcal{J}}\left(ʋ\right)\oplus \widetilde{Ɣ}\left(ȡ\right)\otimes \tilde{\mathcal{J}}\left(ȡ\right),$ $\tilde{\mathcal{N}}\left(ʋ,ȡ\right)=\widetilde{Ɣ}\left(ʋ\right)\otimes \tilde{\mathcal{J}}\left(ȡ\right)\oplus \widetilde{Ɣ}\left(ȡ\right)\otimes \tilde{\mathcal{J}}\left(ʋ\right),$

${\mathcal{M}}_{ƺ}\left(ʋ,ȡ\right)=\left[{\mathcal{M}}_{*}\left(\left(ʋ,ȡ\right), ƺ\right), {\mathcal{M}}^{*}\left(\left(ʋ,ȡ\right), ƺ\right)\right]$, and ${\mathcal{N}}_{ƺ}\left(ʋ,ȡ\right)=\left[{\mathcal{N}}_{*}\left(\left(ʋ,ȡ\right), ƺ\right), {\mathcal{N}}^{*}\left(\left(ʋ,ȡ\right), ƺ\right)\right].$

The interval and fuzzy Aumann’s type integrals are defined as follows for the coordinated $IVM$ $Ɣ\left(⍬,\psi \right)$ and coordinated $FNVM$ $\widetilde{Ɣ}\left(⍬,\psi \right)$:

Theorem 6 

([57]). Let $\widetilde{Ɣ}:∆\left[ծ, ռ\right]\times \left[ʋ, ȡ\right]\subset {\mathbb{R}}^2\to {\mathbb{F}}_0$ be an $FNVM$ on the coordinates, whose $ƺ$-cuts set up the sequence of ${\rm I}VM$s ${Ɣ}_{ƺ}:∆\subset {\mathbb{R}}^2\to {\mathbb{R}}_I$ are given by ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ for all $\left(⍬,\psi \right)\in ∆=\left[ծ, ռ\right]\times \left[ʋ, ȡ\right]$ and for all $ƺ\in \left[0, 1\right].$ Then, $\widetilde{Ɣ}$ is fuzzy double integrable ($FD$-integrable) over $∆$ if and only if ${Ɣ}_{*}\left(⍬,ƺ\right)$ and ${Ɣ}^{*}\left(⍬,ƺ\right)$ are both $D$-integrable over $∆.$ Moreover, if $\widetilde{Ɣ}$ is $FD$-integrable over $∆,$ then

$$\begin{array}{cc}\hfill {\left[\left(FD\right){\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)d\psi d⍬\right]}^{\begin{array}{c} \\ ƺ\end{array}}& =\left[\left(D\right){\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right)d\psi d⍬, \left(D\right){\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)d\psi d⍬\right]\hfill \\ & =\left(ID\right){\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{Ɣ}_{ƺ}\left(⍬,\psi \right)d\psi d⍬,\hfill \end{array}$$

(18)

for all $ƺ\in \left[0, 1\right].$

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

The following is the main definition of the fuzzy Riemann–Liouville fractional integral on the coordinates of the function $\widetilde{Ɣ}\left(⍬,\psi \right)$:

Definition 4. 

([42]). Let $\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ and $\widetilde{Ɣ}\in {\mathcal{F}\mathfrak{O}}_{∆}$. The double fuzzy interval Riemann–Liouville-type integrals ${\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }, {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-} }^{ɤ, \beta }$ $,{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+} }^{ɤ, \beta },{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-} }^{ɤ, \beta }$ of $Ɣ$ order $ɤ, \beta >0$ are defined by the following:

$${\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(⍬,\psi \right)=\frac{1}{\Gamma \left(ɤ\right)\Gamma \left(\beta \right)}{\int }_{ծ}^{⍬}{\int }_{ʋ}^{\psi }{{\left(\mathtt{⍬}-\mathtt{t}\right)}^{ɤ-1}\left(\psi -\mathtt{s}\right)}^{\beta -1}\widetilde{Ɣ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(⍬>ծ, \psi >ʋ\right),$$

(19)

$${\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(⍬,\psi \right)=\frac{1}{\Gamma \left(ɤ\right)\Gamma \left(\beta \right)}{\int }_{ծ}^{⍬}{\int }_{\psi }^{ȡ}{{\left(\mathtt{⍬}-\mathtt{t}\right)}^{ɤ-1}\left(\mathtt{s}-\psi \right)}^{\beta -1}\widetilde{Ɣ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (⍬>ծ, \psi <ȡ),$$

(20)

$${\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(⍬,\psi \right)=\frac{1}{\Gamma \left(ɤ\right)\Gamma \left(\beta \right)}{\int }_{⍬}^{ռ}{\int }_{ʋ}^{\psi }{{\left(\mathtt{t}-\mathtt{⍬}\right)}^{ɤ-1}\left(\psi -\mathtt{s}\right)}^{\beta -1}\widetilde{Ɣ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, \left(⍬<ռ, \psi >ʋ\right),$$

(21)

$${\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(⍬,\psi \right)=\frac{1}{\Gamma \left(ɤ\right)\Gamma \left(\beta \right)}{\int }_{⍬}^{ռ}{\int }_{\psi }^{ȡ}{{\left(\mathtt{t}-\mathtt{⍬}\right)}^{ɤ-1}\left(\mathtt{s}-\psi \right)}^{\beta -1}\widetilde{Ɣ}\left(\mathtt{t},\mathtt{s}\right)d\mathtt{s}d\mathtt{t}, (⍬<ռ, \psi <ȡ).$$

(22)

The following is the newly defined concept of coordinated convexity over fuzzy-number space in the codomain via the $UD$-relation:

Definition 5 

([42]). The $FNVM$ $\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ is referred to as a coordinated $UD$-convex $FNVM$ on$∆$ if

$$\begin{array}{c}\widetilde{Ɣ}\left(\epsilon ծ+\left(1-\epsilon \right)ռ, sʋ+\left(1-s\right)ȡ\right)\hfill \\ {\supseteq }_{\mathbb{F}}\epsilon s\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \epsilon \left(1-s\right)\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \left(1-\epsilon \right)s\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \left(1-\epsilon \right)\left(1-s\right)\widetilde{Ɣ}\left(ռ,ȡ\right),\hfill \end{array}$$

(23)

for all$\left(ծ, ռ\right), \left(ʋ,ȡ\right)\in ∆,$and $\epsilon ,s\in \left[0, 1\right],$ where $\widetilde{Ɣ}\left(⍬\right){\ge }_{\mathbb{F}}\tilde{0}.$ If inequality (23) is reversed, then $\widetilde{Ɣ}$ is referred to as a coordinate concave $FNVM$ on $∆$.

Lemma 1 

([42]). Let $\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ be a coordinated $FNVM$ on $∆$. Then, $\widetilde{Ɣ}$ is a coordinated $UD$-convex $FNVM$ on $∆$ if and only if there are two coordinated $UD$-convex $FNVM$s ${\widetilde{Ɣ}}_{⍬}:\left[ʋ,ȡ\right]\to {\mathbb{F}}_0$, ${\widetilde{Ɣ}}_{⍬}\left(w\right)=\widetilde{Ɣ}\left(⍬,w\right)$ and ${\widetilde{Ɣ}}_{\psi }:\left[ծ,ռ\right]\to {\mathbb{F}}_0$, ${\widetilde{Ɣ}}_{\psi }\left(z\right)=\widetilde{Ɣ}\left(z,\psi \right)$.

Theorem 7 

([42]). Let $\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ be an $FNVM$ on $∆$. Then, from the $ƺ$-levels, we obtain the collection of ${\rm I}VM$s ${Ɣ}_{ƺ}:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ are given by

$${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$$

(24)

for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. Then, $\widetilde{Ɣ}$ is a coordinated $UD$-convex $FNVM$ on $∆,$ if and only if, for all $ƺ\in \left[0, 1\right],$ ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated convex and concave functions, respectively.

Example 1. 

We consider the $FNVM$ $\widetilde{Ɣ}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0$ defined by

$$Ɣ\left(⍬\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma -⍬\psi }{5-⍬\psi }, \sigma \in \left[⍬\psi , 5\right]\\ \frac{\left(6+e^{⍬}\right)\left(6+e^{\psi }\right)-\sigma }{\left(6+e^{⍬}\right)\left(6+e^{\psi }\right)-5}, \sigma \in \left(5, \left(6+e^{⍬}\right)\left(6+e^{\psi }\right)\right]\\ 0 , \mathrm{otherwise}.\end{array}\right.$$

(25)

Then, for each $ƺ\in \left[0, 1\right],$ we have ${Ɣ}_{ƺ}\left(⍬\right)=\left[\left(1-ƺ\right)⍬\psi +5ƺ,\left(1-ƺ\right)\left(6+e^{⍬}\right)\left(6+e^{\psi }\right)+5ƺ\right]$. Since the endpoint functions ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right),$ ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ are coordinate concave functions for each $ƺ\in \left[0, 1\right]$, hence $\widetilde{Ɣ}\left(⍬,\psi \right)$ is a coordinate $UD$-convex $FNVM$.

