A New Class of Coordinated Non-Convex Fuzzy-Number-Valued Mappings with Related Inequalities and Their Applications

Abstract

Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on fuzzy Aumann’s integral theory, the Hermite–Hadamard’s type inequalities are introduced for a newly defined class of nonconvex functions, which is known as $U$·$D$ preinvex fuzzy number-valued mappings ($U$·$D$ preinvex $F$·$N$·$V$·$M$s) on coordinates. Some Pachpatte-type inequalities are also established for the product of two $U$·$D$ preinvex $F$·$N$·$V$·$M$s, and some Hermite–Hadamard–Fejér-type inequalities are also acquired via fuzzy Aumann’s integrals. Additionally, several new generalized inequalities are also obtained for the special situations of the parameters. Additionally, some of the interesting remarks are provided to acquire the classical and new exceptional cases that can be considered as applications of the main outcomes. Lastly, a few suggested uses for these inequalities in numerical integration are made.

1. Introduction

Inequality theory heavily relies on fractional calculus due to its vast content and the constant development of new fractional operators, especially in recent years. Certain fractional operators have some algebraic properties, whereas others lack them, such as the semigroup property. It is always interesting and motivating for us to provide a generalization of inequality that includes all consequences that have been proven thus far for different fractional integrals.

The major theory of inequality began to take shape around the time that the most significant figures in the field—Gauss, Cauchy, and Chebyshev—provided a theoretical foundation for approximate techniques. Many inequalities were demonstrated during the close of the 19th and the start of the 20th centuries; some of these proved to be the modern classics, while the remainder remained singular findings. The book “Inequalities” authored by Hardy et al. [1] stands out as the pioneering work that systematically connects various inequalities, laying the foundation for the field as we recognize it today. As the inaugural book solely dedicated to inequalities, it significantly contributed to the advancement of this area, see [2].

This paper focuses on convex inequalities employing Jensen’s concept of convexity. Numerous inequalities have emerged following Jensen’s identification of the initial convex inequality [3,4], which is explored in this study. Convex inequalities have a wide range of applications, including in the domains of optimization, physics, and numerical analysis. For additional information, consult the books [5,6].

Over time, numerous generalizations have been documented [7,8]. Other convex expansions of Jensen’s inequality can be used to derive the Hermite–Hadamard inequality [9,10]. We use noninteger integral operators and the set-valued function (IVM) in conjunction with convexity characteristics.

L’Hopital corresponded with Leibniz in 1695. A crucial query regarding the derivative’s order surfaced in his message: what could a derivative of order $\frac{1}{2}$ be? Many aspiring mathematicians became interested in learning more about noninteger derivatives as a result of that letter. The derivative was formally introduced in 1822 by Fourier, who proposed an integral representation, which is recognized as the inaugural definition of the derivative of an arbitrary positive order, see [11]. In 1826, Abel’s solution to an integral equation related to a tautochrone problem marked the initial application of FC (noninteger calculus). Subsequently, numerous mathematicians, including Riemann, Grünwald, Letnikov, Hadamard, and Weyl, among others, continued to contribute to the field after Abel’s work. Caputo developed a formulation in the latter part of the 20th century, which was more suitable for discussing issues involving noninteger differential equations with initial conditions, although still more limited compared to the Riemann–Liouville definition, see [12]. Noninteger calculus also found applications in physics; for example, Craft and Meerschaert (2008) elucidated the noninteger conservation of mass and provided equations for acoustic waves in complex media, among other applications. Over time, various forms of noninteger integrals and derivatives have been defined. For further information on the subject, interested readers are encouraged to explore the following [13,14]. In the realm of inequalities, generalizations and the application of noninteger calculus are also prevalent; additional details can be found in [15,16].

On the other hand, exploring how mathematical integration principles adapt to ambiguous regions within fuzzy domains is intriguing. Sugeno initially introduced the theory of fuzzy measures and fuzzy integrals in [17,18]. Developing various types of integral inequalities is a current focus. Recently, numerous valuable studies have been conducted based on different non-additive integrals, including the Sugeno integral [19,20], generalized Sugeno integral [21], pseudo integral [22,23], Choquet integral [24], and others. Set-valued functions [25,26], serving as a generalization of single-valued functions, have become increasingly important both theoretically and practically. They have become essential tools for addressing problems in various fields, particularly in mathematical economics, such as individual demand, mean demand, competitive equilibrium, and coalition production economies. The concept of integrals for set-valued functions originated from Aumann’s research [27], which is based on Lebesgue integrals and is commonly referred to as set-valued Aumann integrals. Moreover, interval-valued Riemanian and fuzzy Aumann’s integrals are discussed in [28,29,30], respectively, and the references are therein. Further main concepts related to fuzzy theory are discussed by Anastassiou [31] and then applied by Bede [32] to introduce gH-differentiability, as well as by Noor [33] to define nonconvex mapping under the umbrella of fuzzy mapping.

It is noteworthy to mention the work by Khan et al. [34], which introduced the concept of fuzzy convex inequalities and was one of the most influential publications published in the past year. The idea itself is a vast area that can be studied in further detail. Recently, Khan et al. [35] introduced new versions of fuzzy integral inequalities via fuzzy fractional integrals and established a relationship between the up and down fuzzy relation and inclusion relation. Moreover, some very interesting examples also support the validity of the results. For further details on this topic, refer to the cited study. For more information related to fuzzy theory, see [36,37,38] and the references therein. These article ideas depend on the coordinates. The concepts of coordinated convexity was initiated by Dragomir [39]. Then, many authors worked on these ideas and discussed different types of convexity, as well as different versions of inequalities, like Latif and Dragomir [40] and Khan et al. [41] for fuzzy convexity and fuzzy nonconvexity. For more concepts related to fuzzy theory, see [42,43] and the reference therein.

This study is primarily concerned with obtaining a generalized definition of convex $F$·$N$·$V$·$M$s, which is known as coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s, as well as new extensions of Hermite–Hadamard–Fejér-type inequalities. Using fuzzy Aumann’s double integral operators, we first prove a Hermite–Hadamard-type inequality. Various generalized Pachpatte-type inequalities are also constructed using fuzzy preinvexity properties based on this new approach. The results are produced as concrete instances, and a validation process is used to confirm the accuracy of the results. A study of the bound estimates is also given. This concept can be extended to fractional-type inequalities. We will extend this concept in to generalized coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s.

2. Preliminaries

We will first go over the basic ideas of fuzzy mathematics. Further details are available from the following sources: Anastassiou [31], Bede [32], and Goetschel and Voxman [43].

