Hermite–Hadamard and Pachpatte Type Inequalities for Coordinated Preinvex Fuzzy-Interval-Valued Functions Pertaining to a Fuzzy-Interval Double Integral Operator

Abstract

Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of preinvex fuzzy-interval-valued functions (preinvex F∙I-V∙F) on coordinates in a rectangle drawn on a plane and show that these functions have Hermite–Hadamard-type inclusions. We also develop Hermite–Hadamard-type inclusions for the combination of two coordinated preinvex functions with interval values. The weighted Hermite–Hadamard-type inclusions for products of coordinated convex interval-valued functions discussed in a recent publication by Khan et al. in 2022 served as the inspiration for our conclusions. Our proven results expand and generalize several previous findings made in the body of literature. Additionally, we offer appropriate examples to corroborate our theoretical main findings.

1. Introduction

The H-H inequality has been a potent instrument to obtain a lot of excellent results in integral inequalities and optimization theory because of its crucial role in convex analysis. It has recently been generalized using other convexity types, particularly s-convex functions [1,2,3,4], log-convex functions [5,6,7], harmonic convexity [8], and particularly for h-convex functions [9]. Since 2007, numerous H-H inequalities for h-convex function extensions and generalizations have been established in [10,11,12,13,14,15,16].

On the other hand, Archimedes’ calculation of the circumference of a circle can be linked to the theory of interval analysis, which has a lengthy history. However, due to a lack of applications to other sciences, it was forgotten for a very long time. Burkill [17] developed several fundamental interval function features in 1924. Kolmogorov’s [18] generalization of Burkill’s findings from single-valued functions to multi-valued functions came shortly after. Of course, throughout the following 20 years, numerous additional outstanding achievements were also obtained. Please take notice that Moore was the first to realize how interval analysis might be used to calculate the error boundaries of computer numerical solutions. The theoretical and applied research on interval analysis has received a lot of attention and has produced useful discoveries during the past 50 years since Moore [19] published the first monograph on the subject in 1966. In more recent years, Nikodem et al. [20] and, particularly, Budak et al. [21], Chalco–Cano et al. [22,23], Costa et al. [24,25,26], Román–Flores et al. [27,28], Flores–Franuli et al. [29], and Zhao et al. [30,31,32,33] have expanded various well-known inequalities. For more information, see [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69] and the references are therein.

We introduce the coordinated preinvex functions in fuzzy interval-valued settings, which are inspired by Dragomir [34], Latif and Dragomir [44], and Khan et al. [37,38]. We also talk about how coordinated fuzzy-interval preinvexity and preinvexity relate to one another. The key findings of this study are new fuzzy-interval versions of Hermite–Hadamard-type inequalities that we develop with the help of newly defined coordinated fuzzy-interval preinvexity. Finally, we provide some examples to highlight our key findings. The current findings can also be seen as instruments for further study into topics like inequalities for fuzzy-interval-valued functions, fuzzy interval optimization, and generalized convexity.

2. Preliminaries

Let ${ℝ}_I$ be the space of all closed and bounded intervals of $ℝ$ and $\mathcal{Q}\in {ℝ}_I$ be defined by

$$\mathcal{Q}=\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]=\left\{\varkappa \in ℝ| {\mathcal{Q}}_{*}\le \varkappa \le {\mathcal{Q}}^{*}\right\},\left({\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\in ℝ\right)$$

(1)

If ${\mathcal{Q}}_{*}={\mathcal{Q}}^{*}$, then $\mathcal{Q}$ is said to be degenerate. In this article, all intervals will be non-degenerate intervals. If ${\mathcal{Q}}_{*}\ge 0$, then $\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]$ is called a positive interval. The set of all positive intervals is denoted by ${ℝ}_I^{+}$ and defined as ${ℝ}_I^{+}=\left\{\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]:\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\in {ℝ}_I \mathrm{and} {\mathcal{Q}}_{*}\ge 0\right\}.$

Let $\lambda \in ℝ$ and $\lambda \cdot \mathcal{Q}$ be defined by

$$\lambda \cdot \mathcal{Q}=\{\begin{array}{c}\left[\lambda {\mathcal{Q}}_{*}, \lambda {\mathcal{Q}}^{*}\right] \mathrm{if} \lambda >0,\\ \left\{0\right\} \mathrm{if} \lambda =0,\\ \left[\lambda {\mathcal{Q}}^{*},\lambda {\mathcal{Q}}_{*}\right] \mathrm{if} \lambda <0. \end{array}$$

(2)

Then, the Minkowski difference $\mathcal{Z}-\mathcal{Q}$, addition $\mathcal{Q}+\mathcal{Z}$, and $\mathcal{Q}\times \mathcal{Z}$ for $\mathcal{Q},\mathcal{Z}\in {ℝ}_I$ are defined by

$$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]+\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]=\left[{\mathcal{Z}}_{*}+{\mathcal{Q}}_{*}, {\mathcal{Z}}^{*}+{\mathcal{Q}}^{*}\right],$$

(3)

$$\begin{array}{c}\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]\times \left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\hfill \\ =\left[\min \left\{{\mathcal{Z}}_{*}{\mathcal{Q}}_{*}, {\mathcal{Z}}^{*}{\mathcal{Q}}_{*}, {\mathcal{Z}}_{*}{\mathcal{Q}}^{*}, {\mathcal{Z}}^{*}{\mathcal{Q}}^{*}\right\}, \max \left\{{\mathcal{Z}}_{*}{\mathcal{Q}}_{*}, {\mathcal{Z}}^{*}{\mathcal{Q}}_{*}, {\mathcal{Z}}_{*}{\mathcal{Q}}^{*}, {\mathcal{Z}}^{*}{\mathcal{Q}}^{*}\right\}\right]\hfill \end{array}$$

(4)

$$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]-\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]=\left[{\mathcal{Z}}_{*}-{\mathcal{Q}}^{*}, {\mathcal{Z}}^{*}-{\mathcal{Q}}_{*}\right],$$

(5)

Remark 1.

(i) For given $\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right], \left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\in {ℝ}_I,$the relation “${\supseteq }_I$” defined on${ℝ}_I$by$\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]{\supseteq }_I\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$if and only if

$${\mathcal{Q}}_{*}\le {\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\le {\mathcal{Q}}^{*},$$

(6)

for all$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right], \left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\in {ℝ}_I,$it is a partial interval inclusion relation. The relation$\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]{\supseteq }_I\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$coincides with$\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\supseteq \left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$on${ℝ}_I.$It can be easily seen that “${\supseteq }_I$” looks like “up and down” on the real line$ℝ,$so we call “${\supseteq }_I$” as “up and down” (or “UD” order, in short) [40].

(ii) For given$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right], \left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\in {ℝ}_I,$we say that$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]{\le }_I\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]$if and only if

$${\mathcal{Z}}_{*}\le {\mathcal{Q}}_{*}, {\mathcal{Z}}^{*}\le {\mathcal{Q}}^{*} or {\mathcal{Z}}_{*}\le {\mathcal{Q}}_{*}, {\mathcal{Z}}^{*}<{\mathcal{Q}}^{*}$$

(7)

it is a partial interval order relation. The relation$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]{\le }_I\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]$is coincident to$\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]\le \left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]$on${ℝ}_I.$It can be easily seen that “${\le }_I$” looks like “left and right” on the real line$ℝ,$so we call “${\le }_I$” as “left and right” (or “LR” order, in short) [39,40].

For $\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right],\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\in {ℝ}_I,$ the Hausdorff–Pompeiu distance between intervals $\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right]$ and $\left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]$ is defined by

$$d_H\left(\left[{\mathcal{Z}}_{*}, {\mathcal{Z}}^{*}\right], \left[{\mathcal{Q}}_{*}, {\mathcal{Q}}^{*}\right]\right)=max\left\{\left|{\mathcal{Z}}_{*}-{\mathcal{Q}}_{*}\right|, \left|{\mathcal{Z}}^{*}-{\mathcal{Q}}^{*}\right|\right\}.$$

(8)

It is a familiar fact that $\left({ℝ}_I,d_H\right)$ is a complete metric space [42,43,44].

Definition 1

([40,41]). A fuzzy subset $L$ of $ℝ$ is distinguished by mapping $\tilde{\psi }:ℝ\to \left[0, 1\right]$ called the membership mapping of $L$. That is, a fuzzy subset $L$ of $ℝ$ is mapping $\tilde{\psi }:ℝ\to \left[0, 1\right]$. So, for further study, we have chosen this notation.We appoint $\mathbb{F}$ to denote the set ofall fuzzy subsets of $ℝ$.

Let$\tilde{\psi }\in \mathbb{F}$. Then,$\tilde{\psi }$is known as a fuzzy number or fuzzy interval if the following properties are satisfied by$\tilde{\psi }$:

(1) 

$\tilde{\psi }$should be normal if there exists$\varkappa \in ℝ$and$\tilde{\psi }\left(\varkappa \right)=1;$

(2) 

$\tilde{\psi }$should be upper semi continuous on$ℝ$if for given$\varkappa \in ℝ,$there exist$\epsilon >0$or there exist$\delta >0$such that$\tilde{\psi }\left(\varkappa \right)-\tilde{\psi }\left(\omega \right)<\epsilon$for all$\omega \in ℝ$with$\left|\varkappa -\omega \right|<\delta ;$

(3) 

$\tilde{\psi }$should be fuzzy convex, that is$\tilde{\psi }\left(\left(1-\phi \right)x+\phi \omega \right)\ge min\left(\tilde{\psi }\left(x\right), \tilde{\psi }\left(\omega \right)\right),$for all$x,\omega \in ℝ$and$\phi \in \left[0,1\right]$

(4) 

$\tilde{\psi }$should be compactly supported, that is$cl\left\{u\in ℝ| \tilde{\psi }\left(\varkappa \right)>0\right\}$is compact.

We appoint${\mathbb{F}}_I$to denote the set ofall fuzzy intervals or fuzzy numbers of$ℝ$.

Definition 2

([40,41]). Given $\tilde{\psi }\in {\mathbb{F}}_I$, the level sets or cut sets are given by ${\left[\tilde{\psi }\right]}^{\lambda }=\left\{\varkappa \in ℝ| \tilde{\psi }\left(\varkappa \right)>\lambda \right\}$ for all $\lambda \in \left[0, 1\right]$ and by ${\left[\tilde{\psi }\right]}^{\lambda }=\left\{\varkappa \in ℝ| \tilde{\psi }\left(\varkappa \right)>0\right\}$. These sets are known as $\lambda$-level sets or $\lambda$-cut sets of $\tilde{\psi }$.

Proposition 1

([39]). Let $\tilde{\psi },\tilde{\varpi }\in {\mathbb{F}}_I$. Then relation “$\preccurlyeq$” given on ${\mathbb{F}}_I$ by $\tilde{\psi }\preccurlyeq \tilde{\varpi }$ when and only when, ${\left[\tilde{\psi }\right]}^{\lambda }{\le }_I{\left[\tilde{\varpi }\right]}^{\lambda }$, for every $\lambda \in \left[0, 1\right],$ it is left and right order relation.

Remember the approaching notions, which are offered in literature. If $\tilde{\psi },\tilde{\varpi }\in {\mathbb{F}}_I$ and $\lambda \in ℝ$, then, for every $\lambda \in \left[0, 1\right],$ the arithmetic operations are defined by

$${\left[\tilde{\psi }\widetilde{+}\tilde{\varpi }\right]}^{\lambda }={\left[\tilde{\psi }\right]}^{\lambda }+{\left[\tilde{\varpi }\right]}^{\lambda },$$

(9)

$${\left[\tilde{\psi }\tilde{\times }\tilde{\varpi }\right]}^{\lambda }={\left[\tilde{\psi }\right]}^{\lambda }\times {\left[ \tilde{\varpi }\right]}^{\lambda },$$

(10)

$${\left[\lambda ·\tilde{\psi }\right]}^{\lambda }=\lambda ·{\left[\tilde{\psi }\right]}^{\lambda }$$

(11)

These operations follow directly from the Equations (2)–(5), respectively.