From Lemma 1 and Example 1, we can easily note that each $UD$-convex $FNVM$ is a coordinated $UD$-convex $FNVM$. But the converse is not true.

Remark 1. 

If one assumes that ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right)={Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ with $ƺ=1$, then $Ɣ$ is referred to as a classical coordinated convex function if $Ɣ$ meets the stated inequality here:

$$\begin{array}{c}Ɣ\left(\epsilon ծ+\left(1-\epsilon \right)ռ, sʋ+\left(1-s\right)ȡ\right)\hfill \\ \hspace{1em}\hspace{1em}\le \epsilon sƔ\left(ծ,ʋ\right)+\epsilon \left(1-s\right)Ɣ\left(ծ,ȡ\right)+\left(1-\epsilon \right)sƔ\left(ռ,ʋ\right)+\left(1-\epsilon \right)\left(1-s\right)Ɣ\left(ռ,ȡ\right).\hfill \end{array}$$

Let one assume that ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ with $ƺ=1$ and ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right)$ is an affine function and ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ is a concave function. If the stated inequality is present (see [25]), then

$$\begin{array}{c}Ɣ\left(\epsilon ծ+\left(1-\epsilon \right)ռ, sʋ+\left(1-s\right)ȡ\right)\hfill \\ \hspace{1em}\hspace{1em}\supseteq \epsilon sƔ\left(ծ,ʋ\right)+\epsilon \left(1-s\right)Ɣ\left(ծ,ȡ\right)+\left(1-\epsilon \right)sƔ\left(ռ,ʋ\right)+\left(1-\epsilon \right)\left(1-s\right)Ɣ\left(ռ,ȡ\right),\hfill \end{array}$$

is true.

Definition 6. 

Let ${Ɣ}_{ƺ}:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ be the collection of IVMs such that for each $ƺ\in \left[0, 1\right]{, Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ with ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated convex (concave) and affine functions on $∆$, respectively. Then,$\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ is called a coordinated left-$UD$-convex (concave) $FNVM$ on $∆$.

Definition 7. 

Let ${Ɣ}_{ƺ}:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$ be the collection of IVMs such that for each $ƺ\in \left[0, 1\right]{, Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ with ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated affine and convex (concave) functions on $∆$, respectively. Then,$\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ is called a coordinated right-$UD$-convex (concave) $FNVM$ on $∆$.

Theorem 8. 

Let $∆$ be a coordinated convex set, and let $\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ be an $FNVM$. Then, from the $ƺ$-levels, we obtain the collection of IVMs ${Ɣ}_{ƺ}:∆\to {\mathbb{R}}_I^{+}\subset {\mathbb{R}}_I$, which are given by

$${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right],$$

(26)

for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. Then, $\widetilde{Ɣ}$ is a coordinated $UD$-concave $FNVM$ on $∆,$ if and only if, for all $ƺ\in \left[0, 1\right],$ ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ are coordinated concave and convex functions, respectively.

Proof. 

The demonstration of proof of Theorem 8 is similar to the demonstration of proof of Theorem 7. □

Example 2. 

We consider the $FNVM$s $\widetilde{Ɣ}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0$ defined by

$$\widetilde{Ɣ}\left(⍬\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma -\left(6-e^{⍬}\right)\left(6-e^{\psi }\right)}{\left(6-e^{⍬}\right)\left(6-e^{\psi }\right)-25}, \sigma \in \left[\left(6-e^{⍬}\right)\left(6-e^{\psi }\right), 25\right]\\ \frac{35⍬\psi -\sigma }{35⍬\psi -25}, \sigma \in \left(25, 35⍬\psi \right]\\ 0, \mathrm{otherwise}.\end{array}\right.$$

(27)

Then, for each $ƺ\in \left[0, 1\right],$ we have ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[\left(1-ƺ\right)\left(6-e^{⍬}\right)\left(6-e^{\psi }\right)+25ƺ,35\left(1-ƺ\right)⍬\psi +25ƺ\right]$. Since the endpoint functions ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right),$ ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ are coordinate concave and convex functions for each $ƺ\in \left[0, 1\right]$, hence $\widetilde{Ɣ}\left(⍬,\psi \right)$ is a coordinated $UD$-concave $FNVM$.

In the next results, 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 Hermite–Hadamard types using a coordinated $UD$-concave $FNVM$. We acquired the most recent estimates for mappings whose products are coordinated $UD$-concave $FNVM$s using the fuzzy fractional operators.

3. Main Results

The following is the first result of coordinated integral inequalities for the Hermite–Hadamard type using the fuzzy fractional operators via coordinated $UD$-concave $FNVM$s.

Theorem 9. 

Let $\widetilde{Ɣ}:∆\to {\mathbb{F}}_0$ be a coordinate $UD$-convex $FNVM$ on $∆$. Then, from the $ƺ$-cuts, we set up the sequence of ${\rm I}VMs$ ${Ɣ}_{ƺ}:∆\to {\mathbb{R}}_I^{+}$ are given by ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. If ${\widetilde{Ɣ}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{cc}\widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill & \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ȡ\right)\right]\hfill \\ & \oplus \frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{ \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

(28)

If $Ɣ\left(⍬\right)$ is a coordinated concave $FNVM$ then,

$$\begin{array}{cc}\hfill \widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \\ & {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ʋ\right)\right]\hfill \\ & {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\right]\hfill \\ & {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ȡ\right)\right]\hfill \\ & \oplus \frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{ \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\right]\hfill \\ & {\subseteq }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

(29)

Proof. 

Let $\widetilde{Ɣ}:\left[ծ, ռ\right]\to {\mathbb{F}}_0$ be a coordinated $UD$-convex $FNVM$. Then, by hypothesis, we have

$$4\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right){\supseteq }_{\mathbb{F}}\widetilde{Ɣ}\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right)\oplus \widetilde{Ɣ}\left(\left(1-\epsilon \right)ծ+\epsilon ռ, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right).$$

Using Theorem 7, for every $ƺ\in \left[0, 1\right]$, we have

$$\begin{array}{c}4{Ɣ}_{*}\left(\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right), ƺ\right) \\ \le {Ɣ}_{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right), ƺ\right)+{Ɣ}_{*}\left(\left(\left(1-\epsilon \right)ծ+\epsilon ռ, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right), ƺ\right), \\ 4{Ɣ}^{*}\left(\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right), ƺ\right) \\ \ge {Ɣ}^{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right), ƺ\right)+{Ɣ}^{*}\left(\left(\left(1-\epsilon \right)ծ+\epsilon ռ, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right),ƺ\right).\end{array}$$

Using Lemma 1, we have

$$\begin{array}{c}2{Ɣ}_{*}\left(\left(⍬,\frac{ʋ+ȡ}{2}\right), ƺ\right)\le {Ɣ}_{*}\left(\left(⍬, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right), ƺ\right)+{Ɣ}_{*}\left(\left(⍬, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right), ƺ\right), \\ 2{Ɣ}^{*}\left(\left(⍬,\frac{ʋ+ȡ}{2}\right), ƺ\right)\ge {Ɣ}^{*}\left(\left(⍬, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right), ƺ\right)+{Ɣ}^{*}\left(\left(⍬, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right),ƺ\right),\end{array}$$

(30)

and

$$\begin{array}{c}2{Ɣ}_{*}\left(\left(\frac{ծ+ռ}{2},\psi \right), ƺ\right)\le {Ɣ}_{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right), ƺ\right)+{Ɣ}_{*}\left(\left(\left(1-\epsilon \right)ծ+tռ, \psi \right), ƺ\right), \\ 2{Ɣ}^{*}\left(\left(\frac{ծ+ռ}{2},\psi \right), ƺ\right)\ge {Ɣ}^{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right), ƺ\right)+{Ɣ}^{*}\left(\left(\left(1-\epsilon \right)ծ+tռ, \psi \right),ƺ\right).\end{array}$$

(31)