Consider ${\mathbb{E}}_C$ as the set comprising all closed and bounded intervals of $\mathfrak{N}$, and let $\mathrm{V}$ belong to ${\mathbb{E}}_C$, defined as follows:

$$\mathrm{V}=\left[{\mathrm{V}}_{*},{\mathrm{V}}^{*}\right]=\left\{\mathfrak{б}\in \mathfrak{N}|{\mathrm{V}}_{*}\le \mathrm{б}\le {\mathrm{V}}^{*}\right\},\left({\mathrm{V}}_{*},{\mathrm{V}}^{*}\in \mathfrak{N}\right)$$

(1)

It is named a positive interval $\left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]$ if ${\mathrm{V}}_{*}\ge 0$. The definition of ${\mathbb{E}}_C^{+}$, which represents the set of all positive intervals, is

$${\mathbb{E}}_C^{+}=\left\{\left[{\mathrm{V}}_{*},{\mathrm{V}}^{*}\right]:\left[{\mathrm{V}}_{*},{\mathrm{V}}^{*}\right]\in {\mathbb{E}}_C and {\mathrm{V}}_{*}\ge 0\right\}.$$

(2)

Let $g\in \mathfrak{N}$ and $g\cdot \mathrm{V}$ be defined by

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

(3)

Subsequently, the Minkowski difference $\mathsf{Ϗ}-\mathrm{V}$, addition $\mathrm{V}+\mathsf{Ϗ}$, and multiplication $\mathrm{V}\times \mathsf{Ϗ}$ for $\mathrm{V},\mathsf{Ϗ}$ belong to ${\mathbb{E}}_C$ and are delineated as follows:

$$\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right]+\left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]=\left[{\mathsf{Ϗ}}_{*}+{\mathrm{V}}_{*}, {\mathsf{Ϗ}}^{*}+{\mathrm{V}}^{*}\right],$$

(4)

$$\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right]\times \left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]=\left[min\left\{{\mathsf{Ϗ}}_{*}{\mathrm{V}}_{*}, {\mathsf{Ϗ}}^{*}{\mathrm{V}}_{*}, {\mathsf{Ϗ}}_{*}{\mathrm{V}}^{*}, {\mathsf{Ϗ}}^{*}{\mathrm{V}}^{*}\right\}, max\left\{{\mathsf{Ϗ}}_{*}{\mathrm{V}}_{*}, {\mathsf{Ϗ}}^{*}{\mathrm{V}}_{*}, {\mathsf{Ϗ}}_{*}{\mathrm{V}}^{*}, {\mathsf{Ϗ}}^{*}{\mathrm{V}}^{*}\right\}\right],$$

(5)

$$\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right]-\left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]=\left[{\mathsf{Ϗ}}_{*}-{\mathrm{V}}^{*}, {\mathsf{Ϗ}}^{*}-{\mathrm{V}}_{*}\right].$$

(6)

Remark 1.

Ref. [37]. (i) For given $\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right], \left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]\in {\mathbb{E}}_C,$ the relation $“{\supseteq }_I”$ defined on ${\mathbb{E}}_C$ by $\left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]{\supseteq }_I\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right]$ if and only if ${\mathrm{V}}_{*}\le {\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}{\le \mathrm{V}}^{*}$ for all $\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right], \left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]\in {\mathbb{E}}_C$ is a partial interval inclusion relation. The relation $\left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]{\supseteq }_I\left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right]$ is coincident to $\left[{\mathrm{V}}_{*}, {\mathrm{V}}^{*}\right]\supseteq \left[{\mathsf{Ϗ}}_{*}, {\mathsf{Ϗ}}^{*}\right]$ on ${\mathbb{E}}_C.$ It can be easily seen that “${\supseteq }_I$ ” looks like “up and down” on the real line $\mathfrak{N},$ so we call $“{\supseteq }_I”$ “up and down” (or “$UD$” order, in short).

Let $\mathfrak{N}$ be the set of real numbers. A fuzzy subset $A$ of $\mathfrak{N}$ is characterized by a mapping $\widetilde{Ʀ}:\mathfrak{N}\to [0, 1]$ called the membership function, for each fuzzy set and $ɤ\in (0, 1]$; then, $ɤ$-level sets of $\widetilde{Ʀ}$ are denoted and defined as follows: ${Ʀ}_{ɤ}=\left\{\varkappa \in \mathfrak{N}|\widetilde{Ʀ}\left(\varkappa \right)\ge ɤ\right\}$. If $ɤ=0$, then $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(\widetilde{Ʀ}\right)=\left\{\varkappa \in \mathfrak{N}|\widetilde{Ʀ}\left(\varkappa \right)>0\right\}$ is called support of $\widetilde{Ʀ}$. By ${\left[\widetilde{Ʀ}\right]}^0$, we define the closure of $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\left(\widetilde{Ʀ}\right)$.

Definition 1.

Ref. [43]. A fuzzy set is said to be fuzzy number if

(1)

$\widetilde{Ʀ}$ is normal, i.e., there exists $\varkappa \in \mathfrak{N}$ such that $Ʀ\left(\varkappa \right)=1;$

(2)

$\widetilde{Ʀ}$ is upper semi-continuous, i.e., for a given $\varkappa \in \mathfrak{N},$ there exists $\epsilon >0$ there exists $\delta >0$ such that $\widetilde{Ʀ}\left(\varkappa \right)-\widetilde{Ʀ}\left(\vartheta \right)<\epsilon$ for all $\vartheta \in \mathfrak{N}$ with $\left|\varkappa -\vartheta \right|<\delta$;

(3)

$\widetilde{\mathrm{Ʀ}}$ is fuzzy convex, i.e., $\widetilde{\mathrm{Ʀ}}\left(\left(1-\tau \right)\varkappa +\tau \vartheta \right)\ge min\left(\widetilde{\mathrm{Ʀ}}\left(\varkappa \right),\widetilde{\mathrm{Ʀ}}\left(\vartheta \right)\right)$ $\forall$ $\varkappa ,\vartheta \in \mathfrak{N}$, $\tau \in [0,1]$;

(4)

${\left[\widetilde{Ʀ}\right]}^0$is compact.

Note that ${\Omega }_0$ denotes the set of all fuzzy numbers.

Proposition 1.

Ref. [41]. Let $\widetilde{Ʀ},\widetilde{Ư}\in {\Omega }_0$. Then relation $“{\supseteq }_{\mathbb{F}}”$ given on ${\Omega }_0$ by $\widetilde{Ʀ}{\supseteq }_{\mathbb{F}}\widetilde{Ư}$ when and only when, ${\left[\widetilde{Ʀ}\right]}^{ɤ}{\supseteq }_I{\left[\widetilde{Ư}\right]}^{ɤ}$, for every $ɤ\in [0, 1],$ it is an up and down relation.