Theorem 1

([40]). The space ${\mathbb{F}}_I$ dealing with a supremum metric, i.e., for $\tilde{\psi }, \tilde{\varpi }\in {\mathbb{F}}_I$

$$d_{\infty }\left(\tilde{\psi }, \tilde{\varpi }\right)=\underset{0\le \lambda \le 1}{\sup }{d}_H\left({\left[\tilde{\psi }\right]}^{\lambda }, {\left[\tilde{\varpi }\right]}^{\lambda }\right),$$

(12)

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

Definition 3

([40]). The F∙I-V∙F $\tilde{\mathfrak{S}}:\left[u,\nu \right]\to {\mathbb{F}}_I$ is said to be convex F∙I-V∙F on$\left[u,\nu \right]$ if

$$\tilde{\mathfrak{S}}\left(\sigma x+\left(1-\sigma \right)\omega \right)\preccurlyeq \sigma \tilde{\mathfrak{S}}\left(x\right) \widetilde{+} \left(1-\sigma \right)\tilde{\mathfrak{S}}\left(\omega \right),$$

(13)

for all$x, \omega \in \left[u,\nu \right], \sigma \in \left[0, 1\right],$ where $\mathfrak{S}\left(x\right)\succcurlyeq \tilde{0}$. If $\tilde{\mathfrak{S}}$ is concave F∙I-V∙F on $\left[u,\nu \right]$, then inequality (14) is reversed.

Definition 4

([37]). Let $h_1, h_2:\left[0, 1] \subseteq [u,\nu \right]\to {ℝ}^{+}$ such that $h_1, h_2\not\equiv 0$. Then, F∙I-V∙F $\tilde{\mathfrak{S}}:\left[u,\nu \right]\to {\mathbb{F}}_I$ is said to be $\left(h_1, h_2\right)$-preinvex F∙I-V∙F on $\left[u,\nu \right]$ if

$$\tilde{\mathfrak{S}}\left(x+\left(1-\sigma \right)\phi \left(\omega ,x\right)\right)\preccurlyeq {\eta }_1\left(\sigma \right){\eta }_2\left(1-\sigma \right)\tilde{\mathfrak{S}}\left(x\right)\widetilde{+}{\eta }_1\left(1-\sigma \right){\eta }_2\left(\sigma \right)\tilde{\mathfrak{S}}\left(\omega \right),$$

(14)

for all$x, \omega \in \left[u,\nu \right], \sigma \in \left[0, 1\right],$ where $\tilde{\mathfrak{S}}\left(x\right)\succcurlyeq \tilde{0}$ and $\phi :\left[u,\nu \right]\times \left[u,\nu \right]\to \left[u,\nu \right]$. If $\tilde{\mathfrak{S}}$ is $\left({\eta }_1, {\eta }_2\right)$-concave on $\left[u,\nu \right]$, then inequality (15) is reversed.

Remark 2

([37]). If ${\eta }_2\left(\sigma \right)\equiv 1,$ then $\left({\eta }_1, {\eta }_2\right)$-preinvex F∙I-V∙F becomes ${\eta }_1$-preinvex F∙I-V∙F, that is

$$\tilde{\mathfrak{S}}\left(x+\left(1-\sigma \right)\phi \left(\omega ,x\right)\right)\preccurlyeq {\eta }_1\left(\sigma \right)\tilde{\mathfrak{S}}\left(x\right)\widetilde{+}{\eta }_1\left(1-\sigma \right)\tilde{\mathfrak{S}}\left(\omega \right), \forall x, \omega \in \left[u,\nu \right], \sigma \in \left[0, 1\right].$$

(15)

If${\eta }_1\left(\sigma \right)=\sigma ,{\eta }_2\left(\sigma \right)\equiv 1,$then$\left({\eta }_1, {\eta }_2\right)$-preinvex F∙I-V∙F becomes preinvex F∙I-V∙F, that is

$$\tilde{\mathfrak{S}}\left(x+\left(1-\sigma \right)\phi \left(\omega ,x\right)\right)\preccurlyeq \sigma \tilde{\mathfrak{S}}\left(x\right)\widetilde{+}\left(1-\sigma \right)\tilde{\mathfrak{S}}\left(\omega \right), \forall x, \omega \in \left[u,\nu \right], \sigma \in \left[0, 1\right].$$

(16)

If${\eta }_1\left(\sigma \right)={\eta }_2\left(\sigma \right)\equiv 1,$then$\left({\eta }_1, {\eta }_2\right)$-preinvex F∙I-V∙F becomes$P$F∙I-V∙F, that is

$$\tilde{\mathfrak{S}}\left(x+\left(1-\sigma \right)\phi \left(\omega ,x\right)\right)\preccurlyeq \tilde{\mathfrak{S}}\left(x\right)\widetilde{+}\tilde{\mathfrak{S}}\left(\omega \right), \forall x, \omega \in \left[u,\nu \right], \sigma \in \left[0, 1\right].$$

(17)

Condition 1

(see [46]). Let $K$ be an invex set with respect to $\theta .$ For any $x, \omega \in K$ and $\xi \in \left[0, 1\right]$,

$$\theta \left(x,x+\xi \theta \left(\omega ,x\right)\right)=-\xi \theta \left(\omega ,x\right),$$

$$\theta \left(\omega ,x+\xi \theta \left(\omega ,x\right)\right)=\left(1-\xi \right)\theta \left(\omega ,x\right).$$

Clearly for$\xi$= 0, we have$\theta \left(\omega ,x\right)$= 0 if and only if,$\omega =x$, for all$x, \omega \in K$. For the applications of Condition 1, see [46,47,48].

Theorem 2

([19]). If $\mathfrak{S}:\left[u,\nu \right]\subset ℝ\to {ℝ}_I$ is an I-V∙F given by $\left(x\right)$ $\left[{\mathfrak{S}}_{*}\left(x\right), {\mathfrak{S}}^{*}\left(x\right)\right]$, then $\mathfrak{S}$ is Riemann integrable over $\left[u,\nu \right]$ if and only if, ${\mathfrak{S}}_{*}$ and ${\mathfrak{S}}^{*}$ both are Riemann integrable over $\left[u,\nu \right]$ such that

$$\left(IR\right){\displaystyle {\int }_u^{\nu }}\mathfrak{S}\left(x\right)dx=[\left(R\right){\displaystyle {\int }_u^{\nu }}{\mathfrak{S}}_{*}\left(x\right)dx, \left(R\right){\displaystyle {\int }_u^{\nu }}{\mathfrak{S}}^{*}\left(x\right)dx]$$

(18)

The collection of all Riemann integrable real-valued functions and Riemann integrable I-V∙F is denoted by${ℛ}_{\left[u, \nu \right]}$and$\mathfrak{T}{ℛ}_{\left[u, \nu \right]},$respectively.

Definition 5

([45]). Let $\mathfrak{S}:\left[\tau , \varsigma \right]\subset ℝ\to {\mathbb{F}}_I$ is fuzzy-number valued mapping. The fuzzy Riemann integral ($\left(FR\right)$-integral) of $\mathfrak{S}$ over $\left[\tau , \varsigma \right],$ denoted by $\left(FR\right){\displaystyle {\int }_{\tau }^{\varsigma }}\mathfrak{S}\left(\varkappa \right)d\varkappa$, is defined level-wise by

$${\left[\left(FR\right){\displaystyle {\int }_{\tau }^{\varsigma }}\mathfrak{S}\left(\varkappa \right)d\varkappa \right]}^{\begin{array}{c} \\ \lambda \end{array}}=\left(IR\right){\displaystyle {\int }_{\tau }^{\varsigma }}{\mathfrak{S}}_{\lambda }\left(\varkappa \right)d\varkappa =\left\{{\displaystyle {\int }_{\tau }^{\varsigma }}\mathfrak{S}\left(\varkappa ,\lambda \right)d\varkappa :\mathfrak{S}\left(\varkappa ,\lambda \right)\in S\left({\mathfrak{S}}_{\lambda }\right)\right\},$$

(19)

where $S\left({\mathfrak{S}}_{\lambda }\right)=\left\{\mathfrak{S}\left(.,\lambda \right)\to ℝ:\mathfrak{S}\left(.,\lambda \right) is integrable and \mathfrak{S}\left(\varkappa ,\lambda \right)={\mathfrak{S}}_{\lambda }\left(\varkappa \right)\right\},$for every $\lambda \in \left[0, 1\right]$. $\mathfrak{S}$ is $\left(FR\right)$-integrable over $\left[\tau , \varsigma \right]$ if $\left(FR\right){\displaystyle {\int }_{\tau }^{\varsigma }}\mathfrak{S}\left(\varkappa \right)d\varkappa \in {ℝ}_I.$

Note that, the Theorem 2 is also true for interval double integrals. The collection of all double integrable I-V∙F is denoted $\mathfrak{T}{\mathfrak{O}}_{\Delta },$ respectively.

Theorem 3

([32]). Let $\Delta =\left[a, b\right]\times \left[u, \nu \right]$. If $\mathfrak{S}:\Delta \to {ℝ}_I$ is $ID$-integrable on $\Delta$, then we have

$$\left(ID\right){\displaystyle {\int }_a^b}{\displaystyle {\int }_u^{\nu }}\mathfrak{S}\left(x,\omega \right)d\omega dx=\left(IR\right){\displaystyle {\int }_a^b}\left(IR\right){\displaystyle {\int }_u^{\nu }}\mathfrak{S}\left(x,\omega \right)d\omega dx$$

(20)

Definition 6

([38]). A fuzzy-interval-valued map$\tilde{\mathfrak{S}}:\Delta =\left[a, b\right]\times \left[u, \nu \right]\to {\mathbb{F}}_I$ is called F∙I-V∙F on coordinates. Then, from $\lambda$-levels, we get the collection of I-V∙Fs ${\mathfrak{S}}_{\lambda }:\Delta \subset {ℝ}^2\to {ℝ}_I$ on coordinates given by ${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$ for all $\left(x,\omega \right)\in \Delta$, where ${\mathfrak{S}}_{*}\left(.,\lambda \right), {\mathfrak{S}}^{*}\left(.,\lambda \right):\left(x,\omega \right)\to ℝ$ are called lower and upper functions of ${\mathfrak{S}}_{\lambda }$.

Definition 7

([38]). Let $\tilde{\mathfrak{S}}:\Delta =\left[a, b\right]\times \left[u, \nu \right]\subset {ℝ}^2\to {\mathbb{F}}_I$ be a coordinated F∙I-V∙F. Then, $\tilde{\mathfrak{S}}\left(x,\omega \right)$ is said to be continuous at $\left(x,\omega \right)\in \Delta =\left[a, b\right]\times \left[u, \nu \right],$ if for each $\lambda \in \left[0, 1\right],$ both end point functions ${\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right)$ and ${\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)$ are continuous at $\left(x,\omega \right)\in \Delta .$

Definition 8

([38]). Let $\tilde{\mathfrak{S}}:\Delta =\left[a, b\right]\times \left[u, \nu \right]\subset {ℝ}^2\to {\mathbb{F}}_I$ be a F∙I-V∙F on coordinates. Then, fuzzy double integral of $\tilde{\mathfrak{S}}$ over $\Delta =\left[a, b\right]\times \left[u, \nu \right],$ denoted by $\left(FD\right){{\displaystyle \int }}_a^b{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx$, it is defined level-wise by

$${\left[\left(FD\right){\displaystyle {\int }_a^b}{\displaystyle {\int }_u^{\nu }}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\right]}^{\lambda }=\left(ID\right){\displaystyle {\int }_a^b}{\displaystyle {\int }_u^{\nu }}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx\left(IR\right){\displaystyle {\int }_a^b}=\left(IR\right){\displaystyle {\int }_u^{\nu }}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx,$$

(21)

for all $\lambda \in \left[0, 1\right],$ $\tilde{\mathfrak{S}}$ is $FD$-integrable over $\Delta$ if $\left(FD\right){\displaystyle {\int }_a^b}{\displaystyle {\int }_u^{\nu }}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\in {\mathbb{F}}_I.$ Note that, if end-point functions are Lebesgue-integrable, then $\tilde{\mathfrak{S}}$ is a fuzzy double Aumann-integrable function over $\Delta$.