From (30) and (31), we have

$$\begin{array}{cc}\hfill 2[{Ɣ}_{*}\left(\left(⍬,\frac{ʋ+ȡ}{2}\right), ƺ\right),& {Ɣ}^{*}\left(\left(⍬,\frac{ʋ+ȡ}{2}\right), ƺ\right)]\hfill \\ & {\supseteq }_I\left[{Ɣ}_{*}\left(\left(⍬, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right), ƺ\right),{Ɣ}^{*}\left(\left(⍬, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right), ƺ\right)\right]\hfill \\ & +\left[{Ɣ}_{*}\left(\left(⍬, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right), ƺ\right),{Ɣ}^{*}\left(\left(⍬, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right),ƺ\right)\right],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill 2[{Ɣ}_{*}\left(\left(\frac{ծ+ռ}{2},\psi \right), ƺ\right),& {Ɣ}^{*}\left(\left(\frac{ծ+ռ}{2},\psi \right), ƺ\right)]\hfill \\ & {\supseteq }_I\left[{Ɣ}_{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right), ƺ\right),{Ɣ}^{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right), ƺ\right)\right]\hfill \\ & +\left[{Ɣ}_{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right), ƺ\right),{Ɣ}^{*}\left(\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right),ƺ\right)\right].\hfill \end{array}$$

It follows that

$${Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\supseteq }_I{Ɣ}_{ƺ}\left(⍬, \epsilon ʋ+\left(1-\epsilon \right)ȡ\right)+{Ɣ}_{ƺ}\left(⍬, \left(1-\epsilon \right)ʋ+\epsilon ȡ\right),$$

(32)

and

$${Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right){\supseteq }_I{Ɣ}_{ƺ}\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right)+{Ɣ}_{ƺ}\left(\epsilon ծ+\left(1-\epsilon \right)ռ, \psi \right).$$

(33)

Since ${Ɣ}_{ƺ}\left(⍬,.\right)$ and ${Ɣ}_{ƺ}\left(.,\psi \right)$, are both coordinated $UD$-convex-${\rm I}VM$s, then from inequality (15), for every $ƺ\in \left[0, 1\right]$, inequalities (32) and (43), we have

$${{Ɣ}_{ƺ}}_{⍬}\left(\frac{ʋ+ȡ}{2}\right){\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{{Ɣ}_{ƺ}}_{⍬}\left(ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } {{Ɣ}_{ƺ}}_{⍬}\left(ʋ\right)\right]{\supseteq }_I\frac{{{Ɣ}_{ƺ}}_{⍬}\left(ʋ\right)+{{Ɣ}_{ƺ}}_{⍬}\left(ȡ\right)}{2},$$

(34)

and

$${{Ɣ}_{ƺ}}_{\psi }\left(\frac{ծ+ռ}{2}\right){\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{{Ɣ}_{ƺ}}_{\psi }\left(ռ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{{Ɣ}_{ƺ}}_{\psi }\left(ծ\right)\right]{\supseteq }_I\frac{{{Ɣ}_{ƺ}}_{\psi }\left(ծ\right)+{{Ɣ}_{ƺ}}_{\psi }\left(ռ\right)}{2}.$$

(35)

Since ${{Ɣ}_{ƺ}}_{⍬}\left(w\right)={Ɣ}_{ƺ}\left(⍬,w\right)$, then (34) can be written as

$${Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(⍬,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(⍬,ʋ\right)\right]{\supseteq }_I\frac{{Ɣ}_{ƺ}\left(⍬,ʋ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)}{2}.$$

(36)

That is

$${Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\supseteq }_I\frac{\beta }{{2\left(ȡ-ʋ\right)}^{\beta }}\left[ {\int }_{ʋ}^{ȡ}{\left(ȡ-s\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,s\right)ds+{\int }_{ʋ}^{ȡ}{\left(s-ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,s\right)ds\right]{\supseteq }_I\frac{{Ɣ}_{ƺ}\left(⍬,ʋ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)}{2}.$$

Multiplying double inequality (36) by $\frac{ɤ{\left(ռ-⍬\right)}^{ɤ-1}}{{2\left(ռ-ծ\right)}^{ɤ}}$ and integrating it with respect to $⍬$ over $\left[ծ, ռ\right],$ we have

$$\begin{array}{cc}\hfill \frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}{\int }_{ծ}^{ռ}{Ɣ}_{ƺ}& \left(⍬,\frac{ʋ+ȡ}{2}\right){\left(ռ-⍬\right)}^{ɤ-1}d⍬\hfill \\ & {\supseteq }_I{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(ռ-⍬\right)}^{ɤ-1}{\left(ȡ-s\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,s\right)dsd⍬+{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(ռ-⍬\right)}^{ɤ-1}{\left(s-ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,s\right)dsd⍬\hfill \\ & {\supseteq }_I\frac{ɤ}{4{\left(ռ-ծ\right)}^{ɤ}}\left[{\int }_{ծ}^{ռ}{\left(ռ-⍬\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,ʋ\right)d⍬+{\int }_{ծ}^{ռ}{\left(ռ-⍬\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,ȡ\right)d⍬\right].\hfill \end{array}$$

(37)

Again, multiplying double inequality (36) by $\frac{ɤ{\left(⍬-ծ\right)}^{ɤ-1}}{{2\left(ռ-ծ\right)}^{ɤ}}$ and integrating it with respect to $⍬$ over $\left[ծ, ռ\right],$ we have

$$\begin{array}{cc}\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}{\int }_{ծ}^{ռ}{Ɣ}_{ƺ}& \left(⍬,\frac{ʋ+ȡ}{2}\right){\left(ռ-⍬\right)}^{ɤ-1}d⍬\hfill \\ & {\supseteq }_I\frac{ɤ\beta }{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(⍬-ծ\right)}^{ɤ-1}{\left(ȡ-s\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,s\right)dsd⍬\hfill \\ & +\frac{ɤ\beta }{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(⍬-ծ\right)}^{ɤ-1}{\left(s-ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,s\right)dsd⍬\hfill \\ & {\supseteq }_I\frac{ɤ}{4{\left(ռ-ծ\right)}^{ɤ}}\left[{\int }_{ծ}^{ռ}{\left(⍬-ծ\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,ʋ\right)d⍬+{\int }_{ծ}^{ռ}{\left(⍬-ծ\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,ȡ\right)d⍬\right].\hfill \end{array}$$

(38)

From (37), we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\right]& \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\right].\hfill \end{array}$$

(39)

From (38), we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]& \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\right].\hfill \end{array}$$

(40)

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

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\right]& \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ȡ\right)\right],\hfill \end{array}$$

(41)

and

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]& \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ȡ\right)\right].\hfill \end{array}$$

(42)

Similarly, since ${\widetilde{Ɣ}}_{\psi }\left(z\right)=\widetilde{Ɣ}\left(z,\psi \right)$ then, from (35), (41), and (42), we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, ȡ\right)\right]& \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+} }^{\beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ʋ}^{+}}^{\beta } \widetilde{Ɣ}\left(ռ,ȡ\right)\right],\hfill \end{array}$$

(43)

and

$$\begin{array}{cc}\hfill \frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ʋ\right)\right]& \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\right].\hfill \end{array}$$

(44)

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

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

$${Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right){\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\right],$$

(45)

and

$${Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right){\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right].$$

(46)

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

$$\begin{gathered} {Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right){\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right] \\ +\frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\right]. \end{gathered}$$

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

$$\begin{array}{c}\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ʋ\right)\right].\hfill \end{array}$$

(47)

The first inequality of (28) is this one.

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

$$\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } {Ɣ}_{ƺ}\left(ծ,ʋ\right)\right]{\supseteq }_I\frac{{Ɣ}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ծ,ȡ\right)}{2}.$$

(48)

$$\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } {Ɣ}_{ƺ}\left(ռ,ʋ\right)\right]{\supseteq }_I\frac{{Ɣ}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)}{2}.$$

(49)

$$\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ} {Ɣ}_{ƺ}\left(ծ,ʋ\right)\right]{\supseteq }_I\frac{{Ɣ}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)}{2}$$

(50)

$$\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ} {Ɣ}_{ƺ}\left(ծ,ȡ\right)\right]{\supseteq }_I\frac{{Ɣ}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)}{2}$$

(51)

Summing inequalities (48)–(51), and then multiplying the resultant with $\frac{1}{4}$, we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)& +{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)]\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\right]\hfill \\ & {\supseteq }_I\frac{{Ɣ}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

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

$$\begin{gathered} \frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ȡ\right)\right] \\ \hspace{1em}\hspace{1em}\oplus \frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\right] \\ {\hspace{1em}\hspace{1em}\supseteq }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)}{4}. \end{gathered}$$

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

Example 3. 