If $\widetilde{Ʀ},\widetilde{Ư}\in {\Omega }_0$ and $\mathfrak{ɤ}\in \mathfrak{R}$, then, for every $g\in \left[0, 1\right],$ the definition of the arithmetic operations is as follows:

$${\left[\widetilde{Ʀ}\oplus \widetilde{Ư}\right]}^{ɤ}={\left[\widetilde{Ʀ}\right]}^{ɤ}+{\left[\widetilde{Ư}\right]}^{ɤ},$$

(7)

$${\left[\widetilde{Ʀ}\otimes \widetilde{Ư}\right]}^{ɤ}={\left[\widetilde{Ʀ}\right]}^{ɤ}\times {\left[ \widetilde{Ư}\right]}^{ɤ},$$

(8)

$${\left[g\odot \widetilde{Ʀ}\right]}^{ɤ}=g·{\left[\widetilde{Ʀ}\right]}^{ɤ}.$$

(9)

Theorem 1.

Ref. [32]. The space ${\Omega }_0$ dealing with a supremum metric, i.e., for $\widetilde{Ʀ}, \widetilde{Ư}\in {\Omega }_0$

$$d_{\infty }\left(\widetilde{Ʀ}, \widetilde{Ư}\right)=\underset{0\le ɤ\le 1}{\sup }{d}_H\left({\left[\widetilde{Ʀ}\right]}^{ɤ}, {\left[\widetilde{Ư}\right]}^{ɤ}\right),$$

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

Definition 2.

Ref. [37]. The $F$·$N$·$V$·$M$   $\tilde{\mathcal{Q}}:\left[\varsigma ,\sigma \right]\to {\Omega }_0$ is said to be convex $F$·$N$·$V$·$M$ on$[\varsigma ,\sigma ]$ if

$$\tilde{\mathcal{Q}}\left(\zeta \varkappa +\left(1-\zeta \right)\omega \right){\supseteq }_{\mathbb{F}}\zeta \odot \tilde{\mathcal{Q}}\left(\varkappa \right)\oplus \left(1-\zeta \right)\odot \tilde{\mathcal{Q}}\left(\omega \right),$$

(10)

for all$\varkappa , \omega \in [\varsigma ,\sigma ], \zeta \in \left[0, 1\right],$ where $\mathcal{Q}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$. If $\tilde{\mathcal{Q}}$ is concave $F$·$N$·$V$·$M$ on $[\varsigma ,\sigma ]$, then inequality (10) is reversed.

Definition 3.

Ref. [41]. The $F$·$N$·$V$·$M$   $\tilde{\mathcal{Q}}:[\varsigma ,\sigma ]\to {\Omega }_0$ is said to be $U$·$D$-preinvex $F$·$N$·$V$·$M$ on $[\varsigma ,\sigma ]$ if

$$\tilde{\mathcal{Q}}\left(\varkappa +\left(1-\zeta \right)\pi \left(\omega ,\varkappa \right)\right){\supseteq }_{\mathbb{F}}\zeta \odot \tilde{\mathcal{Q}}\left(\varkappa \right)\oplus \left(1-\zeta \right)\odot \tilde{\mathcal{Q}}\left(\omega \right),$$

(11)

for all $\varkappa ,\omega \in [\varsigma ,\sigma ],\zeta \in \left[0,1\right],$ where $\tilde{\mathcal{Q}}\left(\varkappa \right){\ge }_{\mathbb{F}}\tilde{0}$ and $\pi :[\varsigma ,\sigma ]\times [\varsigma ,\sigma ]\to [\varsigma ,\sigma ]$ If $\tilde{\mathcal{Q}}$ is $\left({\eta }_1,{\eta }_2\right)$-concave on $[\varsigma ,\sigma ]$, then inequality (11) is reversed.

Condition 1.

(see Ref. [15]) Let $K$ be an invex set with respect to $\pi .$ For any $\varkappa , \omega \in K$ and $\zeta \in \left[0, 1\right]$,

$$\pi \left(\varkappa ,\varkappa +\zeta \pi (\omega ,\varkappa )\right)=-\zeta \pi \left(\omega ,\varkappa \right),$$

$$\pi \left(\omega ,\varkappa +\zeta \pi (\omega ,\varkappa )\right)=\left(1-\zeta \right)\pi \left(\omega ,\varkappa \right).$$

Clearly, for $\zeta$ = 0, we have $\pi \left(\omega ,\varkappa \right)$ = 0 if and only if,$\omega =\varkappa$, for all $\varkappa , \omega \in K$. For the applications of Condition 1, see [27,40,41].

Theorem 2.

Ref. [27]. If $\mathcal{Q}:[\varsigma ,\sigma ]\subset \mathfrak{R}\to {\mathfrak{R}}_I$ is an $I$·$V$·$M$ offered by $\mathcal{Q}\left(\phi \right)$ $\left[{\mathcal{Q}}_{*}\left(\phi \right), {\mathcal{Q}}^{*}\left(\phi \right)\right]$, then $\mathcal{Q}$ is Aumaan’s integrable over $[\varsigma ,\sigma ]$ if and only if, ${\mathcal{Q}}_{*}$ and ${\mathcal{Q}}^{*}$ both are Lebesgue-integrable over $\left[\varsigma ,\sigma \right]$ such that

$$\left(IA\right){\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(\phi \right)d\phi =\left[\left(A\right){\int }_{\varsigma }^{\sigma }{\mathcal{Q}}_{*}\left(\phi \right)d\phi , \left(A\right){\int }_{\varsigma }^{\begin{array}{c} \\ \sigma \end{array}}{\mathcal{Q}}^{*}\left(\phi \right)d\phi \right].$$

The collection of all Lebesgue-integrable real valued functions and Lebesgue-integrable $I$·$V$·$M$ is denoted by ${\mathcal{A}}_{[\varsigma , \sigma ]}$ and ${\mathfrak{T}\mathcal{A}}_{[\varsigma , \sigma ]},$ respectively.

Definition 4.