Theorem 4

([38]). Let $\tilde{\mathfrak{S}}:\Delta \subset {ℝ}^2\to {\mathbb{F}}_I$ be a F∙I-V∙F on coordinates. Then, from $\lambda$-levels, we get the collection of I-V∙Fs ${\mathfrak{S}}_{\lambda }:\Delta \subset {ℝ}^2\to {ℝ}_I$ are given by ${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$ for all $\left(x,\omega \right)\in \Delta =\left[a, b\right]\times \left[u, \nu \right]$ and for all $\lambda \in \left[0, 1\right].$ Then, $\tilde{\mathfrak{S}}$ is $FD$-integrable over $\Delta$ if and only if, ${\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right)$ and ${\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)$ both are $D$-integrable over $\Delta .$ Moreover, if $\tilde{\mathfrak{S}}$ is $FD$-integrable over $\Delta ,$ then

$$\begin{array}{c}{\left[\left(FD\right){\displaystyle {\int }_a^b}{\displaystyle {\int }_u^{\nu }}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\right]}^{\begin{array}{c} \\ \lambda \end{array}}={\left[\left(FR\right){\displaystyle {\int }_a^b}\left(FR\right){\displaystyle {\int }_u^{\nu }}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\right]}^{\lambda }=\hfill \\ \left(IR\right){\displaystyle {\int }_a^b}\left(IR\right){\displaystyle {\int }_u^{\nu }}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx=\left(ID\right){\displaystyle {\int }_a^b}{\displaystyle {\int }_u^{\nu }}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx,\hfill \end{array}$$

(22)

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

The family of all $FD$-integrable F∙I-V∙Fs over coordinates is denoted by $ℱ{\mathfrak{O}}_{\Delta }$ for all $\lambda \in \left[0, 1\right].$

Theorem 5

([38]). Let $\varrho \in ℝ$, and $\tilde{\mathfrak{S}}, \tilde{\mathcal{J}}\in ℱ{\mathfrak{O}}_{\Delta }$. Then,

(1) 

$\varrho \tilde{\mathfrak{S}}\in ℱ{\mathfrak{O}}_{\Delta }$and

$$\left(FD\right){\displaystyle {\iint }_{\Delta }^{ }}\varrho \tilde{\mathfrak{S}}dA=\varrho \left(FD\right){\displaystyle {\iint }_{\Delta }^{ }}\tilde{\mathfrak{S}}dA$$

(23)

(2) 

$\tilde{\mathfrak{S}}\widetilde{+} \tilde{\mathcal{J}}\in ℱ{\mathfrak{O}}_{\Delta }$, and

$$\left(FD\right){\displaystyle {\iint }_{\Delta }^{ }}\left(\tilde{\mathfrak{S}}\widetilde{+}\tilde{\mathcal{J}}\right)dA=\left(FD\right){\displaystyle {\iint }_{\Delta }^{ }}\tilde{\mathfrak{S}}dA\widetilde{+}\left(FD\right){\displaystyle {\iint }_{\Delta }^{ }}\tilde{\mathcal{J}}dA$$

(24)

(3) 

suppose that${\Delta }_1$and${\Delta }_2$are non-overlapping, then

$$\left(FD\right){\displaystyle {\iint }_{{\Delta }_1{{\displaystyle \cup }}^{ }{\Delta }_2}^{ }}\tilde{\mathfrak{S}}dA=\left(FD\right){\displaystyle {\iint }_{{\Delta }_1}^{ }}\tilde{\mathfrak{S}}dA+\left(FD\right){\displaystyle {\iint }_{{\Delta }_2}^{ }}\tilde{\mathfrak{S}}dA$$

(25)

Theorem 6

([37]). Let $\tilde{\mathfrak{S}},\tilde{\mathcal{J}} :\left[u, u+\phi \left(\nu ,u\right)\right]\to {\mathbb{F}}_I$ be two $\left({\eta }_1, {\eta }_2\right)$-preinvex F∙I-V∙Fs with ${\eta }_1, {\eta }_2:\left[0, 1\right]\to {ℝ}^{+}$ and ${\eta }_1\left(\frac{1}{2}\right){\eta }_2\left(\frac{1}{2}\right)\ne 0.$ Then, from $\lambda$-levels, we get the collection of I-V∙Fs ${\mathfrak{S}}_{\lambda }, {\mathcal{J}}_{\lambda }:\left[u, u+\phi \left(\nu ,u\right)\right]\subset ℝ\to {ℝ}_I^{+}$ are given by ${\mathfrak{S}}_{\lambda }\left(x\right)=\left[{\mathfrak{S}}_{*}\left(x,\lambda \right), {\mathfrak{S}}^{*}\left(x,\lambda \right)\right]$ and ${\mathcal{J}}_{\lambda }\left(x\right)=\left[{\mathcal{J}}_{*}\left(x,\lambda \right), {\mathcal{J}}^{*}\left(x,\lambda \right)\right]$ for all $x\in \left[u, u+\phi \left(\nu ,u\right)\right]$ and for all $\lambda \in \left[0, 1\right]$. If $\tilde{\mathfrak{S}}\tilde{\times }\tilde{\mathcal{J}}$ is fuzzy Riemann integrable, then

$$\begin{array}{c}\frac{1}{\phi \left(\nu ,u\right)} \left(FR\right){{\displaystyle \int }}_u^{u+\phi \left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x\right)\tilde{\times }\tilde{\mathcal{J}}\left(x\right)dx\hfill \\ \preccurlyeq \tilde{\beta }\left(u,\nu \right){{\displaystyle \int }}_0^1{\left[{\eta }_1\left(\sigma \right){\eta }_2\left(1-\sigma \right)\right]}^{2}d\sigma \widetilde{+}\tilde{\gamma }\left(u,\nu \right){{\displaystyle \int }}_0^1{\eta }_1\left(\sigma \right){\eta }_2\left(\sigma \right){\eta }_1\left(1-\sigma \right){\eta }_2\left(1-\sigma \right)d\sigma ,\hfill \end{array}$$

(26)

and,

$$\begin{array}{c}\frac{1}{2{\left[{\eta }_1\left(\frac{1}{2}\right){\eta }_2\left(\frac{1}{2}\right)\right]}^2} \tilde{\mathfrak{S}}\left(\frac{2u+\phi \left(\nu ,u\right)}{2}\right)\tilde{\mathcal{J}}\left(\frac{2u+\phi \left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \frac{1}{\phi \left(\nu ,u\right)} \left(FR\right){{\displaystyle \int }}_u^{u+\phi \left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x\right)\tilde{\mathcal{J}}\left(x\right)dx \widetilde{+} \tilde{\gamma }\left(u,\nu \right){{\displaystyle \int }}_0^1{\left[{\eta }_1\left(\sigma \right){\eta }_2\left(1-\sigma \right)\right]}^{2}d\sigma \hfill \\ \widetilde{+}\tilde{\beta }\left(u,\nu \right){{\displaystyle \int }}_0^1{\eta }_1\left(\sigma \right){\eta }_2\left(\sigma \right){\eta }_1\left(1-\sigma \right){\eta }_2\left(1-\sigma \right)d\sigma \hfill \end{array}$$

(27)

where$\tilde{\beta }\left(u,\nu \right)=\tilde{\mathfrak{S}}\left(u\right)\tilde{\times }\tilde{\mathcal{J}}\left(u\right) \widetilde{+} \tilde{\mathfrak{S}}\left(\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(\nu \right),$ $\tilde{\gamma }\left(u,\nu \right)=\tilde{\mathfrak{S}}\left(u\right)\tilde{\times }\tilde{\mathcal{J}}\left(\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(u\right),$ and ${\beta }_{\lambda }\left(u,\nu \right)=\left[{\beta }_{*}\left(\left(u,\nu \right), \lambda \right), {\beta }^{*}\left(\left(u,\nu \right), \lambda \right)\right]$ and ${\gamma }_{\lambda }\left(u,\nu \right)=\left[{\gamma }_{*}\left(\left(u,\nu \right), \lambda \right), {\gamma }^{*}\left(\left(u,\nu \right), \lambda \right)\right].$

Remark 3.

If${\eta }_1\left(\sigma \right)=\sigma$and${\eta }_2\left(\sigma \right)\equiv 1,$then (27) reduces to the result for preinvex F∙I-V∙F:

$$\frac{1}{\phi \left(\nu ,u\right)} \left(FR\right){{\displaystyle \int }}_u^{u+\phi \left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x\right)\tilde{\times }\tilde{\mathcal{J}}\left(x\right)dx\preccurlyeq \frac{1}{3}\tilde{\beta }\left(u,\nu \right)\widetilde{+} \frac{1}{6}\tilde{\gamma }\left(u,\nu \right)$$

(28)

If${\eta }_1\left(\sigma \right)=\sigma$and${\eta }_2\left(\sigma \right)\equiv 1,$then (28) reduces to the result for preinvex F∙I-V∙F:

$$\begin{array}{c}2 \tilde{\mathfrak{S}}\left(\frac{2u+\phi \left(\nu ,u\right)}{2}\right)\tilde{\times }\tilde{\mathcal{J}}\left(\frac{2u+\phi \left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \frac{1}{\phi \left(\nu ,u\right)} \left(FR\right){{\displaystyle \int }}_u^{u+\phi \left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x\right)\tilde{\times }\tilde{\mathcal{J}}\left(x\right)dx \widetilde{+}\frac{1}{6}\tilde{\beta }\left(u,\nu \right)\widetilde{+} \frac{1}{3}\tilde{\gamma }\left(u,\nu \right).\hfill \end{array}$$

(29)

Theorem 7

([37]). Let $\tilde{\mathfrak{S}}:\left[u, u+\phi \left(\nu ,u\right)\right]\to {\mathbb{F}}_I$ be a preinvex F∙I-V∙F with $u<u+\phi \left(\nu ,u\right)$. Then, from $\lambda$-levels, we get the collection of I-V∙Fs ${\mathfrak{S}}_{\lambda }:\left[u, u+\phi \left(\nu ,u\right)\right]\subset ℝ\to {ℝ}_I^{+}$ are given by ${\mathfrak{S}}_{\lambda }\left(x\right)=\left[{\mathfrak{S}}_{*}\left(x,\lambda \right), {\mathfrak{S}}^{*}\left(x,\lambda \right)\right]$ for all $x\in \left[u, u+\phi \left(\nu ,u\right)\right]$ and for all $\lambda \in \left[0, 1\right]$, and Condition 1 for $\phi$ holds. If $\tilde{\mathfrak{S}}\in \mathfrak{S}{ℛ}_{\left(\left[u, u+\phi \left(\nu ,u\right)\right], \lambda \right)}$ and $\psi :\left[u, \nu \right]\to ℝ, \psi \left(x\right)\ge 0,$ symmetric with respect to $\frac{2 u+\phi \left(\nu ,u\right)}{2},$ and ${{\displaystyle \int }}_u^{u, u+\phi \left(\nu ,u\right)}\psi \left(x\right)dx>0$, then

$$\tilde{\mathfrak{S}}\left(\frac{2u+\phi \left(\nu ,u\right)}{2}\right)\preccurlyeq \frac{1}{{\displaystyle {\int }_u^{u+\phi \left(\nu ,u\right)}}\psi \left(x\right)dx} \left(FR\right){\displaystyle {\int }_u^{u+\phi \left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x\right)\psi \left(x\right)dx\preccurlyeq \frac{\tilde{\mathfrak{S}}\left(u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(\nu \right)}{2}.$$

(30)

If$\mathfrak{S}$is preincave F∙I-V∙F, then inequality (31) is reversed.

Note that if$\psi \left(x\right)=1$, then we acquire the following inequality:

$$\tilde{\mathfrak{S}}\left(\frac{2u+\phi \left(\nu ,u\right)}{2}\right)\preccurlyeq \frac{1}{\left(\nu ,u\right)} \left(FR\right){{\displaystyle \int }}_u^{u+\phi \left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x\right)dx\preccurlyeq \frac{\tilde{\mathfrak{S}}\left(u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(\nu \right)}{2}.$$

(31)

Coordinated preinvex fuzzy-interval-valued functions

Definition 9.

The F∙I-V∙F$\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_I$is said to be a coordinated preinvex F∙I-V∙F on$\Delta$if

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\hfill \\ \preccurlyeq \sigma s\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\sigma \left(1-s\right)\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\left(1-\sigma \right)s\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\left(1-\sigma \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(b,\nu \right),\hfill \end{array}$$

(32)

for all$\left(a, b\right), \left(u,\nu \right)\in \Delta ,$and$\sigma ,s\in \left[0, 1\right],$where$\tilde{\mathfrak{S}}\left(x\right)\succcurlyeq \tilde{0}.$If inequality (33) is reversed, then$\tilde{\mathfrak{S}}$is called coordinated concave F∙I-V∙F on$\Delta$.

The proof that Lemma 1 is straightforward will be omitted here.

Lemma 1.