We assume the $FNVM$s $\widetilde{Ɣ}:\left[0, 2\right]\times \left[0, 2\right]\to {\mathbb{F}}_0$ defined by

$$Ɣ\left(⍬,\psi \right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma -\left(2-\sqrt{⍬}\right)\left(2-\sqrt{\psi }\right)}{4-\left(2-\sqrt{⍬}\right)\left(2-\sqrt{\psi }\right)}, \sigma \in \left[\left(2-\sqrt{⍬}\right)\left(2-\sqrt{\psi }\right), 4\right]\\ \frac{\left(2+\sqrt{⍬}\right)\left(2+\sqrt{\psi }\right)-\sigma }{\left(2+\sqrt{⍬}\right)\left(2+\sqrt{\psi }\right)-4}, \sigma \in \left(4, \left(2+\sqrt{⍬}\right)\left(2+\sqrt{\psi }\right)\right]\\ 0, \mathrm{otherwise},\end{array}\right.$$

Then, for each $ƺ\in \left[0, 1\right],$ we have ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[\left(1-ƺ\right)\left(2-\sqrt{⍬}\right)\left(2-\sqrt{\psi }\right)+4ƺ,\left(1-ƺ\right)\left(2+\sqrt{⍬}\right)\left(2+\sqrt{\psi }\right)+4ƺ\right]$. Since the end point functions ${Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right),$ ${Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)$ are coordinate concave functions for each $ƺ\in \left[0, 1\right]$, hence $\widetilde{Ɣ}\left(⍬,\psi \right)$ is a coordinate concave $FNVM$.

$$\begin{array}{c}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)=\left[\left(1-ƺ\right)+4ƺ,9\left(1-ƺ\right)+4ƺ\right],\\ \frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]\oplus \frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},ʋ\right)\right]\\ =\left[\left(1-ƺ\right)\left(2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\pi \right)+4ƺ,\left(1-ƺ\right)\left(2+\frac{\sqrt{2}}{4}+\frac{\sqrt{2}}{8}\pi \right)+4ƺ\right]\\ \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\right]\\ =\left[\left(1-ƺ\right)\left(\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4ƺ,\left(1-ƺ\right)\left(\frac{33}{8}+\sqrt{2}+\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4ƺ\right]\\ \frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus {\mathcal{I}}_{{ծ}^{+}}^{ɤ}\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ռ}^{-}}^{ɤ}\widetilde{Ɣ}\left(ծ,ȡ\right)\right]\\ \oplus \frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ʋ}^{+}}^{\beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\right]\\ =\left[\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}\left(1-ƺ\right)+4ƺ,\frac{34\sqrt{2}+\left(\sqrt{2}+4\right)\pi +24}{8\sqrt{2}}\left(1-ƺ\right)+4ƺ\right]\\ \frac{{Ɣ}_{ƺ}\left(ʋ,ռ\right)+{Ɣ}_{ƺ}\left(\sigma ,ռ\right)+{Ɣ}_{ƺ}\left(ʋ,ȡ\right)+{Ɣ}_{ƺ}\left(\sigma ,ȡ\right)}{4}=\left[\left(1-ƺ\right)\left(\frac{9}{2}-2\sqrt{2}\right)+4ƺ,\left(1-ƺ\right)\left(\frac{9}{2}+2\sqrt{2}\right)+4ƺ\right].\end{array}$$

That is

$$\begin{array}{cc}\hfill [\left(1-ƺ\right)+& 4ƺ,9\left(1-ƺ\right)+4ƺ]{\supseteq }_I\left[\left(1-ƺ\right)\left(2-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{8}\pi \right)+4ƺ,\left(1-ƺ\right)\left(2+\frac{\sqrt{2}}{4}+\frac{\sqrt{2}}{8}\pi \right)+4ƺ\right]\hfill \\ & {\supseteq }_I\left[\left(1-ƺ\right)\left(\frac{33}{8}-\sqrt{2}-\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4ƺ,\left(1-ƺ\right)\left(\frac{33}{8}+\sqrt{2}+\frac{\sqrt{2}}{2}\pi +\frac{\pi }{8}+\frac{{\pi }^2}{32}\right)+4ƺ\right]\hfill \\ & {\supseteq }_I\left[\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}\left(1-ƺ\right)+4ƺ,\frac{34\sqrt{2}+\left(\sqrt{2}+4\right)\pi +24}{8\sqrt{2}}\left(1-ƺ\right)+4ƺ\right]\hfill \\ & {\supseteq }_I\frac{34\sqrt{2}+\left(\sqrt{2}-4\right)\pi -24}{8\sqrt{2}}\left(1-ƺ\right)+4ƺ,.\hfill \end{array}$$

Hence, Theorem 9 has been verified. □

Remark 2. 

If one assumes that $ɤ=1=\beta$, then from (28), as a result, there will be inequity (see [42]):

$$\begin{array}{c}\widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{ռ-ծ}{\int }_{ծ}^{ռ}\widetilde{Ɣ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬\oplus \frac{1}{ȡ-ʋ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \right]{\supseteq }_{\mathbb{F}}\frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)d\psi d⍬\hfill \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4\left(ռ-ծ\right)}\left[{\int }_{ծ}^{ռ}\widetilde{Ɣ}\left(⍬,ʋ\right)d⍬\oplus {\int }_{ծ}^{ռ}\widetilde{Ɣ}\left(⍬,ȡ\right)d⍬\right]\oplus \frac{\begin{array}{c} \\ 1\end{array}}{4\left(ȡ-ʋ\right)}\left[{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(ծ,\psi \right)d\psi \oplus {\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(ռ,\psi \right)d\psi \right]\hfill \\ \hfill {\supseteq }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

If one assumes that $ɤ=1=\beta$ and $\widetilde{Ɣ}$ is a coordinated left-$UD$-convex, then from (28), as a result, there will be inequity (see [27]):

$$\begin{array}{cc}\hfill \widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \\ & {\le }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{ռ-ծ}{\int }_{ծ}^{ռ}\widetilde{Ɣ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬\oplus \frac{1}{ȡ-ʋ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \right]{\le }_{\mathbb{F}}\frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)d\psi d⍬\hfill \\ & {\le }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4\left(ռ-ծ\right)}\left[{\int }_{ծ}^{ռ}\widetilde{Ɣ}\left(⍬,ʋ\right)d⍬\oplus {\int }_{ծ}^{ռ}\widetilde{Ɣ}\left(⍬,ȡ\right)d⍬\right]\oplus \frac{\begin{array}{c} \\ 1\end{array}}{4\left(ȡ-ʋ\right)}\left[{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(ծ,\psi \right)d\psi \oplus {\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(ռ,\psi \right)d\psi \right]\hfill \\ & {\le }_{\mathbb{F}}\frac{\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (28), we succeed in bringing about the upcoming inequity (see [26]):

$$\begin{array}{cc}\hfill Ɣ\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \\ & \supseteq \frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}Ɣ\left(ռ,\frac{ʋ+ȡ}{2}\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}Ɣ\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }Ɣ\left(\frac{ծ+ռ}{2},ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }Ɣ\left(\frac{ծ+ռ}{2},ʋ\right)\right]\hfill \\ & \supseteq \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }Ɣ\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }Ɣ\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }Ɣ\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }Ɣ\left(ծ,ʋ\right)\right]\hfill \\ & \supseteq \frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+} }^{ɤ}Ɣ\left(ռ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+}}^{ɤ}Ɣ\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}Ɣ\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}Ɣ\left(ծ,ȡ\right)\right]\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+} }^{ \beta }Ɣ\left(ծ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }Ɣ\left(ծ,ʋ\right)+{\mathcal{I}}_{{ʋ}^{+}}^{\beta }Ɣ\left(ռ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }Ɣ\left(ռ,ʋ\right)\right]\hfill \\ & \supseteq \frac{Ɣ\left(ծ,ʋ\right)+Ɣ\left(ռ,ʋ\right)+Ɣ\left(ծ,ȡ\right)+Ɣ\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (28), we succeed in bringing about the upcoming inequity (see [25]):

$$\begin{array}{cc}\hfill Ɣ\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \\ & \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{ռ-ծ}{\int }_{ծ}^{ռ}Ɣ\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬+\frac{1}{ȡ-ʋ}{\int }_{ʋ}^{ȡ}Ɣ\left(\frac{ծ+ռ}{2},\psi \right)d\psi \right]\subseteq \frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)} {\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}Ɣ\left(⍬,\psi \right)d\psi d⍬\hfill \\ & \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{4\left(ռ-ծ\right)}\left[{\int }_{ծ}^{ռ}Ɣ\left(⍬,ʋ\right)d⍬+{\int }_{ծ}^{ռ}Ɣ\left(⍬,ȡ\right)d⍬\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(ȡ-ʋ\right)}\left[{\int }_{ʋ}^{ȡ}Ɣ\left(ծ,\psi \right)d\psi +{\int }_{ʋ}^{ȡ}Ɣ\left(ռ,\psi \right)d\psi \right]\hfill \\ & \supseteq \frac{Ɣ\left(ծ,ʋ\right)+Ɣ\left(ռ,ʋ\right)+Ɣ\left(ծ,ȡ\right)+Ɣ\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