Ref. [39]. Let $\mathcal{Q}:[ⱱ, ⱳ]\subset \mathfrak{R}\to {\Omega }_0$ is $F$·$N$·$V$·$M$. The fuzzy Aumaan’s integral ($\left(FA\right)$-integral) of $\mathcal{Q}$ over $\left[ⱱ, ⱳ\right],$ denoted by $\left(FA\right){\int }_{ⱱ}^{ⱳ}\mathcal{Q}\left(\varkappa \right)d\varkappa$, is defined level-wise by

$${\left[\left(FA\right){\int }_{ⱱ}^{ⱳ}\mathcal{Q}\left(\varkappa \right)d\varkappa \right]}^{\begin{array}{c} \\ ɤ\end{array}}=\left(IA\right){\int }_{ⱱ}^{ⱳ}{\mathcal{Q}}_{ɤ}\left(\varkappa \right)d\varkappa =\left\{{\int }_{ⱱ}^{ⱳ}\mathcal{Q}\left(\varkappa ,ɤ\right)d\varkappa :\mathcal{Q}\left(\varkappa ,ɤ\right)\in S\left({\mathcal{Q}}_{ɤ}\right)\right\},$$

where $S\left({\mathcal{Q}}_{\mathrm{ɤ}}\right)=\left\{\mathcal{Q}\left(.,\mathrm{ɤ}\right)\to \mathfrak{R}:\mathcal{Q}\left(.,\mathrm{ɤ}\right) is integrable and \mathcal{Q}\left(\varkappa ,\mathrm{ɤ}\right)={\mathcal{Q}}_{\mathrm{ɤ}}\left(\varkappa \right)\right\}$, for every $\mathrm{ɤ}\in \left[0,1\right]$. $\mathcal{Q}$ is $\left(FA\right)$-integrable over $[\mathrm{ⱱ},\mathrm{ⱳ}]$ if $\left(FA\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}\mathcal{Q}\left(\varkappa \right)d\varkappa \in {\mathfrak{R}}_I.$

Note that Theorem 3 is also true for interval double integrals. The collection of all double integrable $I$·$V$·$M$ is denoted by ${\mathfrak{T}\mathfrak{O}}_{∆},$ respectively.

Theorem 3.

Ref. [36]. Let $∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right]$. If $\mathcal{Q}:∆\to {\mathfrak{R}}_I$ is $ID$-integrable on $∆$, then we have

$$\left(ID\right){\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(\phi ,\omega \right)d\omega d\phi =\left(IA\right){\int }_{ⱱ}^{ⱳ}\left(IA\right){\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(\phi ,\omega \right)d\omega d\phi .$$

(12)

Definition 5.

Ref. [37]. A fuzzy-interval-valued map$\tilde{\mathcal{Q}}:∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right]\to {\Omega }_0$ is called $F$·$N$·$V$·$M$ on coordinates. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:∆\subset {\mathfrak{R}}^2\to {\mathfrak{R}}_I$ on coordinates and offered by ${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ for all $\left(\phi ,\omega \right)\in ∆$, where ${\mathcal{Q}}_{*}\left(.,ɤ\right), {\mathcal{Q}}^{*}\left(.,ɤ\right):\left(\phi ,\omega \right)\to \mathfrak{R}$ are called lower and upper functions of ${\mathcal{Q}}_{ɤ}$.

Definition 6.

Ref. [37]. Let $\tilde{\mathcal{Q}}:∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right]\subset {\mathfrak{R}}^2\to {\Omega }_0$ be a coordinated $F$·$N$·$V$·$M$. Then, $\tilde{\mathcal{Q}}\left(\phi ,\omega \right)$ is said to be continuous at $\left(\phi ,\omega \right)\in ∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right],$ if for each $ɤ\in \left[0, 1\right],$ both end point functions ${\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right)$ and ${\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)$ are continuous at $\left(\phi ,\omega \right)\in ∆.$

Definition 7.

Ref. [37]. Let $\tilde{\mathcal{Q}}:∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right]\subset {\mathfrak{R}}^2\to {\Omega }_0$ be a $F$·$N$·$V$·$M$ on coordinates. Then, fuzzy double integral of $\tilde{\mathcal{Q}}$ over $∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right],$ denoted by $\left(FD\right){\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi$, it is defined level-wise by

$$\begin{array}{ll}{\left[\left(FD\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \right]}^{\mathrm{ɤ}}& =\left(ID\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}{\int }_{\varsigma }^{\sigma }{\mathcal{Q}}_{\mathrm{ɤ}}\left(\phi ,\omega \right)d\omega d\phi \\ & =\left(IA\right){\int }_{\mathrm{ⱱ}}^{\mathrm{ⱳ}}\left(IA\right){\int }_{\varsigma }^{\sigma }{\mathcal{Q}}_{\mathrm{ɤ}}\left(\phi ,\omega \right)d\omega d\phi ,\end{array}$$

(13)

for all $ɤ\in \left[0, 1\right],$ $\tilde{\mathcal{Q}}$ is $FD$-integrable over $∆$ if $\left(FD\right){\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \in {\Omega }_0.$ Note that if end point functions are Lebesgue-integrable, then $\tilde{\mathcal{Q}}$ is fuzzy double Aumann-integrable function over $∆$.

Theorem 4.

Ref. [37]. Let $\tilde{\mathcal{Q}}:∆\subset {\mathfrak{R}}^2\to {\Omega }_0$ be a $F$·$N$·$V$·$M$ on coordinates. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:∆\subset {\mathfrak{R}}^2\to {\mathfrak{R}}_I$ and offered by ${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ for all $\left(\phi ,\omega \right)\in ∆=\left[ⱱ, ⱳ\right]\times \left[\varsigma , \sigma \right]$ and for all $ɤ\in \left[0, 1\right].$ Then, $\tilde{\mathcal{Q}}$ is $FD$-integrable over $∆$ if and only if, ${\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right)$ and ${\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)$ both are $D$-integrable over $∆.$ Moreover, if $\tilde{\mathcal{Q}}$ is $FD$-integrable over $∆,$ then

$$\begin{array}{ll}{\left[\left(FD\right){\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \right]}^{\begin{array}{c} \\ ɤ\end{array}}& ={\left[\left(FA\right){\int }_{ⱱ}^{ⱳ}\left(FA\right){\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \right]}^{ɤ}\\ & =\left(IA\right){\int }_{ⱱ}^{ⱳ}\left(IA\right){\int }_{\varsigma }^{\sigma }{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ & =\left(ID\right){\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi ,\end{array}$$

(14)

for all $ɤ\in \left[0, 1\right].$

The family of all $FD$-integrable $F$·$N$·$V$·$M$s over coordinates is denoted by ${\mathcal{F}\mathfrak{O}}_{∆}$ for all $ɤ\in \left[0, 1\right].$

Theorem 5.