Let$\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_I$be a coordinated F∙I-V∙F on$\Delta$. Then,$\tilde{\mathfrak{S}}$is a coordinated preinvex F∙I-V∙F on$\Delta ,$if and only if there exist two coordinated preinvex F∙I-V∙Fs${\widetilde{ \mathfrak{S}}}_x:\left[u,\nu \right]\to {\mathbb{F}}_I$,${\tilde{\mathfrak{S}}}_x\left(w\right)=\tilde{\mathfrak{S}}\left(x,w\right)$and${\tilde{\mathfrak{S}}}_{\omega }:\left[a,b\right]\to {\mathbb{F}}_I$,${\tilde{\mathfrak{S}}}_{\omega }\left(q\right)=\tilde{\mathfrak{S}}\left(q,\omega \right)$.

Proof. 

From the Definition 9 of coordinated preinvex F∙I-V∙F, it can be easily proved. □

From Lemma 1, we can easily note each preinvex F∙I-V∙F is a coordinated preinvex F∙I-V∙F. However, the converse is not true, see Example 1.

Theorem 8.

Let$\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_I$be a F∙I-V∙F on$\Delta$. Then, from$\lambda$-levels, we get the collection of I-V∙Fs${\mathfrak{S}}_{\lambda }:\Delta \to {ℝ}_I^{+}\subset {ℝ}_I$which are given by

$${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right],$$

(33)

For all$\left(x,\omega \right)\in \Delta$and for all$\lambda \in \left[0, 1\right]$. Then,$\tilde{\mathfrak{S}}$is coordinated preinvex F∙I-V∙F on$\Delta ,$if and only if, for all$\lambda \in \left[0, 1\right],$ ${\mathfrak{S}}_{*}\left(\left(x,\omega \right), \lambda \right)$and${\mathfrak{S}}^{*}\left(\left(x,\omega \right), \lambda \right)$are coordinated preinvex functions$.$

Proof. 

Assume that for each $\lambda \in \left[0, 1\right],$${\mathfrak{S}}_{*}\left(x, \lambda \right)$ and ${\mathfrak{S}}^{*}\left(x, \lambda \right)$ are coordinated preinvex on $\Delta .$ Then, from (33), for all $\left(a, b\right), \left(u,\nu \right)\in \Delta , \sigma$ and $s\in \left[0, 1\right]$, we have

$$\begin{array}{c}{\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\hfill \\ \le \sigma s{\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right)+t\left(1-s\right){\mathfrak{S}}_{*}\left(\left(a,\nu \right), \lambda \right)\hfill \\ +s\left(1-t\right){\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right)+\left(1-\sigma \right)\left(1-s\right){\mathfrak{S}}_{*}\left(\left(a,\nu \right), \lambda \right)\hfill \end{array}$$

and

$$\begin{array}{c}{\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\hfill \\ \le \sigma s{\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right)+t\left(1-s\right){\mathfrak{S}}^{*}\left(\left(a,\nu \right), \lambda \right)\hfill \\ +s\left(1-t\right){\mathfrak{S}}^{*}\left(\left(a,u\right), \lambda \right)+\left(1-\sigma \right)\left(1-s\right){\mathfrak{S}}^{*}\left(\left(a,\nu \right), \lambda \right)\hfill \end{array}$$

Then, by (33), (10), and (12), we obtain

$$\begin{array}{c}{\mathfrak{S}}_{\lambda }\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\right) \hfill \\ =\left[{\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\right]\hfill \\ {\le }_I\sigma s\left[{\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,u\right), \lambda \right)\right]+t\left(1-s\right)[{\mathfrak{S}}_{*}\left(\left(a,\nu \right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,\nu \right), \lambda \right)]\hfill \\ +s\left(1-\sigma \right)\left[{\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,u\right), \lambda \right)\right]+\left(1-\sigma \right)\left(1-s\right)\left[{\mathfrak{S}}_{*}\left(\left(a,\nu \right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,\nu \right), \lambda \right)\right]\hfill \end{array}$$

That is

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\hfill \\ \preccurlyeq \sigma s\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\sigma \left(1-s\right)\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\left(1-\sigma \right)s\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\left(1-\sigma \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(b,\nu \right),\hfill \end{array}$$

hence, $\mathfrak{S}$ is a coordinated preinvex F∙I-V∙F on $\Delta$

Conversely, let $\mathfrak{S}$ be a coordinated preinvex F∙I-V∙F on $\Delta .$ Then, for all $\left(a, b\right), \left(u,\nu \right)\in \Delta , \sigma$ and $s\in \left[0, 1\right],$ we have

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\hfill \\ \preccurlyeq \sigma s\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\sigma \left(1-s\right)\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\left(1-\sigma \right)s\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\left(1-\sigma \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(b,\nu \right)\hfill \end{array}$$

Therefore, again from (34), for each $\lambda \in \left[0, 1\right]$, we have

$$\begin{array}{c}{\mathfrak{S}}_{\lambda }\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\right)\hfill \\ =\left[{\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\right]\hfill \end{array}$$

Again, (10) and (12), we obtain

$$\begin{array}{c}\sigma s{\mathfrak{S}}_{\lambda }\left(a,u\right)+\sigma \left(1-s\right){\mathfrak{S}}_{\lambda }\left(a,\nu \right)+\left(1-\sigma \right)s{\mathfrak{S}}_{\lambda }\left(b,u\right)+\left(1-\sigma \right)\left(1-s\right){\mathfrak{S}}_{\lambda }\left(b,\nu \right)\hfill \\ =\sigma s\left[{\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,u\right), \lambda \right)\right]\hfill \\ +t\left(1-s\right)\left[{\mathfrak{S}}_{*}\left(\left(a,\nu \right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,\nu \right), \lambda \right)\right]\hfill \\ +s\left(1-\sigma \right)\left[{\mathfrak{S}}_{*}\left(\left(a,u\right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,u\right), \lambda \right)\right]\hfill \\ +\left(1-\sigma \right)\left(1-s\right)\left[{\mathfrak{S}}_{*}\left(\left(a,\nu \right), \lambda \right), {\mathfrak{S}}^{*}\left(\left(a,\nu \right), \lambda \right)\right],\hfill \end{array}$$

for all $x,\omega \in \Delta$ and $\sigma \in \left[0, 1\right].$ Then, by coordinated preinvexity of $\mathfrak{S}$, we have for all $x,\omega \in \Delta$ and $\sigma \in \left[0, 1\right]$ such that

$$\begin{array}{c}{\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\hfill \\ \le \sigma s{\mathfrak{S}}_{*}\left(a,u\right)+\sigma \left(1-s\right){\mathfrak{S}}_{*}\left(a,\nu \right)+\left(1-\sigma \right)s{\mathfrak{S}}_{*}\left(b,u\right)\hfill \\ +\left(1-\sigma \right)\left(1-s\right){\mathfrak{S}}_{*}\left(b,\nu \right),\hfill \end{array}$$

and

$$\begin{array}{c}{\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\hfill \\ \le \sigma s{\mathfrak{S}}^{*}\left(a,u\right)+\sigma \left(1-s\right){\mathfrak{S}}^{*}\left(a,\nu \right)+\left(1-\sigma \right)s{\mathfrak{S}}^{*}\left(b,u\right)\hfill \\ +\left(1-\sigma \right)\left(1-s\right){\mathfrak{S}}^{*}\left(b,\nu \right),\hfill \end{array}$$

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

Remark 4.

If one takes ${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$, then$\tilde{\mathfrak{S}}$is known as aconvex F∙I-V∙F on coordinates if$\tilde{\mathfrak{S}}$satisfies the following inequality:

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\sigma a+\left(1-\sigma \right)b, su+\left(1-s\right)\nu \right)\hfill \\ \preccurlyeq \sigma s\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\sigma \left(1-s\right)\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\left(1-\sigma \right)s\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\left(1-\sigma \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(b,\nu \right),\hfill \end{array}$$

(34)

which is valid defined by Khan et al. [38].

If one takes${\mathfrak{S}}_{*}\left(x,\omega \right)={\mathfrak{S}}^{*}\left(x,\omega \right)$with$\lambda =1$, then$\mathfrak{S}$is known as a preinvex function on coordinates if$\mathfrak{S}$satisfies the following inequality

$$\begin{array}{c}\mathfrak{S}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_1\left(\nu ,u\right)\right)\hfill \\ \le \sigma s\mathfrak{S}\left(a,u\right)+\sigma \left(1-s\right)\mathfrak{S}\left(a,\nu \right)+\left(1-\sigma \right)s\mathfrak{S}\left(b,u\right)\hfill \\ +\left(1-\sigma \right)\left(1-s\right)\mathfrak{S}\left(b,\nu \right),\hfill \end{array}$$

(35)

which is defined by Latif and Dragomir [44].

If one takes${\mathfrak{S}}_{*}\left(x,\omega \right)={\mathfrak{S}}^{*}\left(x,\omega \right)$with$\lambda =1$, then$\mathfrak{S}$is known as a convex function on coordinates if$\mathfrak{S}$satisfies the following inequality

$$\begin{array}{c}\mathfrak{S}\left(a\sigma a+\left(1-\sigma \right)b, su+\left(1-s\right)\nu \right)\hfill \\ \le \sigma s\mathfrak{S}\left(a,u\right)+\sigma \left(1-s\right)\mathfrak{S}\left(a,\nu \right)+\left(1-\sigma \right)s\mathfrak{S}\left(b,u\right)\hfill \\ +\left(1-\sigma \right)\left(1-s\right)\mathfrak{S}\left(b,\nu \right),\hfill \end{array}$$

(36)

is valid, then$\mathfrak{S}$is named as IVFon coordinates, which is defined by Dragomir [34].

Example 1.

We consider the F∙I-V∙Fs $\tilde{\mathfrak{S}}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_I$defined by,

$$\mathfrak{S}\left(x, \omega \right)\left(\sigma \right)=\left\{\begin{array}{cc}\frac{\sigma }{\sigma \omega }& \sigma \in \left[0, \sigma \omega \right]\\ \frac{2\sigma \omega -\sigma }{\sigma \omega }& \sigma \in \left(\sigma \omega , 2\sigma \omega \right]\\ 0& \mathrm{otherwise},\end{array}\right.$$

Then, for each$\lambda \in \left[0, 1\right],$we have${\mathfrak{S}}_{\lambda }\left(x\right)=\left[\lambda x\omega ,\left(2-\lambda \right)x\omega \right]$. End-point functions${\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right),$ ${\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)$are coordinated concave functions with respect to${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$for each$\lambda \in \left[0, 1\right]$. Hence,$\tilde{\mathfrak{S}}\left(x,\omega \right)$is a coordinated concave F∙I-V∙F.

From Example 1, it can be easily seen that each coordinated preinvex F∙I-V∙F is not a preinvex F∙I-V∙F.

Theorem 9.

Let$\Delta$be a coordinated preinvex set, and let$\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_I$be a F∙I-V∙F. Then, from$\lambda$-levels, we obtain the collection of I-V∙Fs${\mathfrak{S}}_{\lambda }:\Delta \to {ℝ}_I^{+}\subset {ℝ}_I$are given by

$${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right],$$

(37)

for all$\left(x,\omega \right)\in \Delta$and for all$\lambda \in \left[0, 1\right]$. Then,$\tilde{\mathfrak{S}}$is a coordinated preinvex F∙I-V∙F on$\Delta ,$if and only if, for all$\lambda \in \left[0, 1\right],$ ${\mathfrak{S}}_{*}\left(\left(x,\omega \right), \lambda \right)$and${\mathfrak{S}}^{*}\left(\left(x,\omega \right), \lambda \right)$are coordinated preinvex functions$.$

Proof. 

The proof of Theorem 9 is similar to that of Theorem 8. □

Example 2.

We consider the F∙I-V∙Fs $\tilde{\mathfrak{S}}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_I$defined by,

$$\tilde{\mathfrak{S}}\left(x\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma }{2\left(6-e^x\right)\left(6-e^{\omega }\right)}, \sigma \in \left[0, 2\left(6-e^x\right)\left(6-e^{\omega }\right)\right]\hfill \\ \frac{4\left(6-e^x\right)\left(6-e^{\omega }\right)-\sigma }{2\left(6-e^x\right)\left(6-e^{\omega }\right)}, \sigma \in \left(2\left(6-e^x\right)\left(6-e^{\omega }\right), 4\left(6-e^x\right)\left(6-e^{\omega }\right)\right]\hfill \\ 0, \mathrm{otherwise}.\hfill \end{array}\right.$$

Then, for each$\lambda \in \left[0, 1\right],$we have${\mathfrak{S}}_{\lambda }\left(x\right)=\left[2\lambda \left(6-e^x\right)\left(6-e^{\omega }\right),\left(4-2\lambda \right)\left(6-e^x\right)\left(6-e^{\omega }\right) \right]$. End-point functions${\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right),$ ${\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)$are coordinated preincave functions with respect to${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$for each$\lambda \in \left[0, 1\right]$. Hence,$\tilde{\mathfrak{S}}\left(x,\omega \right)$is a coordinated preincave F∙I-V∙F.