If $\widetilde{Ɣ}$ is coordinated right-$UD$-convex and ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)={Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (28), we succeed in bringing about the upcoming inequity (see [37]):

$$\begin{array}{cc}\hfill Ɣ\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \\ & \le \frac{\Gamma \left(ɤ+1\right)}{{4\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ} Ɣ\left(ռ,\frac{ʋ+ȡ}{2}\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}Ɣ\left(ծ,\frac{ʋ+ȡ}{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{{4\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta } Ɣ\left(\frac{ծ+ռ}{2},ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } Ɣ\left(\frac{ծ+ռ}{2},ʋ\right)\right]\hfill \\ & \le \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }Ɣ\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta } Ɣ\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+} }^{ɤ, \beta }Ɣ\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta } Ɣ\left(ծ,ʋ\right)\right]\hfill \\ & \le \frac{\Gamma \left(ɤ+1\right)}{{8\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+} }^{ɤ}Ɣ\left(ռ,ʋ\right)Ɣ{\mathcal{I}}_{{ծ}^{+}}^{ɤ} Ɣ\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}Ɣ\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ} Ɣ\left(ծ,ȡ\right)\right].\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{8\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+} }^{ \beta }Ɣ\left(ծ,ȡ\right)\widetilde{+}{\mathcal{I}}_{{ȡ}^{-}}^{\beta } Ɣ\left(ծ,ʋ\right)+{\mathcal{I}}_{{ʋ}^{+} }^{\beta }Ɣ\left(ռ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } Ɣ\left(ռ,ʋ\right)\right]\hfill \\ & \le \frac{Ɣ\left(ծ,ʋ\right)+Ɣ\left(ռ,ʋ\right)+Ɣ\left(ծ,ȡ\right)+Ɣ\left(ռ,ȡ\right)}{4}.\hfill \end{array}$$

Theorem 10. 

Let $\widetilde{Ɣ}, \tilde{\mathcal{J}}:∆\to {\mathbb{F}}_0$ be a coordinated $UD$-convex $FNVMFNVM$ on $∆$. Then, from the $ƺ$-cuts, we set up the sequence of ${\rm I}VM$s ${Ɣ}_{ƺ},{\mathcal{J}}_{ƺ}:∆\to {\mathbb{R}}_I^{+}$ are given by ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ and ${\mathcal{J}}_{ƺ}\left(⍬,\psi \right)=\left[{\mathcal{J}}_{*}\left(\left(⍬,\psi \right),ƺ\right), {\mathcal{J}}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. If${ \widetilde{Ɣ}\otimes \tilde{\mathcal{J}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} & \left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\right]\hfill \\ & \oplus \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\oplus \frac{ɤ}{(ɤ+1)(ɤ+2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\oplus \frac{\beta }{(\beta +1)(\beta +2)}\frac{ɤ}{(ɤ+1)(ɤ+2)}\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(52)

If $\widetilde{Ɣ}$ and $\tilde{\mathcal{J}}$ are both coordinated concave $FNVM$s on $∆$, then the above inequality can be expressed as follows:

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}& \left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\right]\hfill \\ & \oplus \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\right]\hfill \\ & {\subseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\oplus \frac{ɤ}{(ɤ+1)(ɤ+2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\oplus \frac{\beta }{(\beta +1)(\beta +2)}\frac{ɤ}{(ɤ+1)(ɤ+2)}\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \tilde{K}\left(ծ,ռ,ʋ,ȡ\right)& =\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\hfill \\ & \oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right),\hfill \\ \hfill \tilde{L}\left(ծ,ռ,ʋ,ȡ\right)& =\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)\hfill \\ & \otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right),\hfill \\ \hfill \tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)& =\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\hfill \\ & \oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right),\hfill \\ \tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right)\hfill & =\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\oplus \widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\hfill \\ & \oplus \widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\oplus \widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right),\hfill \end{array}$$

and for each $ƺ\in \left[0, 1\right],$ $\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)$, $\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)$, $\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)$, and $\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right)$ are defined as follows:

$$\begin{array}{c}{K}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)=\left[K_{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right), K^{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right)\right],\\ L_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)=\left[L_{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right), L^{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right)\right],\\ {\mathcal{M}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)=\left[{\mathcal{M}}_{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right), {\mathcal{M}}^{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right)\right],\\ {\mathcal{N}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)=\left[{\mathcal{N}}_{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right), {\mathcal{N}}^{*}\left(\left(ծ,ռ,ʋ,ȡ\right), ƺ\right)\right]\end{array}$$

Proof. 

Let $\widetilde{Ɣ}$ and $\tilde{\mathcal{J}}$ be two coordinated $UD$-convex $FNVM$s on $\left[ծ, ռ\right]\times \left[ʋ, ȡ\right]$. Then

$$\begin{array}{c}\widetilde{Ɣ}\left(\epsilon ծ+\left(1-\epsilon \right)ռ, sʋ+\left(1-s\right)ȡ\right)\hfill \\ {\supseteq }_{\mathbb{F}}\epsilon s\widetilde{Ɣ}\left(ծ,ʋ\right)\oplus \epsilon \left(1-s\right)\widetilde{Ɣ}\left(ծ,ȡ\right)\oplus \left(1-\epsilon \right)s\widetilde{Ɣ}\left(ռ,ʋ\right)\oplus \left(1-\epsilon \right)\left(1-s\right)\widetilde{Ɣ}\left(ռ,ȡ\right),\hfill \end{array}$$

And

$$\begin{array}{c}\tilde{\mathcal{J}}\left(\epsilon ծ+\left(1-\epsilon \right)ռ, sʋ+\left(1-s\right)ȡ\right)\hfill \\ {\supseteq }_{\mathbb{F}}\epsilon s\tilde{\mathcal{J}}\left(ծ,ʋ\right)\oplus \epsilon \left(1-s\right)\tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus \left(1-\epsilon \right)s\tilde{\mathcal{J}}\left(ռ,ʋ\right)\oplus \left(1-\epsilon \right)\left(1-s\right)\tilde{\mathcal{J}}\left(ռ,ȡ\right).\hfill \end{array}$$

Since $\widetilde{Ɣ}$ and $\tilde{\mathcal{J}}$ are both coordinated $UD$-convex $FNVM$s, Lemma 1 states that

$${\widetilde{Ɣ}}_{⍬}:\left[ʋ,ȡ\right]\to {\mathbb{F}}_0, {\widetilde{Ɣ}}_{⍬}\left(\psi \right)=\widetilde{Ɣ}\left(⍬,\psi \right), {\tilde{\mathcal{J}}}_{⍬}:\left[ʋ,ȡ\right]\to {\mathbb{F}}_0, {\tilde{\mathcal{J}}}_{⍬}\left(\psi \right)=\tilde{\mathcal{J}}\left(⍬,\psi \right).$$

Since ${\widetilde{Ɣ}}_{⍬}$ and ${\tilde{\mathcal{J}}}_{⍬}$ are $FNVM$s, then by inequality (16), we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}& \left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{\widetilde{Ɣ}}_{⍬}\left(ȡ\right)\otimes {\tilde{\mathcal{J}}}_{⍬}\left(ȡ\right)\oplus {\mathcal{I}}_{{ȡ}^{-}}^{\beta }{\widetilde{Ɣ}}_{⍬}\left(ʋ\right)\otimes {\tilde{\mathcal{J}}}_{⍬}\left(ʋ\right)\right]\hfill \\ & {\supseteq }_{\mathbb{F}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\widetilde{Ɣ}}_{⍬}\left(ʋ\right)\otimes {\tilde{\mathcal{J}}}_{⍬}\left(ʋ\right)\oplus {\widetilde{Ɣ}}_{⍬}\left(ȡ\right)\otimes {\tilde{\mathcal{J}}}_{⍬}\left(ȡ\right)\right)\hfill \\ & \oplus \left(\frac{\beta }{(\beta +1)(\beta +2)}\right)\left({\widetilde{Ɣ}}_{⍬}\left(ʋ\right)\otimes {\tilde{\mathcal{J}}}_{⍬}\left(ȡ\right)\oplus {\widetilde{Ɣ}}_{⍬}\left(ȡ\right)\otimes {\tilde{\mathcal{J}}}_{⍬}\left(ʋ\right)\right).\hfill \end{array}$$