Ref. [37] Let $\tilde{\mathcal{Q}},\tilde{\mathcal{T}}: \left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right]\to {\Omega }_0$ be two $U$·$D$ preinvex $F$·$N$·$V$·$M$s. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}, {\mathcal{T}}_{ɤ}:\left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right]\subset \mathfrak{R}\to {\mathfrak{R}}_I^{+}$ and offered by ${\mathcal{Q}}_{ɤ}\left(\phi \right)=\left[{\mathcal{Q}}_{*}\left(\phi ,ɤ\right), {\mathcal{Q}}^{*}\left(\phi ,ɤ\right)\right]$ and ${\mathcal{T}}_{ɤ}\left(\phi \right)=\left[{\mathcal{T}}_{*}\left(\phi ,ɤ\right), {\mathcal{T}}^{*}\left(\phi ,ɤ\right)\right]$ for all $\phi \in \left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right]$ and for all $ɤ\in \left[0, 1\right]$. If$\tilde{\mathcal{Q}}\otimes \tilde{\mathcal{T}}$ is fuzzy Riemann integrable, then

$$\frac{1}{\pi \left(\sigma ,\varsigma \right)}\odot \left(FA\right){\int }_{\varsigma }^{\varsigma +\pi \left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi \right)\otimes \tilde{\mathcal{T}}\left(\phi \right)d\phi {\supseteq }_{\mathbb{F}}\frac{1}{3}\odot \tilde{\beta }\left(\varsigma ,\sigma \right)\oplus \frac{1}{6}\odot \tilde{\gamma }\left(\varsigma ,\sigma \right).$$

(15)

and,

$$\begin{gathered} 2\odot \tilde{\mathcal{Q}}\left(\frac{2\varsigma +\pi \left(\sigma ,\varsigma \right)}{2}\right)\otimes \tilde{\mathcal{T}}\left(\frac{2\varsigma +\pi \left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\frac{1}{\pi \left(\sigma ,\varsigma \right)}\odot \left(FA\right){\int }_{\varsigma }^{\varsigma +\pi \left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi \right)\otimes \tilde{\mathcal{T}}\left(\phi \right)d\phi \oplus \frac{1}{6}\odot \tilde{\beta }\left(\varsigma ,\sigma \right)\oplus \frac{1}{3}\odot \tilde{\gamma }\left(\varsigma ,\sigma \right). \end{gathered}$$

(16)

where $\tilde{\beta }\left(\varsigma ,\sigma \right)=\tilde{\mathcal{Q}}\left(\varsigma \right)\otimes \tilde{\mathcal{T}}\left(\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(\sigma \right)\otimes \tilde{\mathcal{T}}\left(\sigma \right),$ $\tilde{\gamma }\left(\varsigma ,\sigma \right)=\tilde{\mathcal{Q}}\left(\varsigma \right)\otimes \tilde{\mathcal{T}}\left(\sigma \right)\oplus \tilde{\mathcal{Q}}\left(\sigma \right)\otimes \tilde{\mathcal{T}}\left(\varsigma \right),$ and ${\beta }_{\mathrm{ɤ}}\left(\varsigma ,\sigma \right)=\left[{\beta }_{*}\left(\left(\varsigma ,\sigma \right),\mathrm{ɤ}\right),{\beta }^{*}\left(\left(\varsigma ,\sigma \right),\mathrm{ɤ}\right)\right]$ and ${\gamma }_{\mathrm{ɤ}}\left(\varsigma ,\sigma \right)=\left[{\gamma }_{*}\left(\left(\varsigma ,\sigma \right),\mathrm{ɤ}\right),{\gamma }^{*}\left(\left(\varsigma ,\sigma \right),\mathrm{ɤ}\right)\right].$

Theorem 6.

Ref. [37] Let $\tilde{\mathcal{Q}}:\left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right]\to {\Omega }_0$ be an $U$·$D$ preinvex $F$·$N$·$V$·$M$ with $\varsigma <\varsigma +\pi \left(\sigma ,\varsigma \right)$. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:\left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right]\subset \mathfrak{R}\to {\mathfrak{R}}_I^{+}$ and offered by ${\mathcal{Q}}_{ɤ}\left(\phi \right)=\left[{\mathcal{Q}}_{*}\left(\phi ,ɤ\right), {\mathcal{Q}}^{*}\left(\phi ,ɤ\right)\right]$ for all $\phi \in \left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right]$ and for all $ɤ\in \left[0, 1\right]$, and Condition 1 for $\pi$ holds. If $\tilde{\mathcal{Q}}\in {\mathcal{F}\mathcal{A}}_{\left(\left[\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)\right], ɤ\right)}$ and $\mathsf{ƕ}:\left[\varsigma , \sigma \right]\to \mathfrak{R},\mathsf{ƕ}\left(\phi \right)\ge 0,$ symmetric with respect to $\frac{2\varsigma +\pi \left(\sigma ,\varsigma \right)}{2},$ and ${\int }_{\varsigma }^{\varsigma , \varsigma +\pi \left(\sigma ,\varsigma \right)}\mathsf{ƕ}\left(\phi \right)d\phi >0$, then

$$\tilde{\mathcal{Q}}\left(\frac{2\varsigma +\pi \left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{\varsigma }^{\varsigma +\pi \left(\sigma ,\varsigma \right)}\mathsf{ƕ}\left(\phi \right)d\phi }\odot \left(FA\right){\int }_{\varsigma }^{\varsigma +\pi \left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi \right)\mathsf{ƕ}\left(\phi \right)d\phi {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(\sigma \right)}{2}.$$

(17)

If $\mathcal{Q}$ is $U$·$D$-preincave $F$·$N$·$V$·$M$, then inequality (17) is reversed.

Note that if $\mathsf{ƕ}\left(\phi \right)=1$, then we acquire the following inequality:

$$\tilde{\mathcal{Q}}\left(\frac{2\varsigma +\pi \left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{\pi \left(\sigma ,\varsigma \right)}\odot \left(FA\right){\int }_{\varsigma }^{\varsigma +\pi \left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi \right)d\phi {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(\sigma \right)}{2}.$$

(18)

Coordinated $U$·$D$ preinvex fuzzy-interval-valued functions

Definition 8.

The $F$·$N$·$V$·$M$   $\tilde{\mathcal{Q}}:∆\to {\Omega }_0$is said to be coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on$∆$ if

$$\begin{gathered} \tilde{\mathcal{Q}}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right) \\ {\supseteq }_{\mathbb{F}}\zeta \mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \zeta \left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \left(1-\zeta \right)\mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right), \end{gathered}$$

(19)

for all $\left(ⱱ, ⱳ\right), \left(\varsigma ,\sigma \right)\in ∆,$ and $\zeta ,\mathsf{ϛ}\in \left[0, 1\right],$ where $\tilde{\mathcal{Q}}\left(\phi \right){\ge }_{\mathbb{F}}\tilde{0}.$ If inequality (19) is reversed, then $\tilde{\mathcal{Q}}$ is called coordinated concave $F$·$N$·$V$·$M$ on $∆$.

The proof of Lemma 1 is straightforward and will be omitted here.

Lemma 1.