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.

3. Fuzzy-Interval Hermite-Hadamard Inequalities

In this section, we propose HH- and HH–Fejér inequalities for coordinated preinvex F∙I-V∙Fs, and verify with the help of some nontrivial example.

Theorem 10.

Let$\tilde{\mathfrak{S}}:\Delta =\left[a, a+{\phi }_1\left(b,a\right)\right]\times \left[u, u+{\phi }_2\left(\nu ,u\right)\right]\to {\mathbb{F}}_I$be a coordinated preinvex F∙I-V∙F on$\Delta$. Then, from$\lambda$-levels, we get the collection of I-V∙Fs${\mathfrak{S}}_{\lambda }:\Delta \to {ℝ}_I^{+}$are given by${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$for all$\left(x,\omega \right)\in \Delta$and for all$\lambda \in \left[0, 1\right]$, and Condition 1 for${\phi }_1$and${\phi }_2$holds. Then, the following inequality holds:

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\widetilde{+}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]\hfill \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\preccurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4}.\hfill \end{array}$$

(38)

If$\tilde{\mathfrak{S}}\left(x\right)$preincave F∙I-V∙F, then inequality (38) is reversed such that,

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \succcurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\widetilde{+}\frac{1}{{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \\ \succcurlyeq \frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\hfill \\ \succcurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\hfill \\ \succcurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4} \hfill \end{array}$$

(39)

Proof. 

Let $\tilde{\mathfrak{S}}:\left[a, a+{\phi }_1\left(b,a\right)\right]\to {\mathbb{F}}_I$ be a coordinated preinvex F∙I-V∙F. Then, by hypotheses, we have

$$\begin{array}{c}4\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \tilde{\mathfrak{S}}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b+\sigma {\phi }_1\left(b,a\right), \nu +s{\phi }_2\left(\nu ,u\right)\right).\hfill \end{array}$$

By using Theorem 10, for every $\lambda \in \left[0, 1\right]$, we have

$$\begin{array}{c}4{\mathfrak{S}}_{*}\left(\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right), \lambda \right)\hfill \\ \le {\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\hfill \\ +{\mathfrak{S}}_{*}\left(\left(b+\sigma {\phi }_1\left(b,a\right), \nu +s{\phi }_2\left(\nu ,u\right)\right), \lambda \right),\hfill \\ 4{\mathfrak{S}}^{*}\left(\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right), \lambda \right)\hfill \\ \le {\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\hfill \\ +{\mathfrak{S}}^{*}\left(\left(b+\sigma {\phi }_1\left(b,a\right), \nu +s{\phi }_2\left(\nu ,u\right)\right),\lambda \right).\hfill \end{array}$$

By using Lemma 1, we have

$$\begin{array}{c}2{\mathfrak{S}}_{*}\left(\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right), \lambda \right)\le {\mathfrak{S}}_{*}\left(\left(x, u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)+{\mathfrak{S}}_{*}\left(\left(x, \nu +s{\phi }_2\left(\nu ,u\right)\right), \lambda \right), \hfill \\ 2{\mathfrak{S}}^{*}\left(\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right), \lambda \right)\le {\mathfrak{S}}^{*}\left(\left(x, u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)+{\mathfrak{S}}^{*}\left(\left(x, \nu +s{\phi }_2\left(\nu ,u\right)\right),\lambda \right),\hfill \end{array}$$

(40)

and

$$\begin{array}{c}2{\mathfrak{S}}_{*}\left(\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right), \lambda \right)\le {\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right), \lambda \right)+{\mathfrak{S}}_{*}\left(\left(\nu +s{\phi }_2\left(\nu ,u\right), \omega \right), \lambda \right), \hfill \\ 2{\mathfrak{S}}^{*}\left(\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right), \lambda \right)\le {\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right), \lambda \right)+{\mathfrak{S}}^{*}\left(\left(\nu +s{\phi }_2\left(\nu ,u\right), \omega \right),\lambda \right).\hfill \end{array}$$

(41)

From (41) and (42), we have

$$\begin{array}{c}2\left[{\mathfrak{S}}_{*}\left(\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right), \lambda \right),{\mathfrak{S}}^{*}\left(\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right), \lambda \right)\right]\hfill \\ {\le }_I\left[{\mathfrak{S}}_{*}\left(\left(x, u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right),{\mathfrak{S}}^{*}\left(\left(x, u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right), \lambda \right)\right]\hfill \\ +\left[{\mathfrak{S}}_{*}\left(\left(x, \nu +s{\phi }_2\left(\nu ,u\right)\right), \lambda \right),{\mathfrak{S}}^{*}\left(\left(x, \nu +s{\phi }_2\left(\nu ,u\right)\right),\lambda \right)\right],\hfill \end{array}$$

and

$$\begin{array}{c}2\left[{\mathfrak{S}}_{*}\left(\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right), \lambda \right),{\mathfrak{S}}^{*}\left(\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right), \lambda \right)\right]\hfill \\ {\le }_I\left[{\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right), \lambda \right),{\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right), \lambda \right)\right]\hfill \\ +\left[{\mathfrak{S}}_{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right), \lambda \right),{\mathfrak{S}}^{*}\left(\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right),\lambda \right)\right],\hfill \end{array}$$

It follows that

$${\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I{\mathfrak{S}}_{\lambda }\left(x, u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)+{\mathfrak{S}}_{\lambda }\left(x, \nu +s{\phi }_2\left(\nu ,u\right)\right)$$

(42)

and

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right){\le }_I{\mathfrak{S}}_{\lambda }\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), \omega \right)+{\mathfrak{S}}_{\lambda }\left(b+\sigma {\phi }_1\left(b,a\right), \omega \right)$$

(43)

Since ${\mathfrak{S}}_{\lambda }\left(x,.\right)$ And ${\mathfrak{S}}_{\lambda }\left(.,\omega \right)$, both are coordinated preinvex-I-V∙Fs, then from inequality (32), for every $\lambda \in \left[0, 1\right]$, inequality (42) and (43), we have

$${\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega {\le }_I \frac{\begin{array}{c} \\ {\mathfrak{S}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\end{array}}{2}.$$

(44)

and

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right){\le }_I\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)dx{\le }_I \frac{{\mathfrak{S}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)}{2}.$$

(45)

Dividing double inequality (44) by ${\phi }_1\left(b,a\right)$, and integrating with respect to $x$ over $\left[a, a+{\phi }_1\left(b,a\right)\right],$ we have

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ {\le }_I \frac{\begin{array}{c} \\ 1\end{array}}{2{\phi }_1\left(b,a\right)}\left[{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)dx+{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)dx.\right]\hfill \end{array}$$

(46)

Similarly, dividing double inequality (46) by ${\phi }_2\left(\nu ,u\right)$, and integrating with respect to $x$ over $\left[u, u+{\phi }_2\left(\nu ,u\right)\right],$ we have

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(\frac{2a + {\phi }_1\left(b,a\right)}{2},\omega \right)d\omega {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ {\le }_I \frac{\begin{array}{c} \\ 1\end{array}}{2{\phi }_2\left(\nu ,u\right)}\left[{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)d\omega +{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)d\omega \right].\hfill \end{array}$$

(47)

By adding (46) and (47), we have

$$\begin{array}{c}\frac{1}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u + {\phi }_2\left(\nu ,u\right)}{2}\right)dx+\frac{1}{{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ {\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)dx+{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)d\omega +{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)d\omega \right]\hfill \end{array}$$

(48)

Since $\mathfrak{S}$ is F∙I-V∙F, then inequality (48), we have

$$\begin{array}{c}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\widetilde{+}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]\widetilde{+}\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\hfill \end{array}$$

(49)

From the left side of inequality (32), for each $\lambda \in \left[0, 1\right]$, we have

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx,$$

(50)

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega .$$

(51)

Taking addition of inequality (50) with inequality (51), we have

$$\begin{array}{c}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx+\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \end{array}$$

(52)

Since $\tilde{\mathfrak{S}}$ is a F∙I-V∙F, then it follows that

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx \widetilde{+}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \end{array}$$

(53)

Now from right side of inequality (32), for every $\lambda \in \left[0, 1\right]$, we have

$$\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)dx{\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)}{2}$$

(54)

$$\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)dx{\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)}{2}$$

(55)

$$\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)d\omega {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,u\right)}{2}$$

(56)

$$\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)d\omega {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)}{2}$$

(57)

By adding inequalities (54)–(57), we have

$$\begin{array}{c}\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)dx+{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)d\omega +{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)d\omega \right]\hfill \\ {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)}{4}\hfill \end{array}$$

Since $\mathfrak{S}$ is a F∙I-V∙F, then it follows that

$$\begin{array}{c}\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4}\hfill \end{array}$$

(58)

By combining inequalities (50), (53), and (58), we get the desired result. □

Remark 5.

If one takes${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$, then from (39), we acquire the following inequality, see [38]:

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{a+b}{2}, \frac{u+\nu }{2}\right)\preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{b-a}{{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(x,\frac{u+\nu }{2}\right)dx\widetilde{+}\frac{1}{\nu -u}{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(\frac{a+b}{2},\omega \right)d\omega \right]\preccurlyeq \frac{1}{\left(b-a\right)\left(\nu -u\right)} {{\displaystyle \int }}_a^b{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(x,\omega \right)d\omega dx\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4\left(b-a\right)}\left[{{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+}{{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]\widetilde{+}\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\nu -u\right)}\left[{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+}{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\preccurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4}.\hfill \end{array}$$

(59)

If${\mathfrak{S}}_{*}\left(x,\omega \right)={\mathfrak{S}}^{*}\left(x,\omega \right)$with$\lambda =1$, then from (39), we acquire the following inequality, see [44]:

$$\begin{array}{c}\mathfrak{S}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \le \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\mathfrak{S}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx+\frac{1}{{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathfrak{S}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \right]\hfill \\ \le \frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathfrak{S}\left(x,\omega \right)d\omega dx\hfill \\ \le \frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_1\left(b,a\right)}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\mathfrak{S}\left(x,u\right)dx+{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\mathfrak{S}\left(x,\nu \right)dx\right]\hfill \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\phi }_2\left(\nu ,u\right)}\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathfrak{S}\left(a,\omega \right)d\omega +{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathfrak{S}\left(b,\omega \right)d\omega \right]\hfill \\ \le \frac{\mathfrak{S}\left(a,u\right)+\mathfrak{S}\left(b,u\right)+\mathfrak{S}\left(a,\nu \right)+\mathfrak{S}\left(b,\nu \right)}{4}\hfill \end{array}$$

(60)

If${\mathfrak{S}}_{*}\left(x,\omega \right)={\mathfrak{S}}^{*}\left(x,\omega \right)$with$\lambda =1$and,${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$, then from (39), we acquire the following inequality, see [34]:

$$\begin{array}{c}\mathfrak{S}\left(\frac{a+b}{2}, \frac{u+\nu }{2}\right)\le \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{b-a}{{\displaystyle \int }}_a^b\mathfrak{S}\left(x,\frac{u+\nu }{2}\right)dx+\frac{1}{\nu -u}{{\displaystyle \int }}_u^{\nu }\mathfrak{S}\left(\frac{a+b}{2},\omega \right)d\omega \right]\hfill \\ \le \frac{1}{\left(b-a\right)\left(\nu -u\right)} {{\displaystyle \int }}_a^b{{\displaystyle \int }}_u^{\nu }\mathfrak{S}\left(x,\omega \right)d\omega dx\hfill \\ \le \frac{\begin{array}{c} \\ 1\end{array}}{4\left(b-a\right)}\left[{{\displaystyle \int }}_a^b\mathfrak{S}\left(x,u\right)dx+{{\displaystyle \int }}_a^b\mathfrak{S}\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\nu -u\right)}\left[{{\displaystyle \int }}_u^{\nu }\mathfrak{S}\left(a,\omega \right)d\omega +{{\displaystyle \int }}_u^{\nu }\mathfrak{S}\left(b,\omega \right)d\omega \right]\hfill \\ \le \frac{\mathfrak{S}\left(a,u\right)+\mathfrak{S}\left(b,u\right)+\mathfrak{S}\left(a,\nu \right)+\mathfrak{S}\left(b,\nu \right)}{4}. \hfill \end{array}$$

(61)

Example 3.