Now, for all $ƺ\in \left[0, 1\right]$, we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}& \left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta } {{Ɣ}_{ƺ}}_{⍬}\left(ȡ\right)\times {{\mathcal{J}}_{ƺ}}_{⍬}\left(ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{{Ɣ}_{ƺ}}_{⍬}\left(ʋ\right)\times {{\mathcal{J}}_{ƺ}}_{⍬}\left(ʋ\right)\right]\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({{Ɣ}_{ƺ}}_{⍬}\left(ʋ\right)\times {{\mathcal{J}}_{ƺ}}_{⍬}\left(ʋ\right)+{{Ɣ}_{ƺ}}_{⍬}\left(ȡ\right)\times {{\mathcal{J}}_{ƺ}}_{⍬}\left(ȡ\right)\right)\hfill \\ & +\left(\frac{\beta }{(\beta +1)(\beta +2)}\right)\left({{Ɣ}_{ƺ}}_{⍬}\left(ʋ\right)\times {{\mathcal{J}}_{ƺ}}_{⍬}\left(ȡ\right)+{{Ɣ}_{ƺ}}_{⍬}\left(ȡ\right)\times {{\mathcal{J}}_{ƺ}}_{⍬}\left(ʋ\right)\right).\hfill \end{array}$$

That is

$$\begin{array}{cc}\hfill \frac{\beta }{{2\left(ȡ-ʋ\right)}^{\beta }}& \left[{\int }_{ʋ}^{ȡ}{\left(ȡ-\psi \right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,\psi \right)\times {\mathcal{J}}_{ƺ}\left(⍬,\psi \right)d\psi +{\int }_{ʋ}^{ȡ}{\left(\psi -ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,\psi \right)\times {\mathcal{J}}_{ƺ}\left(⍬,\psi \right)d\psi \right]\\ & {\supseteq }_I\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(⍬,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ʋ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ȡ\right)\right)\hfill \\ & +\left(\frac{\beta }{(\beta +1)(\beta +2)}\right)\left({Ɣ}_{ƺ}\left(⍬,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ȡ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ʋ\right)\right).\hfill \end{array}$$

(53)

Multiplying double inequality (53) by $\frac{ɤ{\left(ռ-⍬\right)}^{ɤ-1}}{{2\left(ռ-ծ\right)}^{ɤ}}$ and integrating it with respect to $⍬$ over $\left[ծ, ռ\right],$ we obtain

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} & \left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta } {Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\frac{\beta }{(\beta +1)(\beta +2)}\left({\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{{\mathcal{I}}_{{ծ}^{+} }^{ɤ}Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right).\hfill \end{array}$$

(54)

Again, multiplying inequality (54) by $\frac{ɤ{\left(⍬-ծ\right)}^{ɤ-1}}{{2\left(ռ-ծ\right)}^{ɤ}}$ and integrating it with respect to $⍬$ over $\left[ծ, ռ\right],$ we obtain

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}& \left[\begin{array}{c}{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\frac{\beta }{(\beta +1)(\beta +2)}\left({\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{{\mathcal{I}}_{{ռ}^{-} }^{ɤ}Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right).\hfill \end{array}$$

(55)

Summing (54) and (55), we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}& \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\\ +\begin{array}{c}{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\end{array}\end{array}\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right) \right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\frac{\beta }{(\beta +1)(\beta +2)}\left({{\mathcal{I}}_{{ծ}^{+} }^{ɤ}Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{{\mathcal{I}}_{{ռ}^{-} }^{ɤ}Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right).\hfill \end{array}$$

(56)

Now, once more with the aid of integral inequality (16), we obtain the following relationship for the first two integrals on the right-hand side of (56):

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}& \left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & +\left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right).\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}& \left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right).\hfill \end{array}$$

(58)

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}& \left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right) \right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right).\hfill \end{array}$$

(59)

And

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}& \left({{\mathcal{I}}_{{ծ}^{+} }^{ɤ}Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{{\mathcal{I}}_{{ռ}^{-} }^{ɤ}Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & +\left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right).\hfill \end{array}$$

(60)

From (57)–(60), inequality (54) we have

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}& \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\\ +{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)L_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}{\mathcal{M}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{ɤ}{(ɤ+1)(ɤ+2)}{\mathcal{N}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

Since we obtain the collection of ${\rm I}VM$s ${Ɣ}_{ƺ}, {\mathcal{J}}_{ƺ}:∆\to {\mathbb{R}}_I^{+}$ from the $ƺ$-cuts, the aforementioned inequality can be expressed as an inequality (52). A conclusion has, therefore, been established. □

Remark 3. 

If one assumes that $ɤ=1=\beta$, then from (52), as a result, there will be inequity (see [42]):

$$\begin{array}{cc}\hfill \frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}& {\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)\otimes \tilde{\mathcal{J}}\left(⍬,\psi \right)d\psi d⍬\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{1}{9}\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\oplus \frac{1}{18}\left[\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\oplus \tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\right]\oplus \frac{1}{36}\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If $\widetilde{Ɣ}$ is coordinated left-$UD$-convex and one assumes that $ɤ=1=\beta$, then from (52), as a result, there will be inequity (see [27]):

$$\begin{array}{cc}\hfill \frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}& {\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)\otimes \tilde{\mathcal{J}}\left(⍬,\psi \right)d\psi d⍬\hfill \\ & {\le }_{\mathbb{F}}\frac{1}{9}\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\oplus \frac{1}{18}\left[\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\oplus \tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\right]\oplus \frac{1}{36}\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (52), we succeed in bringing about the upcoming inequity (see [26]):

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}& \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }Ɣ\left(ռ,ȡ\right)\times J\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }Ɣ\left(ռ,ʋ\right)\times J\left(ռ,ʋ\right)\end{array}\right]\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }Ɣ\left(ծ,ȡ\right)\times J\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }Ɣ\left(ծ,ʋ\right)\times J\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & \supseteq \left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K\left(ծ,ռ,ʋ,ȡ\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)L\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\mathcal{M}\left(ծ,ռ,ʋ,ȡ\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{ɤ}{(ɤ+1)(ɤ+2)}\mathcal{N}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (52), we succeed in bringing about the upcoming inequity (see [25]):

$$\begin{array}{cc}\hfill \frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}{\int }_{ծ}^{ռ}& {\int }_{ʋ}^{ȡ}Ɣ\left(⍬,\psi \right)\times \mathcal{J}\left(⍬,\psi \right)d\psi d⍬\hfill \\ & \supseteq \frac{1}{9}K\left(ծ,ռ,ʋ,ȡ\right)+\frac{1}{18}\left[L\left(ծ,ռ,ʋ,ȡ\right)+\mathcal{M}\left(ծ,ռ,ʋ,ȡ\right)\right]+\frac{1}{36}\mathcal{N}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)={Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${\mathcal{J}}_{*}\left(\left(⍬,\psi \right), ƺ\right)={\mathcal{J}}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (52), we succeed in bringing about the upcoming inequity (see [41]):

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}& \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }Ɣ\left(ռ,ȡ\right)\times J\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }Ɣ\left(ռ,ʋ\right)\times J\left(ռ,ʋ\right)\end{array}\right]\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}+{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }Ɣ\left(ծ,ȡ\right)\times J\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }Ɣ\left(ծ,ʋ\right)\times J\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & \le \left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K\left(ծ,ռ,ʋ,ȡ\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)L\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\frac{\beta }{(\beta +1)(\beta +2)}\mathcal{M}\left(ծ,ռ,ʋ,ȡ\right)+\frac{\beta }{(\beta +1)(\beta +2)}\frac{ɤ}{(ɤ+1)(ɤ+2)}\mathcal{N}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

Theorem 11. 