Let $\tilde{\mathcal{Q}}:∆\to {\Omega }_0$ be a coordinated $F$·$N$·$V$·$M$ on $∆$. Then, $\tilde{\mathcal{Q}}$ is coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆,$ if and only if there exist two coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s ${\tilde{\mathcal{Q}}}_{\phi }:\left[\varsigma ,\sigma \right]\to {\Omega }_0$, ${\tilde{\mathcal{Q}}}_{\phi }\left(w\right)=\tilde{\mathcal{Q}}\left(\phi ,w\right)$ and ${\tilde{\mathcal{Q}}}_{\omega }:\left[ⱱ,ⱳ\right]\to {\Omega }_0$, ${\tilde{\mathcal{Q}}}_{\omega }\left(q\right)=\tilde{\mathcal{Q}}\left(q,\omega \right)$.

Proof. 

From the Definition 8 of coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$, it can be easily proved.  □

From Lemma 1, we can easily note that each $U$·$D$ preinvex $F$·$N$·$V$·$M$ is coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$. But the converse is not true, see Example 1.

Theorem 7.

Let $\tilde{\mathcal{Q}}:∆\to {\Omega }_0$ be a $F$·$N$·$V$·$M$ on $∆$. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:∆\to {\mathfrak{R}}_I^{+}\subset {\mathfrak{R}}_I$ and offered by

$${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right],$$

(20)

for all $\left(\phi ,\omega \right)\in ∆$ and for all $ɤ\in \left[0, 1\right]$. Then, $\tilde{\mathcal{Q}}$ is coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆,$ if and only if, for all $ɤ\in \left[0, 1\right],$ ${\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right), ɤ\right)$ and ${\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right), ɤ\right)$ are a coordinated preinvex function.

Proof. 

Assume that for each $ɤ\in \left[0, 1\right],$ ${\mathcal{Q}}_{*}\left(\phi , ɤ\right)$ and ${\mathcal{Q}}^{*}\left(\phi , ɤ\right)$ are coordinated preinvex on $∆.$ Then, from (19), for all$\left(ⱱ, ⱳ\right), \left(\varsigma ,\sigma \right)\in ∆, \zeta$ and $\mathsf{ϛ}\in \left[0, 1\right]$ we have

$$\begin{gathered} {\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right) \\ \le \zeta {\mathcal{ϛ}\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)+\zeta \left(1-\mathsf{ϛ}\right){\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right) \\ +{\mathsf{ϛ}\left(1-\zeta \right)\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right){\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right), \end{gathered}$$

and

$$\begin{gathered} {\mathcal{Q}}^{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right) \\ \ge \zeta \mathsf{ϛ}{\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)+\zeta \left(1-\mathsf{ϛ}\right){\mathcal{Q}}^{*}\left(\left(ⱱ,\sigma \right), ɤ\right) \\ +\mathsf{ϛ}\left(1-\zeta \right){\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right){\mathcal{Q}}^{*}\left(\left(ⱱ,\sigma \right), ɤ\right), \end{gathered}$$

 □

Then, by (19), (7), and (9), we obtain

$$\begin{gathered} {\mathcal{Q}}_{ɤ}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right)\right) \\ =\left[{\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right)\right] \\ {\supseteq }_I\zeta \mathsf{ϛ}\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)\right]+\zeta \left(1-\mathsf{ϛ}\right)\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right), {\mathcal{Q}}^{\begin{array}{c} \\ \\ \begin{array}{c} \\ *\end{array}\end{array}}\left(\left(ⱱ,\sigma \right), ɤ\right)\right] \\ +\mathsf{ϛ}\left(1-\zeta \right)\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)\right]+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\sigma \right), ɤ\right)\right]. \end{gathered}$$

That is

$$\begin{gathered} \tilde{\mathcal{Q}}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right) \\ {\supseteq }_{\mathbb{F}}\zeta \mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \zeta \left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \left(1-\zeta \right)\mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right), \end{gathered}$$

hence, $\tilde{\mathcal{Q}}$ is coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆$.

Conversely, let $\tilde{\mathcal{Q}}$ be coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆.$ Then, for all$\left(ⱱ, ⱳ\right), \left(\varsigma ,\sigma \right)\in ∆,$ $\zeta$,$\mathsf{ϛ}\in \left[0, 1\right],$ we have

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\mathrm{ⱱ}+\left(1-\zeta \right){\pi }_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right),\varsigma +\left(1-\mathrm{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right) \\ {\supseteq }_{\mathbb{F}}\zeta \mathrm{ϛ}\odot \tilde{\mathcal{Q}}\left(\mathrm{ⱱ},\varsigma \right)\oplus \zeta \left(1-\mathrm{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(\mathrm{ⱱ},\sigma \right)\oplus \left(1-\zeta \right)\mathrm{ϛ}\odot \tilde{\mathcal{Q}}\left(\mathrm{ⱳ},\varsigma \right)\oplus \left(1-\zeta \right)\left(1-\mathrm{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(\mathrm{ⱳ},\sigma \right) \end{gathered}$$

Therefore, again from (20), for each $ɤ\in \left[0, 1\right]$, we have

Again, (7) and (9), we obtain

$$\begin{gathered} \zeta \mathsf{ϛ}{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+\zeta \left(1-\mathsf{ϛ}\right){\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+\left(1-\zeta \right)\mathsf{ϛ}{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right){\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right) \\ =\zeta \mathsf{ϛ}\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)\right]+\zeta \left(1-\mathsf{ϛ}\right)\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\sigma \right), ɤ\right)\right] \\ +\mathsf{ϛ}\left(1-\zeta \right)\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)\right] \\ +\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\left[{\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right), {\mathcal{Q}}^{*}\left(\left(ⱱ,\sigma \right), ɤ\right)\right], \end{gathered}$$

for all $\left(\phi ,\omega \right)\in ∆$ and $\zeta \in \left[0, 1\right].$ Then, by coordinated preinvexity of $\tilde{\mathcal{Q}}$, we have for all $\left(\phi ,\omega \right)\in ∆$ and $\zeta \in \left[0, 1\right]$such that

$$\begin{gathered} {\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right) \\ \le \zeta \mathsf{ϛ}{\mathcal{Q}}_{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)+\zeta \left(1-\mathsf{ϛ}\right){\mathcal{Q}}_{*}\left(\left(ⱱ,\sigma \right), ɤ\right)+\left(1-\zeta \right)\mathsf{ϛ}{\mathcal{Q}}_{*}\left(\left(ⱳ,\varsigma \right), ɤ\right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right){\mathcal{Q}}_{*}\left(\left(ⱳ,\sigma \right), ɤ\right), \end{gathered}$$

and

$$\begin{array}{ll}{\mathcal{Q}}^{*}((ⱱ+\left(1-\zeta \right)& {\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)), ɤ)\\ & \ge \zeta \mathsf{ϛ}{\mathcal{Q}}^{*}\left(\left(ⱱ,\varsigma \right), ɤ\right)+\zeta \left(1-\mathsf{ϛ}\right){\mathcal{Q}}^{*}\left(\left(ⱱ,\sigma \right), ɤ\right)\\ & +\left(1-\zeta \right)\mathsf{ϛ}{\mathcal{Q}}^{*}\left(\left(ⱳ,\varsigma \right), ɤ\right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right){\mathcal{Q}}^{*}\left(\left(ⱳ,\sigma \right), ɤ\right),\end{array}$$

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

Remark 2.