We consider the F∙I-V∙Fs$\tilde{\mathfrak{S}}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_I$defined by,

$$\mathfrak{S}\left(x\right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma }{2\left(6+e^x\right)\left(6+e^{\omega }\right)}, \sigma \in \left[0, 2\left(6+e^x\right)\left(6+e^{\omega }\right)\right]\\ \frac{4\left(6+e^x\right)\left(6+e^{\omega }\right)-\sigma }{2\left(6+e^x\right)\left(6+e^{\omega }\right)}, \sigma \in \left(2\left(6+e^x\right)\left(6+e^{\omega }\right), 4\left(6+e^x\right)\left(6+e^{\omega }\right)\right]\\ 0 , \mathrm{otherwise},\end{array}\right.$$

Then, for each$\lambda \in \left[0, 1\right],$we have${\mathfrak{S}}_{\lambda }\left(x\right)=\left[2\lambda \left(6+e^x\right)\left(6+e^{\omega }\right),\left(4+2\lambda \right)\left(6+e^x\right)\left(6+e^{\omega }\right) \right]$. End-point functions${\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right),$${\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)$are coordinated preinvex functions with respect to${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$for each$\lambda \in \left[0, 1\right]$. Hence,$\tilde{\mathfrak{S}}\left(x,\omega \right)$. is a coordinated preinvex F∙I-V∙F.

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}, \frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)=\left[2\lambda {\left(5+e^{\frac{1}{2}}\right)}^2,2\left(2+\lambda \right){\left(6+e^{\frac{1}{2}}\right)}^{\begin{array}{c} \\ 2\end{array}} \right]$$

$$\begin{array}{c}\frac{1}{2}[\frac{1}{{\phi }_1\left(b,a\right)}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\hfill \\ +\frac{\begin{array}{c} \\ 1\end{array}}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega )d\omega ]\hfill \\ =\left[4\lambda \left(6+e^{\frac{1}{2}}\right)\left(5+e\right),4\left(2+\lambda \right)\left(6+e^{\frac{1}{2}}\right)\left(5+e\right) \right]\hfill \end{array}$$

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ =\left[2\lambda {\left(5+e\right)}^2,2\left(2+\lambda \right){\left(5+e\right)}^2 \right]\hfill \end{array}$$

$$\begin{array}{c} \frac{1}{4{\phi }_1\left(b,a\right)}\left[{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)dx+{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)dx\right]\hfill \\ +\frac{1}{4{\phi }_2\left(\nu ,u\right)}\left[{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)d\omega +{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)d\omega \right]\hfill \\ =\left[\lambda \left(5+e\right)\left(13+e\right),\left(2+\lambda \right)\left(5+e\right)\left(13+e\right)\right]\hfill \end{array}$$

$$\begin{array}{c}\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)}{4}\hfill \\ =\left[\lambda \frac{\left(6+e\right)\left(20+e\right)+49}{2},2\left(2+\lambda \right)\frac{\left(6+e\right)\left(20+e\right)+49}{2}\right]\hfill \end{array}$$

That is

$$\begin{array}{c}\left[2\lambda {\left(5+e^{\frac{1}{2}}\right)}^{\begin{array}{c} \\ 2\end{array}},2\left(2+\lambda \right){\left(6+e^{\frac{1}{2}}\right)}^2 \right]\hfill \\ {\le }_I\left[4\lambda \left(6+e^{\frac{1}{2}}\right)\left(5+e\right),4\left(2+\lambda \right)\left(6+e^{\frac{1}{2}}\right)\left(5+e\right) \right]\hfill \\ {\le }_I\left[2\lambda {\left(5+e\right)}^2,2\left(2+\lambda \right){\left(5+e\right)}^2 \right]\hfill \\ {\le }_I\left[\lambda \left(5+e\right)\left(13+e\right),\left(2+\lambda \right)\left(5+e\right)\left(13+e\right)\right]\hfill \\ {\le }_I\left[\lambda \frac{\left(6+e\right)\left(20+e\right)+49}{2},2\left(2+\lambda \right)\frac{\left(6+e\right)\left(20+e\right)+49}{2}\right]\hfill \end{array}$$

Hence, Theorem 10 has been verified.

We now obtain some HH-inequalities for the product of coordinated preinvex F∙I-V∙Fs which are known as Pachpatte Type inequalities. These inequalities are refinements of some known inequalities; see [34,37,38,44].

Theorem 11.

Let$\tilde{\mathfrak{S}},\tilde{\mathcal{J}} :\Delta =\left[a, a+{\phi }_1\left(b,a\right)\right]\times \left[u, u+{\phi }_2\left(\nu ,u\right)\right]\subset {ℝ}^2\to {\mathbb{F}}_I$be two coordinated preinvex F∙I-V∙Fs on$\Delta$ $,$whose$\lambda$-levels${\mathfrak{S}}_{\lambda }, {\mathcal{J}}_{\lambda }:\left[a, a+{\phi }_1\left(b,a\right)\right]\times \left[u, u+{\phi }_2\left(\nu ,u\right)\right]\to {ℝ}_I^{+}$are defined by${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$and${\mathcal{J}}_{\lambda }\left(x,\omega \right)=\left[{\mathcal{J}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathcal{J}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$for all$\left(x,\omega \right)\in \Delta$and for all$\lambda \in \left[0, 1\right]$. If Condition 1 for${\phi }_1$and${\phi }_2$is fulfilled, then following inequality hold:

$$\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x,\omega \right)\tilde{\times }\tilde{\mathcal{J}}\left(x,\omega \right)d\omega dx\preccurlyeq \frac{1}{9}\tilde{\alpha }\left(a,b,u,\nu \right)\widetilde{+}\frac{1}{18}\tilde{\beta }\left(a,b,u,\nu \right)\widetilde{+}\frac{1}{36}\tilde{\gamma }\left(a,b,u,\nu \right),$$

(62)

where

$$\tilde{\alpha }\left(a,b,u,\nu \right)=\tilde{\mathfrak{S}}\left(a,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(a,u\right) \widetilde{+} \tilde{\mathfrak{S}}\left(a,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(b,u\right) \widetilde{+} \tilde{\mathfrak{S}}\left(b,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(b,\nu \right),$$

$$\tilde{\beta }\left(a,b,u,\nu \right)=\tilde{\mathfrak{S}}\left(a,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(a,\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(a,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(b,\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(b,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(b,u\right),$$

$$\widetilde{+}\tilde{\mathfrak{S}}\left(a,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(b,u\right) \widetilde{+} \tilde{\mathfrak{S}}\left(b,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(a,u\right) \widetilde{+} \tilde{\mathfrak{S}}\left(a,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(b,\nu \right)$$

$$\tilde{\gamma }\left(a,b,u,\nu \right)=\tilde{\mathfrak{S}}\left(a,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(b,\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(b,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(a,\nu \right)\widetilde{+} \tilde{\mathfrak{S}}\left(b,\nu \right)\tilde{\times }\tilde{\mathcal{J}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\tilde{\times }\tilde{\mathcal{J}}\left(a,\nu \right)$$

and for each$\lambda \in \left[0, 1\right],$$\tilde{\alpha }\left(a,b,u,\nu \right)$, $\tilde{\beta }\left(a,b,u,\nu \right)$and$\tilde{\gamma }\left(a,b,u,\nu \right)$are defined as follows:

$${\alpha }_{\lambda }\left(a,b,u,\nu \right)=\left[{\alpha }_{*}\left(\left(a,b,u,\nu \right), \lambda \right), {\alpha }^{*}\left(\left(a,b,u,\nu \right), \lambda \right)\right]$$

$${\beta }_{\lambda }\left(a,b,u,\nu \right)=\left[{\beta }_{*}\left(\left(a,b,u,\nu \right), \lambda \right), {\beta }^{*}\left(\left(a,b,u,\nu \right), \lambda \right)\right]$$

$${\gamma }_{\lambda }\left(a,b,u,\nu \right)=\left[{\gamma }_{*}\left(\left(a,b,u,\nu \right), \lambda \right), {\gamma }^{*}\left(\left(a,b,u,\nu \right), \lambda \right)\right].$$

Proof. 

Let $\tilde{\mathfrak{S}}$ and $\tilde{\mathcal{J}}$ both are coordinated preinvex F∙I-V∙Fs on $\left[a, a+{\phi }_1\left(b,a\right)\right]\times \left[u, u+{\phi }_2\left(\nu ,u\right)\right]$. Then

$$\tilde{\mathfrak{S}}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\preccurlyeq \sigma s\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\sigma \left(1-s\right)\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\left(1-\sigma \right)s\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\left(1-\sigma \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(b,\nu \right),$$

and

$$\tilde{\mathcal{J}}\left(a+\left(1-\sigma \right){\phi }_1\left(b,a\right), u+\left(1-s\right){\phi }_2\left(\nu ,u\right)\right)\preccurlyeq \sigma s\tilde{\mathcal{J}}\left(a,u\right)\widetilde{+}\sigma \left(1-s\right)\tilde{\mathcal{J}}\left(a,\nu \right)\widetilde{+}\left(1-\sigma \right)s\tilde{\mathcal{J}}\left(b,u\right)\widetilde{+}\left(1-\sigma \right)\left(1-s\right)\tilde{\mathcal{J}}\left(b,\nu \right).$$

Since $\tilde{\mathfrak{S}}$ and $\tilde{\mathcal{J}}$ both are coordinated preinvex F∙I-V∙Fs, then by Lemma 1, there exist

$${\tilde{\mathfrak{S}}}_x:\left[u,\nu \right]\to {\mathbb{F}}_I$$

$${\tilde{\mathfrak{S}}}_x\left(\omega \right)=\tilde{\mathfrak{S}}\left(x,\omega \right)$$

$${\tilde{\mathcal{J}}}_x:\left[u,\nu \right]\to {\mathbb{F}}_I$$

$${\tilde{\mathcal{J}}}_x\left(\omega \right)=\tilde{\mathcal{J}}\left(x,\omega \right)$$

and

$${\tilde{\mathfrak{S}}}_{\omega }:\left[a,b\right]\to {\mathbb{F}}_I$$

$${\tilde{\mathfrak{S}}}_{\omega }\left(x\right)=\tilde{\mathfrak{S}}\left(x,\omega \right)$$

$${\tilde{\mathcal{J}}}_{\omega }:\left[a,b\right]\to {\mathbb{F}}_I$$

$${\tilde{\mathcal{J}}}_{\omega }\left(x\right)=\tilde{\mathcal{J}}\left(x,\omega \right)$$

Since ${\tilde{\mathfrak{S}}}_x$, ${\tilde{\mathcal{J}}}_x, {\tilde{\mathfrak{S}}}_{\omega }$ and ${\tilde{\mathcal{J}}}_{\omega }$ are F∙I-V∙Fs, then by inequality (29), we have

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\tilde{\mathfrak{S}}}_{\omega }\left(x\right)\times {\tilde{\mathcal{J}}}_{\omega }\left(x\right)dx\hfill \\ \preccurlyeq \frac{1}{3}\left[{\tilde{\mathfrak{S}}}_{\omega }\left(a\right)\times {\tilde{\mathcal{J}}}_{\omega }\left(a\right)+{\tilde{\mathfrak{S}}}_{\omega }\left(b\right)\times {\tilde{\mathcal{J}}}_{\omega }\left(b\right)\right]+\frac{1}{6}\left[{\tilde{\mathfrak{S}}}_{\omega }\left(a\right)\times {\tilde{\mathcal{J}}}_{\omega }\left(b\right)+{\tilde{\mathfrak{S}}}_{\omega }\left(b\right)\times {\tilde{\mathcal{J}}}_{\omega }\left(a\right)\right],\hfill \end{array}$$

and

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\tilde{\mathfrak{S}}}_x\left(\omega \right)\times {\tilde{\mathcal{J}}}_x\left(\omega \right)d\omega \hfill \\ \preccurlyeq \frac{1}{3}\left[{\tilde{\mathfrak{S}}}_x\left(u\right)\times {\tilde{\mathcal{J}}}_x\left(u\right)+{\tilde{\mathfrak{S}}}_x\left(\nu \right)\times {\tilde{\mathcal{J}}}_x\left(\nu \right)\right]+\frac{1}{6}\left[{\tilde{\mathfrak{S}}}_x\left(u\right)\times {\tilde{\mathcal{J}}}_x\left(\nu \right)+{\tilde{\mathfrak{S}}}_x\left(u\right)\times {\tilde{\mathcal{J}}}_x\left(\nu \right)\right].\hfill \end{array}$$