Let $\widetilde{Ɣ}, \tilde{\mathcal{J}}:∆\to {\mathbb{F}}_0$ be a coordinated $UD$-convex $FNVMFNVM$ on $∆$. Then, from the $ƺ$-cuts, we set up the sequence of ${\rm I}VM$s${Ɣ}_{ƺ},{\mathcal{J}}_{ƺ}:∆\to {\mathbb{R}}_I^{+}$ that are given by ${Ɣ}_{ƺ}\left(⍬,\psi \right)=\left[{Ɣ}_{*}\left(\left(⍬,\psi \right),ƺ\right), {Ɣ}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ and ${\mathcal{J}}_{ƺ}\left(⍬,\psi \right)=\left[{\mathcal{J}}_{*}\left(\left(⍬,\psi \right),ƺ\right), {\mathcal{J}}^{*}\left(\left(⍬,\psi \right),ƺ\right)\right]$ for all $\left(⍬,\psi \right)\in ∆$ and for all $ƺ\in \left[0, 1\right]$. If ${\widetilde{Ɣ}\otimes \tilde{\mathcal{J}}\in \mathcal{F}\mathfrak{O}}_{∆}$, then the following inequalities hold:

$$\begin{array}{cc}\hfill 4\widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \otimes \tilde{\mathcal{J}}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\end{array}\right]\hfill \\ & \oplus \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & \oplus \left[\frac{ɤ}{2\left(ɤ+1\right)\left(ɤ+2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\right]\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{4}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(61)

If $\widetilde{Ɣ}$ and $\tilde{\mathcal{J}}$ are both coordinated concave $FNVM$s on $∆$, then the above inequality can be expressed as follows:

$$\begin{array}{cc}\hfill 4\widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \otimes \tilde{\mathcal{J}}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ & {\subseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\right]\hfill \\ & \oplus \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\right]\hfill \\ & \oplus \left[\frac{ɤ}{2\left(ɤ+1\right)\left(ɤ+2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\right]\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{4}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right),\hfill \end{array}$$

(62)

where $\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)$, $\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)$, $\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right),$ and $\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right)$ are given in Theorem 10.

Proof. 

Since $\widetilde{Ɣ},\tilde{\mathcal{J}} :∆\to {\mathbb{F}}_0$ are two $UD$-convex $FNVM$s, then from inequality (17) and for each $ƺ\in \left[0, 1\right],$ we have

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}} \left[\begin{array}{c}{\int }_{ծ}^{ռ}{\left(ռ-⍬\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\times \mathcal{J}}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬\\ +{\int }_{ծ}^{ռ}{\left(⍬-ծ\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\times \mathcal{J}}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬ \end{array}\right]\hfill \\ & +\left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left({Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)+{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)+{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right),\hfill \end{array}$$

(63)

and

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\beta }{{2\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}{\int }_{ʋ}^{ȡ}{\left(ȡ-\psi \right)}^{\beta -1}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \\ +{\int }_{ʋ}^{ȡ}{\left(\psi -ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \end{array}\right]\hfill \\ & +\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)+{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)+{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\right)\hfill \end{array}$$

(64)

By adding (63) and (64) and multiplying the result by $2$, we obtain:

$$\begin{array}{cc}\hfill 8{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}} \left[\begin{array}{c}{\int }_{ծ}^{ռ}2{\left(ռ-⍬\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\times \mathcal{J}}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬\\ +{\int }_{ծ}^{ռ}2{\left(⍬-ծ\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\times \mathcal{J}}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬ \end{array}\right]\hfill \\ & +\frac{\beta }{{2\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\int }_{ʋ}^{ȡ}2{\left(ȡ-\psi \right)}^{\beta -1}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \\ +{\int }_{ʋ}^{ȡ}2{\left(\psi -ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \end{array}\right]\hfill \\ & +\left(\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left(2{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)+2{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left(2{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)+2{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\times {\mathcal{J}}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\right)\hfill \\ & +\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left(2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)+2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left(2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)+2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\right).\hfill \end{array}$$

(65)

Using Lemma 1 for each integral on the right-hand side of (65) and the integral inequality (17) once more leads us to obtain the following:

$$\begin{array}{cc}\hfill \frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}& {\int }_{ծ}^{ռ}2{\left(ռ-⍬\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\times \mathcal{J}}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬\hfill \\ & {\supseteq }_I\frac{ɤ\beta }{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(ռ-⍬\right)}^{ɤ-1}{\left(ȡ-\psi \right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,\psi \right)d\psi d⍬\right]\hfill \\ & +\frac{ɤ\beta }{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(ռ-⍬\right)}^{ɤ-1}{\left(\psi -ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,\psi \right)d\psi d⍬\right]\hfill \\ & +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}{\int }_{ծ}^{ռ}{\left(ռ-⍬\right)}^{ɤ-1}\left({Ɣ}_{ƺ}\left(⍬,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ʋ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ȡ\right)\right)d⍬\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}{\int }_{ծ}^{ռ}{\left(ռ-⍬\right)}^{ɤ-1}\left({Ɣ}_{ƺ}\left(⍬,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ȡ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ʋ\right)\right)d⍬,\hfill \\ & =\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta } {Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\end{array}\right]\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+} }^{ɤ} {Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right).\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill \frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}& {\int }_{ծ}^{ռ}2{\left(⍬-ծ\right)}^{ɤ-1}{Ɣ}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right){\times \mathcal{J}}_{ƺ}\left(⍬,\frac{ʋ+ȡ}{2}\right)d⍬\hfill \\ & {\supseteq }_I\frac{ɤ\beta }{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(⍬-ծ\right)}^{ɤ-1}{\left(ȡ-\psi \right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,\psi \right)d\psi d⍬\right]\hfill \\ & +\frac{ɤ\beta }{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}{\left(⍬-ծ\right)}^{ɤ-1}{\left(\psi -ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(⍬,\psi \right)d\psi d⍬\right]\hfill \\ & +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}{\int }_{ծ}^{ռ}{\left(⍬-ծ\right)}^{ɤ-1}\left({Ɣ}_{ƺ}\left(⍬,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ʋ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ȡ\right)\right)d⍬\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\frac{ɤ}{{2\left(ռ-ծ\right)}^{ɤ}}{\int }_{ծ}^{ռ}{\left(⍬-ծ\right)}^{ɤ-1}\left({Ɣ}_{ƺ}\left(⍬,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ȡ\right)+{Ɣ}_{ƺ}\left(⍬,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(⍬,ʋ\right)\right)d⍬,\hfill \\ & =\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta } {Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right).\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill \frac{\beta }{{2\left(ȡ-ʋ\right)}^{\beta }} & \left[\begin{array}{c}{\int }_{ʋ}^{ȡ}2{\left(ȡ-\psi \right)}^{\beta -1}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \end{array}\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\end{array}\right]\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left(\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right).\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill \frac{\beta }{{2\left(ȡ-ʋ\right)}^{\beta }}& \left[\begin{array}{c}{\int }_{ʋ}^{ȡ}2{\left(\psi -ʋ\right)}^{\beta -1}{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)\times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\psi \right)d\psi \end{array}\right]\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-} }^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta } {Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\end{array}\right]\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left(\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right).\hfill \end{array}$$

(69)

And

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ} {Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ} {Ɣ}_{ƺ}\left(ծ, ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right]\hfill \\ & +\frac{ɤ}{(ɤ+1)(ɤ+2)}\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right),\hfill \end{array}$$

(70)

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ, ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right]\hfill \\ & +\frac{ɤ}{(ɤ+1)(ɤ+2)}\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right),\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ} {Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ} {Ɣ}_{ƺ}\left(ծ, ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right]\hfill \\ & +\frac{ɤ}{(ɤ+1)(ɤ+2)}\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right),\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},ȡ\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},ʋ\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left[{\mathcal{I}}_{{ծ}^{+}}^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-}}^{ɤ}{Ɣ}_{ƺ}\left(ծ, ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right]\hfill \\ & +\frac{ɤ}{(ɤ+1)(ɤ+2)}\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right),\hfill \end{array}$$

(73)

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{Ɣ}_{ƺ}\left(ծ, ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right]\hfill \\ & +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right),\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta } {Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } {Ɣ}_{ƺ}\left(ռ, ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right]\hfill \\ & +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right),\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta } {Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } {Ɣ}_{ƺ}\left(ծ, ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right]\hfill \\ & +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right),\hfill \end{array}$$

(76)

and

$$\begin{array}{cc}\hfill 2{Ɣ}_{ƺ}\left(ռ,\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(ծ,\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left[{\mathcal{I}}_{{ʋ}^{+}}^{\beta } {Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta } {Ɣ}_{ƺ}\left(ռ, ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right]\hfill \\ & +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right).\hfill \end{array}$$

(77)