If one takes ${\pi }_1\left(ⱳ,ⱱ\right)=ⱳ-ⱱ$ and ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$, then $\tilde{\mathcal{Q}}$ is known as convex $F$·$N$·$V$·$M$ on coordinates if $\tilde{\mathcal{Q}}$ satisfies the coming inequality

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\zeta ⱱ+\left(1-\zeta \right)ⱳ, \mathsf{ϛ}\varsigma +\left(1-\mathsf{ϛ}\right)\sigma \right) \\ {\supseteq }_{\mathbb{F}}\zeta \mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \zeta \left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \left(1-\zeta \right)\mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right), \end{gathered}$$

(21)

is valid, which is defined by Khan et al. [39].

If one takes ${\mathcal{Q}}_{*}\left(\phi ,\omega \right)={\mathcal{Q}}^{*}\left(\phi ,\omega \right)$ with $ɤ=1$, then $\mathcal{Q}$ is known as a preinvex function on coordinates if $\mathcal{Q}$ satisfies the coming inequality

$$\begin{gathered} \mathcal{Q}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_1\left(\sigma ,\varsigma \right)\right) \\ \le \zeta \mathsf{ϛ}\mathcal{Q}\left(ⱱ,\varsigma \right)+\zeta \left(1-\mathsf{ϛ}\right)\mathcal{Q}\left(ⱱ,\sigma \right)+\left(1-\zeta \right)\mathsf{ϛ}\mathcal{Q}\left(ⱳ,\varsigma \right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\mathcal{Q}\left(ⱳ,\sigma \right), \end{gathered}$$

(22)

is valid, which is defined by Latif and Dragomir [40].

If one takes ${\mathcal{Q}}_{*}\left(\phi ,\omega \right)={\mathcal{Q}}^{*}\left(\phi ,\omega \right)$ with $ɤ=1$, then $\mathcal{Q}$ is known as a convex function on coordinates if $\mathcal{Q}$ satisfies the coming inequality

$$\begin{gathered} \mathcal{Q}\left(\zeta ⱱ+\left(1-\zeta \right)ⱳ, \mathsf{ϛ}\varsigma +\left(1-\mathsf{ϛ}\right)\sigma \right) \\ \le \zeta \mathsf{ϛ}\mathcal{Q}\left(ⱱ,\varsigma \right)+\zeta \left(1-\mathsf{ϛ}\right)\mathcal{Q}\left(ⱱ,\sigma \right)+\left(1-\zeta \right)\mathsf{ϛ}\mathcal{Q}\left(ⱳ,\varsigma \right)+\left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\mathcal{Q}\left(ⱳ,\sigma \right), \end{gathered}$$

(23)

is valid, then $\mathcal{Q}$ is named as IVF on coordinates, which is defined by Dragomir [39].

Example 1.

We consider the $F$·$N$·$V$·$M$s $\tilde{\mathcal{Q}}:\left[0, 1\right]\times \left[0, 1\right]\to {\Omega }_0$ defined as

$$\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\left(m\right)=\left\{\begin{array}{c}\frac{\begin{array}{c} \\ m-\phi \omega \end{array}}{5-\phi \omega } \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m\in \left[\phi \omega , 5\right]\\ \frac{\left(6+e^{\phi }\right)\left(6+e^{\omega }\right)-m}{\left(6+e^{\phi }\right)\left(6+e^{\omega }\right)-5}, m\in \left(5, \left(6+e^{\phi }\right)\left(6+e^{\omega }\right)\right]\\ 0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}otherwise,\end{array}\right.$$

(24)

and then, for each $\mathrm{ɤ}\in \left[0,1\right],$ we obtain ${\mathcal{Q}}_{\mathrm{ɤ}}\left(\phi ,\omega \right)=\left[\left(1-\mathrm{ɤ}\right)\phi \omega +5\mathrm{ɤ},\left(1-\mathrm{ɤ}\right)\left(6+e^{\phi }\right)\left(6+e^{\omega }\right)+5\mathrm{ɤ}\right]$. The end-point functions ${\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),\mathrm{ɤ}\right)$and ${\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),\mathrm{ɤ}\right)$ are coordinated preinvex and preincave functions with respect to ${\pi }_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)=\mathrm{ⱳ}-\mathrm{ⱱ}$ and ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$, for each $\mathrm{ɤ}\in \left[0,1\right],respectively$. Hence, $\tilde{\mathcal{Q}}\left(\phi ,\omega \right)$ is an up and down coordinated preinvex $F$·$N$·$V$·$M$.

From Example 1, it can be easily seen that each coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ is not a preinvex $F$·$N$·$V$·$M$.

Theorem 8.

Let $∆$ be an invex set, and let $\tilde{\mathcal{Q}}:∆\to {\Omega }_0$ be a $F$·$N$·$V$·$M$. Then, from $ɤ$-levels, we obtain the collection of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:∆\to {\mathfrak{R}}_I^{+}\subset {\mathfrak{R}}_I$ and offered by

$${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right],$$

(25)

for all $\left(\phi ,\omega \right)\in ∆$ and for all $ɤ\in \left[0, 1\right]$. Then, $\tilde{\mathcal{Q}}$ is coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆,$ if and only if, for all $ɤ\in \left[0, 1\right],$ ${\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right), ɤ\right)$ and ${\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right), ɤ\right)$ are coordinated preinvex functions.

Proof. 

The proof of Theorem 8 is similar to that of Theorem 7.  □

In the next results, to avoid confusion, we will not include the symbols $\left(A\right)$, $\left(IA\right)$, $\left(FA\right)$, $\left(ID\right)$, and $\left(FD\right)$ before the integral sign.

3. Main Outcomes

In this section, new H·H-type inequalities are obtained in the following, and the results presented in the recent literature follow from the aforementioned generalization in the $F$·$N$·$V$·$M$ sense and validated with the support of nontrivial examples.

Theorem 9.