For each $\lambda \in \left[0, 1\right],$ we have

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }{}_{\omega }\left(x\right)\times {\mathcal{J}}_{\lambda }{}_{\omega }\left(x\right)dx\hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }{}_{\omega }\left(a\right)\times {\mathcal{J}}_{\lambda }{}_{\omega }\left(a\right)+{\mathfrak{S}}_{\lambda }{}_{\omega }\left(b\right)\times {\mathcal{J}}_{\lambda }{}_{\omega }\left(b\right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }{}_{\omega }\left(a\right)\times {\mathcal{J}}_{\lambda }{}_{\omega }\left(b\right)+{\mathfrak{S}}_{\lambda }{}_{\omega }\left(b\right)\times {\mathcal{J}}_{\lambda }{}_{\omega }\left(a\right)\right],\hfill \end{array}$$

and

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }{}_x\left(\omega \right)\times {\mathcal{J}}_{\lambda }{}_x\left(\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }{}_x\left(u\right)\times {\mathcal{J}}_{\lambda }{}_x\left(u\right)+{\mathfrak{S}}_{\lambda }{}_x\left(\nu \right)\times {\mathcal{J}}_{\lambda }{}_x\left(\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }{}_x\left(u\right)\times {\mathcal{J}}_{\lambda }{}_x\left(\nu \right)+{\mathfrak{S}}_{\lambda }{}_x\left(u\right)\times {\mathcal{J}}_{\lambda }{}_x\left(\nu \right)\right]\hfill \end{array}$$

The above inequalities can be written as

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)dx\hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)\right],\hfill \end{array}$$

(63)

and

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)\right].\hfill \end{array}$$

(64)

Firstly, we solve inequality (63), taking integration on the both sides of inequality with respect to $\omega$ over interval $\left[u, u+{\phi }_2\left(\nu ,u\right)\right]$ and dividing both sides by ${\phi }_2\left(\nu ,u\right)$, we have

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ {\le }_I\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)\right]d\omega \hfill \\ +\frac{1}{6{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)\right]d\omega . \hfill \end{array}$$

(65)

Now again by inequality (29), for each $\lambda \in \left[0, 1\right],$ we have

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]d\omega +\frac{1}{6}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]d\omega .\hfill \end{array}$$

(66)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]d\omega +\frac{1}{6}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]d\omega \hfill \end{array}$$

(67)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]d\omega +\frac{1}{6}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]d\omega .\hfill \end{array}$$

(68)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]d\omega +\frac{1}{6}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right]d\omega .\hfill \end{array}$$

(69)

From (66)–(69), inequality (65) we have

$$\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx{\le }_I\frac{1}{9}{\alpha }_{\lambda }\left(a,b,u,\nu \right)+\frac{1}{18}{\beta }_{\lambda }\left(a,b,u,\nu \right)+\frac{1}{36}{\gamma }_{\lambda }\left(a,b,u,\nu \right).$$

That is

$$\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x,\omega \right)\tilde{\times }\tilde{\mathcal{J}}\left(x,\omega \right)d\omega dx\preccurlyeq \frac{1}{9}\tilde{\alpha }\left(a,b,u,\nu \right)\widetilde{+}\frac{1}{18}\tilde{\beta }\left(a,b,u,\nu \right)\widetilde{+}\frac{1}{36}\tilde{\gamma }\left(a,b,u,\nu \right).$$

Hence, this concludes the proof of theorem. □

Theorem 12.

Let$\tilde{\mathfrak{S}},\tilde{\mathcal{J}} :\Delta =\left[a, a+{\phi }_1\left(b,a\right)\right]\times \left[u, u+{\phi }_2\left(\nu ,u\right)\right]\subset {ℝ}^2\to {\mathbb{F}}_I$be two coordinated preinvex F∙I-V∙Fs. Then, from$\lambda$-levels, we get the collection of I-V∙Fs${\mathfrak{S}}_{\lambda }, {\mathcal{J}}_{\lambda }:\Delta \subset {ℝ}^2\to {ℝ}_I^{+}$are given by${\mathfrak{S}}_{\lambda }\left(x\right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$and${\mathcal{J}}_{\lambda }\left(x\right)=\left[{\mathcal{J}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathcal{J}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$for all$\left(x,\omega \right)\in \Delta$and for all$\lambda \in \left[0, 1\right]$. If Condition 1 for${\phi }_1$and${\phi }_2$is fulfilled, then following inequality hold:

$$\begin{array}{c}4 \tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\tilde{\times }\tilde{\mathcal{J}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x,\omega \right)\tilde{\times }\tilde{\mathcal{J}}\left(x,\omega \right)d\omega dx\widetilde{+}\frac{5}{36}\tilde{\alpha }\left(a,b,u,\nu \right)\widetilde{+}\frac{7}{36}\tilde{\beta }\left(a,b,u,\nu \right)\widetilde{+}\frac{2}{9}\tilde{\gamma }\left(a,b,u,\nu \right),\hfill \end{array}$$

(70)

where$\tilde{\alpha }\left(a,b,u,\nu \right)$,$\tilde{\beta }\left(a,b,u,\nu \right)$, and$\tilde{\gamma }\left(a,b,u,\nu \right)$are given in Theorem 11.

Proof. 

$\tilde{\mathfrak{S}},\tilde{\mathcal{J}} :\Delta \to {\mathbb{F}}_I$ are two coordinated preinvex F∙I-V∙Fs, and then from inequality (30) and for each $\lambda \in \left[0, 1\right],$ we have

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)+{\mathfrak{S}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\right]+{\mathfrak{S}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \end{array}$$

(71)

and

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)d\omega \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)+{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)+{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\right]\hfill \end{array}$$

(72)

Summing the inequalities (71) and (72), then taking the multiplication of the resultant one by 2, we obtain

$$\begin{array}{c}8{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{2}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx\hfill \\ +\frac{2}{{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)dx \hfill \\ +\frac{1}{6}\left[2{\mathfrak{S}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)+2{\mathfrak{S}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\right]\hfill \\ +\frac{1}{6}\left[2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)+2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\right]\hfill \\ +\frac{1}{3}\left[2{\mathfrak{S}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)+2{\mathfrak{S}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\right]\hfill \\ +\frac{1}{3}[2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)+2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)]\hfill \end{array}.$$

(73)

Now, with the help of integral inequality (30) for each integral on the right-hand side of (73), we have

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right]\hfill \end{array}$$

(74)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]\hfill \end{array}$$

(75)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]\hfill \end{array}$$

(76)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(b,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(a,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right]\hfill \end{array}$$

(77)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)+{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\right]\hfill \end{array}$$

(78)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)+{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\right]\hfill \end{array}$$

(79)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)+{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\right]\hfill \end{array}$$

(80)

$$\begin{array}{c}2{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx \hfill \\ +\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]\hfill \\ +\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)+{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\nu \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},u\right)\right]\hfill \end{array}$$

(81)

From (74)–(81), we have

$$\begin{array}{c}8{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{2}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx \hfill \\ +\frac{2}{{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)dx +\frac{1}{6{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ +\frac{1}{6{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega +\frac{1}{6{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\begin{array}{c} \\ {\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx\end{array}\hfill \\ +\frac{1}{6{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega +\frac{1}{3{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx\hfill \\ +\frac{1}{3{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx ,+\frac{1}{18}{\alpha }_{\lambda }\left(a,b,u,\nu \right)+\frac{1}{9}{\beta }_{\lambda }\left(a,b,u,\nu \right)+\frac{2}{9}{\gamma }_{\lambda }\left(a,b,u,\nu \right)\hfill \end{array}$$

(82)

Now, again with the help of integral inequality (30) for first two integrals on the right-hand side of (82), we have the following relation:

$$\begin{array}{c}\frac{2}{{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)dx \hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ +\frac{1}{3{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)\right]dx \hfill \\ +\frac{1}{6{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)\right]dx,\hfill \end{array}$$

(83)

$$\begin{array}{c}\frac{2}{{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)dx \hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)\right]d\omega \hfill \\ +\frac{1}{6{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)\right]d\omega \hfill \end{array}$$

(84)

From (83) and (84), we have

$$\begin{array}{c}8{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ +\frac{1}{3{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)\right]dx \hfill \\ +\frac{1}{6{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)\right]dx\hfill \\ +\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)\right]d\omega \hfill \\ +\frac{1}{6{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)\right]d\omega ,+\frac{1}{6{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ +\frac{1}{6{\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ +\frac{1}{6{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx+\frac{1}{6{\phi }_1\left(b,a\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx\hfill \\ +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ +\frac{1}{3{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx+\frac{1}{3{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx ,\hfill \\ + \frac{1}{18}{\alpha }_{\lambda }\left(a,b,u,\nu \right)+\frac{1}{9}{\beta }_{\lambda }\left(a,b,u,\nu \right)+\frac{2}{9}{\gamma }_{\lambda }\left(a,b,u,\nu \right).\hfill \end{array}$$

It follows that

$$\begin{array}{c}8{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{2}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ +\frac{2}{3{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)\right]dx \hfill \\ +\frac{1}{3{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\left[{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)+{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)\right]dx\hfill \\ +\frac{2}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)\right]d\omega \hfill \\ +\frac{1}{3{\phi }_2\left(\nu ,u\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\left[{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)\right]d\omega \hfill \\ +\frac{1}{18}{\alpha }_{\lambda }\left(a,b,u,\nu \right)+\frac{1}{9}{\beta }_{\lambda }\left(a,b,u,\nu \right)+\frac{2}{9}{\gamma }_{\lambda }\left(a,b,u,\nu \right).\hfill \end{array}$$

(85)

Now, using integral inequality (25) for integrals on the right-hand side of (85), we have the following relation:

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right],\hfill \end{array}$$

(86)

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right], \hfill \end{array}$$

(87)

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)\times {\mathcal{J}}_{\lambda }\left(x,\nu \right)dx \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right], \hfill \end{array}$$

(88)

$$\begin{array}{c}\frac{1}{{\phi }_1\left(b,a\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\times {\mathcal{J}}_{\lambda }\left(x,u\right)dx \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right], \hfill \end{array}$$

(89)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right], \hfill \end{array}$$

(90)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right], \hfill \end{array}$$

(91)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\times {\mathcal{J}}_{\lambda }\left(b,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(a,u\right)\times {\mathcal{J}}_{\lambda }\left(b,\nu \right)+{\mathfrak{S}}_{\lambda }\left(a,\nu \right)\times {\mathcal{J}}_{\lambda }\left(b,u\right)\right], \hfill \end{array}$$

(92)

$$\begin{array}{c}\frac{1}{{\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\times {\mathcal{J}}_{\lambda }\left(a,\omega \right)d\omega \hfill \\ {\le }_I\frac{1}{3}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)\right]+\frac{1}{6}\left[{\mathfrak{S}}_{\lambda }\left(b,u\right)\times {\mathcal{J}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)\times {\mathcal{J}}_{\lambda }\left(a,u\right)\right]. \hfill \end{array}$$

(93)

From (86)–(93), inequality (95), we have

$$\begin{array}{c}4 {\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\times {\mathcal{J}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ {\le }_I\frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\times {\mathcal{J}}_{\lambda }\left(x,\omega \right)d\omega dx\hfill \\ +\frac{5}{36}{\alpha }_{\lambda }\left(a,b,u,\nu \right)+\frac{7}{36}{\beta }_{\lambda }\left(a,b,u,\nu \right)+\frac{2}{9}{\gamma }_{\lambda }\left(a,b,u,\nu \right)\hfill \end{array}$$

That is

$$\begin{array}{c}4 \tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\tilde{\times }\tilde{\mathcal{J}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \tilde{\preccurlyeq \frac{1}{{\phi }_1\left(b,a\right){\phi }_2\left(\nu ,u\right)} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\tilde{\mathfrak{S}}\left(x,\omega \right)\tilde{\times }\tilde{\mathcal{J}}\left(x,\omega \right)d\omega dx+} \frac{5}{36}\tilde{\alpha }\left(a,b,u,\nu \right)\widetilde{+}\frac{7}{36}\tilde{\beta }\left(a,b,u,\nu \right)\widetilde{+}\frac{2}{9}\tilde{\gamma }\left(a,b,u,\nu \right).\hfill \end{array}$$

We now give HH-Fejér inequality for coordinated preinvex F∙I-V∙Fs by means of FOR in the following result. □

Theorem 13.