From inequalities (66)–(77) and inequality (65), we have

$$\begin{array}{cc}\hfill 8{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{2\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\end{array}\\ +{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & +\left(\frac{2ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left[\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left({\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ʋ}^{+}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left({\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ȡ}^{-}}^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\end{array}\right]\hfill \\ & +2\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\left[\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left({\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left({\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\end{array}\right]\hfill \\ & +2\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left[\begin{array}{c}\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+} }^{ɤ} {Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\\ +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\end{array}\right]\hfill \\ & +2\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)\left[\begin{array}{c}\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\\ +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right)\end{array}\right]\hfill \\ & +\frac{2ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}{K}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)++\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\frac{2\beta }{\left(\beta +1\right)\left(\beta +2\right)}{L}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\frac{2ɤ}{(ɤ+1)(ɤ+2)}\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right){\mathcal{M}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +2\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right){\mathcal{N}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(78)

Using Lemma 1 for each integral on the right-hand side of (78) and the integral inequality (16) once more leads us to arrive at:

$$\begin{array}{cc}\hfill \frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}& \left({\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ʋ}^{+} }^{\beta } {Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left({\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)K_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)+\frac{\beta }{(\beta +1)(\beta +2)}{\mathcal{M}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill \frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}& \left({\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ʋ}^{+} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(\beta +1\right)}{{2\left(ȡ-ʋ\right)}^{\beta }}\left({\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ȡ}^{-} }^{\beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{\beta }{(\beta +1)(\beta +2)}\right)L_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)+\frac{\beta }{(\beta +1)(\beta +2)}{\mathcal{N}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(80)

$$\begin{array}{cc}\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\hfill & \left({\mathcal{I}}_{{ծ}^{+} }^{ɤ}{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)+{\mathcal{I}}_{{ծ}^{+} }^{ɤ} {Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)\right)\hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right)K_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}{L}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill \frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}& \left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right) \hfill \\ & +\frac{\Gamma \left(ɤ+1\right)}{{2\left(ռ-ծ\right)}^{ɤ}}\left({\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-} }^{ɤ}{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\right)\hfill \\ & {\supseteq }_I\left(\frac{1}{2}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\right){\mathcal{M}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}{\mathcal{N}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(82)

From (79)–(82) we have

$$\begin{array}{cc}\hfill 4{Ɣ}_{ƺ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \times {\mathcal{J}}_{ƺ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_I\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ռ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ռ,ʋ\right)\\ +{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ȡ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }{Ɣ}_{ƺ}\left(ծ,ʋ\right)\times {\mathcal{J}}_{ƺ}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & +\left[\frac{ɤ}{2\left(ɤ+1\right)\left(ɤ+2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\right]K_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]L_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]{\mathcal{M}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{4}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]{\mathcal{N}}_{ƺ}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

(83)

That is

$$\begin{array}{cc}\hfill 4\widetilde{Ɣ}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \otimes \tilde{\mathcal{J}}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }}\left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ȡ\right)\oplus {\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ռ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ռ,ʋ\right)\\ \oplus {\mathcal{I}}_{{ռ}^{-},{ʋ}^{+}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ȡ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ȡ\right)\oplus {\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta }\widetilde{Ɣ}\left(ծ,ʋ\right)\otimes \tilde{\mathcal{J}}\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & \oplus \left[\frac{ɤ}{2\left(ɤ+1\right)\left(ɤ+2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\right]\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \left[\frac{1}{4}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

A conclusion has, therefore, been established.     □

Remark 4. 

If one assumes that $ɤ=1=\beta$, then from (61), as a result, there will be inequity (see [42]):

$$\begin{array}{cc}\hfill 4\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \otimes \tilde{\mathcal{J}}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\supseteq }_{\mathbb{F}}\frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)\otimes \tilde{\mathcal{J}}\left(⍬,\psi \right)d\psi d⍬\oplus \frac{5}{36}\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \frac{7}{36}\left[\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\widetilde{+}\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\right]\oplus \frac{2}{9}\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If $\widetilde{Ɣ}$ is coordinated left-$UD$-convex and one assumes that $ɤ=1=\beta$, then from (61), as a result, there will be inequity (see [27]):

$$\begin{array}{cc}\hfill 4\widetilde{Ɣ}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \otimes \tilde{\mathcal{J}}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & {\le }_{\mathbb{F}}\frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)}{\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}\widetilde{Ɣ}\left(⍬,\psi \right)\otimes \tilde{\mathcal{J}}\left(⍬,\psi \right)d\psi d⍬\oplus \frac{5}{36}\tilde{K}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & \oplus \frac{7}{36}\left[\tilde{L}\left(ծ,ռ,ʋ,ȡ\right)\widetilde{+}\tilde{\mathcal{M}}\left(ծ,ռ,ʋ,ȡ\right)\right]\oplus \frac{2}{9}\tilde{\mathcal{N}}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (61), we succeed in bringing about the upcoming inequity (see [25]):

$$\begin{array}{cc}\hfill 4 Ɣ\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)& \times \mathcal{J}\left(\frac{ծ+ռ}{2},\frac{ʋ+ȡ}{2}\right)\hfill \\ & \supseteq \frac{1}{\left(ռ-ծ\right)\left(ȡ-ʋ\right)} {\int }_{ծ}^{ռ}{\int }_{ʋ}^{ȡ}Ɣ\left(⍬,\psi \right)\times \mathcal{J}\left(⍬,\psi \right)d\psi d⍬+\frac{5}{36}K\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\frac{7}{36}\left[L\left(ծ,ռ,ʋ,ȡ\right)+\mathcal{M}\left(ծ,ռ,ʋ,ȡ\right)\right]+\frac{2}{9}\mathcal{N}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)\ne {Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (61), we succeed in bringing about the upcoming inequity (see [26]):

$$\begin{array}{cc}\hfill 4Ɣ\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \times \mathcal{J}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ & \supseteq \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }Ɣ\left(ռ,ȡ\right)\times J\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta } Ɣ\left(ռ,ʋ\right)\times J\left(ռ,ʋ\right)\\ +{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+} }^{ɤ, \beta }Ɣ\left(ծ,ȡ\right)\times J\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta } Ɣ\left(ծ,ʋ\right)\times J\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & +\left[\frac{ɤ}{2\left(ɤ+1\right)\left(ɤ+2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\right]K\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]L\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{M}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{4}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(ծ,ռ,ʋ,ȡ\right).\hfill \end{array}$$

If ${Ɣ}_{*}\left(\left(⍬,\psi \right), ƺ\right)={Ɣ}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ and ${\mathcal{J}}_{*}\left(\left(⍬,\psi \right), ƺ\right)={\mathcal{J}}^{*}\left(\left(⍬,\psi \right), ƺ\right)$ with $ƺ=1$, then from (61), we succeed in bringing about the upcoming inequity (see [41]):

$$\begin{array}{cc}\hfill 4Ɣ\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)& \times \mathcal{J}\left(\frac{ծ+ռ}{2}, \frac{ʋ+ȡ}{2}\right)\hfill \\ & \le \frac{\Gamma \left(ɤ+1\right)\Gamma \left(\beta +1\right)}{{4\left(ռ-ծ\right)}^{ɤ}{\left(ȡ-ʋ\right)}^{\beta }} \left[\begin{array}{c}{\mathcal{I}}_{{ծ}^{+},{ʋ}^{+} }^{ɤ, \beta }Ɣ\left(ռ,ȡ\right)\times J\left(ռ,ȡ\right)+{\mathcal{I}}_{{ծ}^{+},{ȡ}^{-}}^{ɤ, \beta } Ɣ\left(ռ,ʋ\right)\times J\left(ռ,ʋ\right)\\ +{\mathcal{I}}_{{ռ}^{-},{ʋ}^{+} }^{ɤ, \beta }Ɣ\left(ծ,ȡ\right)\times J\left(ծ,ȡ\right)+{\mathcal{I}}_{{ռ}^{-},{ȡ}^{-}}^{ɤ, \beta } Ɣ\left(ծ,ʋ\right)\times J\left(ծ,ʋ\right)\end{array}\right]\hfill \\ & +\left[\frac{ɤ}{2\left(ɤ+1\right)\left(ɤ+2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)\right]K\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{ɤ}{\left(ɤ+1\right)\left(ɤ+2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]L\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{M}\left(ծ,ռ,ʋ,ȡ\right)\hfill \\ & +\left[\frac{1}{4}-\frac{ɤ}{(ɤ+1)(ɤ+2)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(ծ,ռ,ʋ,ȡ\right)\hfill \end{array}$$

4. Conclusions and Future Plans

In this study, Hermite–Hadamard-type inequalities for coordinated $UD$-convex $FNVM$ were established. These inequalities are very important in the field of inequalities because the findings in this research constitute an expansion of a number of earlier findings. A coordinated fuzzy-number-valued convexity is a novel type of class and, by using this class and other fractional integrals, new fractional inequalities can be found that are available to interested authors.