Let$\tilde{\mathfrak{S}}:\Delta =\left[a, a+{\phi }_1\left(b,a\right)\right]\times \left[u, u+{\phi }_2\left(\nu ,u\right)\right]\to {\mathbb{F}}_I$be a coordinated preinvex F∙I-V∙F with$a<b$and$u<\nu .$Then, from$\lambda$-levels, we get the collection of I-V∙Fs${\mathfrak{S}}_{\lambda }:\Delta \to {ℝ}_I^{+}$are given by${\mathfrak{S}}_{\lambda }\left(x,\omega \right)=\left[{\mathfrak{S}}_{*}\left(\left(x,\omega \right),\lambda \right), {\mathfrak{S}}^{*}\left(\left(x,\omega \right),\lambda \right)\right]$for all$\left(x,\omega \right)\in \Delta$and for all$\lambda \in \left[0, 1\right]$. Let$\psi :\left[a, a+{\phi }_1\left(b,a\right)\right]\to ℝ$with$\left(x\right)\ge 0,$ ${\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx>0$, and$\mathcal{K}:\left[u, u+{\phi }_2\left(\nu ,u\right)\right]\to ℝ$with$\mathcal{K}\left(\omega \right)\ge 0,$${\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega >0,$be two symmetric functions with respect to$\frac{2a+{\phi }_1\left(b,a\right)}{2}$and$\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}$, respectively. If Condition 1 for${\phi }_1$and${\phi }_2$holds, then following inequality hold:

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\hfill \\ \preccurlyeq \frac{1}{2}\left[\begin{array}{c}\frac{1}{{{\displaystyle \int }}_a^b\psi \left(x\right)dx} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx\\ \widetilde{+}\frac{1}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega } {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \end{array}\right]\hfill \\ \preccurlyeq \frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)d\omega dx\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]\hfill \\ \widetilde{+}\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4}\hfill \end{array}$$

(94)

Proof. 

Since $\tilde{\mathfrak{S}}$ both is a coordinated preinvex F∙I-V∙F on $\Delta$, it follows that for functions, then by Lemma 1, there exist

$${\tilde{\mathfrak{S}}}_x:\left[u,\nu \right]\to {\mathbb{F}}_I, {\tilde{\mathfrak{S}}}_x\left(\omega \right)=\tilde{\mathfrak{S}}\left(x,\omega \right), {\tilde{\mathfrak{S}}}_{\omega }:\left[a,b\right]\to {\mathbb{F}}_I, {\tilde{\mathfrak{S}}}_{\omega }\left(x\right)=\tilde{\mathfrak{S}}\left(x,\omega \right)$$

Thus, from inequality (31), for each $\lambda \in \left[0, 1\right],$ we have

$${\mathfrak{S}}_{\lambda }{}_x\left(\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega }{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }{}_x\left(\omega \right)\mathcal{K}\left(\omega \right)d\omega {\le }_I\frac{{\mathfrak{S}}_{\lambda }{}_x\left(u\right)+{\mathfrak{S}}_{\lambda }{}_x\left(\nu \right)}{2},$$

and

$${\mathfrak{S}}_{\lambda }{}_{\omega }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2}\right){\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }{}_{\omega }\left(x\right)\psi \left(x\right)dx{\le }_I\frac{{\mathfrak{S}}_{\lambda }{}_{\omega }\left(a\right)+{\mathfrak{S}}_{\lambda }{}_{\omega }\left(b\right)}{2}$$

The above inequalities can be written as

$${\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega }{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\mathcal{K}\left(\omega \right)d\omega {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x, \nu \right)}{2},$$

(95)

and

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right){\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\psi \left(x\right)dx{\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)}{2}$$

(96)

Multiplying (95) by $\psi \left(x\right)$ and then integrating the resultant with respect to $x$ over $\left[a, a+{\phi }_1\left(b,a\right)\right]$, we have

$$\begin{array}{c}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx\hfill \\ {\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)d\omega dx{\le }_I{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\frac{{\mathfrak{S}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x, \nu \right)}{2}\psi \left(x\right)dx.\hfill \end{array}$$

(97)

Now, multiplying (96) by $\mathcal{K}\left(\omega \right)$ and then integrating the resultant with respect to $\omega$ over $\left[u, u+{\phi }_2\left(\nu ,u\right)\right]$, we have

$$\begin{array}{c}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \hfill \\ {\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)dxd\omega {\le }_I{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\frac{{\mathfrak{S}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)}{2}\mathcal{K}\left(\omega \right)d\omega \hfill \end{array}$$

(98)

Since ${\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx>0$ and ${\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega >0,$ then dividing (97) and (98) by ${\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx>0$ and ${\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega >0$, respectively, we get

$$\begin{array}{c}\frac{1}{2}\left[\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx} {\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx+\frac{1}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega } {\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \right]\hfill \\ {\le }_I\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)d\omega dx\hfill \\ {\le }_I\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\frac{{\mathfrak{S}}_{\lambda }\left(x,u\right)+{\mathfrak{S}}_{\lambda }\left(x, \nu \right)}{4}\psi \left(x\right)dx\hfill \\ +\frac{\begin{array}{c} \\ 1\end{array}}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\frac{{\mathfrak{S}}_{\lambda }\left(a,\omega \right)+{\mathfrak{S}}_{\lambda }\left(b,\omega \right)}{4}\mathcal{K}\left(\omega \right)d\omega \hfill \end{array}$$

(99)

Now, from the left part of double inequalities (95) and (96), we obtain

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega }{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega ,$$

(100)

and

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx$$

(101)

Summing the inequalities (100) and (101), we get

$${\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{1}{2}\left[\begin{array}{c}\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx\\ +\frac{\begin{array}{c} \\ 1\end{array}}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \end{array}\right]$$

(102)

Similarly, from the right part of (101) and (102), we can obtain

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega }{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\mathcal{K}\left(\omega \right)d\omega {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(a, \nu \right)}{2},$$

(103)

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}\mathcal{K}\left(\omega \right)d\omega }{\displaystyle {\int }_u^{u+{\phi }_2\left(\nu ,u\right)}}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\mathcal{K}\left(\omega \right)d\omega {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b, \nu \right)}{2},$$

(104)

and

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,u\right)\psi \left(x\right)dx{\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)}{2}$$

(105)

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}\psi \left(x\right)dx}{\displaystyle {\int }_a^{a+{\phi }_1\left(b,a\right)}}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\psi \left(x\right)dx{\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)}{2}$$

(106)

Adding (103)–(106) and dividing by 4, we get

$$\begin{array}{c}\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\mathcal{K}\left(\omega \right)d\omega +{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\mathcal{K}\left(\omega \right)d\omega \right]\hfill \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)\psi \left(x\right)dx+{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\psi \left(x\right)dx\right]\hfill \\ {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(a, \nu \right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b, \nu \right)}{4}\hfill \end{array}$$

(107)

Combing inequalities (99), (102), and (107), we obtain

$$\begin{array}{c}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right){\le }_I\frac{1}{2}\left[\begin{array}{c}\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx\\ +\frac{\begin{array}{c} \\ 1\end{array}}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \end{array}\right]\hfill \\ {\le }_I\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathfrak{S}\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)d\omega dx\hfill \\ {\le }_I\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(a,\omega \right)\mathcal{K}\left(\omega \right)d\omega +{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}{\mathfrak{S}}_{\lambda }\left(b,\omega \right)\mathcal{K}\left(\omega \right)d\omega \right]\hfill \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,u\right)\psi \left(x\right)dx+{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{\mathfrak{S}}_{\lambda }\left(x,\nu \right)\psi \left(x\right)dx\right]\hfill \\ {\le }_I\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(a, \nu \right)}{2}+\frac{{\mathfrak{S}}_{\lambda }\left(b,u\right)+{\mathfrak{S}}_{\lambda }\left(b, \nu \right)}{2}+\frac{{\mathfrak{S}}_{\lambda }\left(a,u\right)+{\mathfrak{S}}_{\lambda }\left(b,u\right)}{2}+\frac{{\mathfrak{S}}_{\lambda }\left(a,\nu \right)+{\mathfrak{S}}_{\lambda }\left(b,\nu \right)}{2}\hfill \end{array}$$

That is

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\preccurlyeq \frac{1}{2}\left[\begin{array}{c}\frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx} {{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,\frac{2u+{\phi }_2\left(\nu ,u\right)}{2}\right)\psi \left(x\right)dx\\ \widetilde{+}\frac{1}{{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega } {{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(\frac{2a+{\phi }_1\left(b,a\right)}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \end{array}\right]\hfill \\ \preccurlyeq \frac{1}{{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)d\omega dx\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\psi \left(x\right)dx}\left[{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+}{{\displaystyle \int }}_a^{a+{\phi }_1\left(b,a\right)}\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]\hfill \\ \widetilde{+}\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\mathcal{K}\left(\omega \right)d\omega }\left[{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+}{{\displaystyle \int }}_u^{u+{\phi }_2\left(\nu ,u\right)}\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4}\hfill \end{array}$$

Hence, this concludes the proof. □

Remark 6.

If one takes$\mathcal{K}\left(\omega \right)=1=\psi \left(x\right)$, then from (94) we achieve (39).

If one takes${\phi }_1\left(b,a\right)=b-a$and${\phi }_2\left(\nu ,u\right)=\nu -u$, then from (94), we acquire the following inequality, see [38]:

$$\begin{array}{c}\tilde{\mathfrak{S}}\left(\frac{a+b}{2},\frac{u+\nu }{2}\right)\preccurlyeq \frac{1}{2}\left[\frac{1}{{{\displaystyle \int }}_a^b\psi \left(x\right)dx} {{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(x,\frac{u+\nu }{2}\right)\psi \left(x\right)dx\widetilde{+}\frac{1}{{{\displaystyle \int }}_a^b\mathcal{K}\left(\omega \right)d\omega } {{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(\frac{a+b}{2},\omega \right)\mathcal{K}\left(\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{1}{{{\displaystyle \int }}_a^b\psi \left(x\right)dx{{\displaystyle \int }}_a^b\mathcal{K}\left(\omega \right)d\omega }{{\displaystyle \int }}_a^b{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(x,\omega \right)\psi \left(x\right)\mathcal{K}\left(\omega \right)d\omega dx\hfill \\ \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_a^b\psi \left(x\right)dx}\left[{{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(x,u\right)dx\widetilde{+}{{\displaystyle \int }}_a^b\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]\widetilde{+}\frac{\begin{array}{c} \\ 1\end{array}}{4{{\displaystyle \int }}_a^b\mathcal{K}\left(\omega \right)d\omega }\left[{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(a,\omega \right)d\omega \widetilde{+}{{\displaystyle \int }}_u^{\nu }\tilde{\mathfrak{S}}\left(b,\omega \right)d\omega \right]\hfill \\ \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(a,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,u\right)\widetilde{+}\tilde{\mathfrak{S}}\left(a,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(b,\nu \right)}{4}\hfill \end{array}$$

(108)

If one takes${\phi }_1\left(b,a\right)=b-a$,${\phi }_2\left(\nu ,u\right)=\nu -u$and$\mathcal{K}\left(\omega \right)=1=\psi \left(x\right)$, then from (94), we acquire the inequality (59), see [38].

4. Conclusions

As an extension of convex fuzzy-interval-valued functions on coordinates, we have proposed the idea of fuzzy interval-valued preinvex functions in this article. For coordinated preinvex fuzzy interval-valued functions, we have created H–H-type inequalities. The product of two coordinated preinvex fuzzy-interval-valued functions was also examined, which are known as Pachpatte Type inequalities, as well as several new H–H-type inclusions. Other types of interval-valued preinvex functions on the coordinates may be included in the results produced in this study. Future work will explore fuzzy interval-valued fractional integrals on coordinates to study H–H-type and H–H–Fejér-type inequalities with the help of fuzzy-order relation for coordinated preinvex fuzzy interval-valued functions. We hope that the concepts and findings presented in this article will inspire readers to conduct more research.