Let $\tilde{\mathcal{Q}}:∆=\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\to {\Omega }_0$ be a coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆$. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:∆\to {\mathfrak{R}}_I^{+}$ and offered by ${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ for all $\left(\phi ,\omega \right)\in ∆$ and for all $ɤ\in \left[0, 1\right]$, and Condition 1 for ${\pi }_1$ and ${\pi }_2$ holds. Next, the inequality that follows is true:

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \oplus \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\odot \left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\odot \left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(26)

If $\tilde{\mathcal{Q}}\left(\phi \right)$ coordinated $U$·$D$ preincave $F$·$N$·$V$·$M$, then inequality (26) is reversed such that

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\subseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \oplus \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right] \\ {\subseteq }_{\mathbb{F}}\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \\ {\subseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\odot \left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\odot \left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\subseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(27)

Proof. 

Let $\tilde{\mathcal{Q}}:\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\to {\Omega }_0$ be a coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$. Then, by the hypothesis, we have

$$\begin{gathered} 4\odot \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\tilde{\mathcal{Q}}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right)\oplus \tilde{\mathcal{Q}}\left(ⱳ+\zeta {\pi }_1\left(ⱳ,ⱱ\right), \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right).\hspace{1em}\square \end{gathered}$$

By using Theorem 7, for every $ɤ\in \left[0, 1\right]$, we have

$$\begin{array}{c}4{\mathcal{Q}}_{*}\left(\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right), ɤ\right)\le {\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right)\\ +{\mathcal{Q}}_{*}\left(\left(ⱳ+\zeta {\pi }_1\left(ⱳ,ⱱ\right), \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right), \\ 4{\mathcal{Q}}^{*}\left(\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right), ɤ\right)\ge {\mathcal{Q}}^{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right)\\ +{\mathcal{Q}}^{*}\left(\left(ⱳ+\zeta {\pi }_1\left(ⱳ,ⱱ\right), \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right),ɤ\right).\end{array}$$

By using Lemma 1, we have

$$\begin{array}{c}2{\mathcal{Q}}_{*}\left(\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right), ɤ\right)\le {\mathcal{Q}}_{*}\left(\left(\phi , \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right)+{\mathcal{Q}}_{*}\left(\left(\phi , \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right), \\ 2{\mathcal{Q}}^{*}\left(\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right), ɤ\right)\ge {\mathcal{Q}}^{*}\left(\left(\phi , \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right)+{\mathcal{Q}}^{*}\left(\left(\phi , \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right),ɤ\right),\end{array}$$

(28)

and

$$\begin{array}{c}2{\mathcal{Q}}_{*}\left(\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right), ɤ\right)\le {\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right), ɤ\right)+{\mathcal{Q}}_{*}\left(\left(\sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right), \omega \right), ɤ\right), \\ 2{\mathcal{Q}}^{*}\left(\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right), ɤ\right)\ge {\mathcal{Q}}^{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right), ɤ\right)+{\mathcal{Q}}^{*}\left(\left(\sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right), \omega \right),ɤ\right).\end{array}$$

(29)

From (28) and (29), we have

$$\begin{gathered} 2\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right), ɤ\right),{\mathcal{Q}}^{*}\left(\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right), ɤ\right)\right] \\ {\supseteq }_I\left[{\mathcal{Q}}_{*}\left(\left(\phi , \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right),{\mathcal{Q}}^{*}\left(\left(\phi , \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right)\right] \\ +\left[{\mathcal{Q}}_{*}\left(\left(\phi , \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right), ɤ\right),{\mathcal{Q}}^{*}\left(\left(\phi , \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right),ɤ\right)\right], \end{gathered}$$

and

$$\begin{gathered} 2\left[{\mathcal{Q}}_{*}\left(\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right), ɤ\right),{\mathcal{Q}}^{*}\left(\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right), ɤ\right)\right] \\ {\supseteq }_I\left[{\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right), ɤ\right),{\mathcal{Q}}^{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right), ɤ\right)\right] \\ +\left[{\mathcal{Q}}_{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right), ɤ\right),{\mathcal{Q}}^{*}\left(\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right),ɤ\right)\right] \end{gathered}$$

It follows that

$${\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I{\mathcal{Q}}_{ɤ}\left(\phi , \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right)+{\mathcal{Q}}_{ɤ}\left(\phi , \sigma +\mathsf{ϛ}{\pi }_2\left(\sigma ,\varsigma \right)\right),$$

(30)

and

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right){\supseteq }_I{\mathcal{Q}}_{ɤ}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ+\zeta {\pi }_1\left(ⱳ,ⱱ\right), \omega .\right)$$

(31)

Since ${\mathcal{Q}}_{ɤ}\left(\phi ,.\right)$ and ${\mathcal{Q}}_{ɤ}\left(.,\omega \right)$, both are coordinated $U$·$D$ preinvex-$I$·$V$·$M$s, then from inequality (18), for every $ɤ\in \left[0, 1\right]$, inequality (30) and (31) we have

$${\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega {\supseteq }_I \frac{\begin{array}{c} \\ {\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\end{array}}{2}.$$

(32)

and

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right){\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\phi {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)}{2}.$$

(33)

Dividing double inequality (31) by ${\pi }_1\left(ⱳ,ⱱ\right)$, and integrating with respect to $\phi$ over $\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right],$ we have

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{2{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)d\phi \right]. \end{gathered}$$

(34)

Similarly, dividing double inequality (33) by ${\pi }_2\left(\sigma ,\varsigma \right)$, and integrating with respect to $\phi$ over $\left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right],$ we have

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{2{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)d\omega \right]. \end{gathered}$$

(35)

By adding (34) and (35), we have

$$\begin{gathered} \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi +\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right] \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)d\phi \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)d\omega \right]. \end{gathered}$$

(36)

Since $\mathcal{Q}$ is $F$·$N$·$V$·$M$, then inequality (36), we have

$$\begin{gathered} \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \oplus \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\odot \left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\odot \left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right]. \end{gathered}$$

(37)

From the left side of inequality (18), for each $ɤ\in \left[0, 1\right]$, we have

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi ,$$

(38)

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega .$$

(39)

Taking the addition of inequality (38) with inequality (39), we have

$$\begin{gathered} {\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi +\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right]. \end{gathered}$$

Since $\tilde{\mathcal{Q}}$ is a $F$·$N$·$V$·$M$, then it follows that

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \oplus \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right]. \end{gathered}$$

(40)

Now, from the right side of inequality (18), for every $ɤ\in \left[0, 1\right]$, we have

$$\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)d\phi {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)}{2},$$

(41)

$$\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)d\phi {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)}{2},$$

(42)

$$\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)d\omega {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)}{2},$$

(43)

$$\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)d\omega {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)}{2}.$$

(44)

By adding inequalities (41)–(44), we have

$$\begin{gathered} \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)d\phi \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

Since $\mathcal{Q}$ is a $F$·$N$·$V$·$M$, then it follows that

$$\begin{gathered} \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\odot \left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\odot \left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}, \end{gathered}$$

(45)

we obtain the desired conclusion by combining inequalities (37), (40), and (45).

Remark 3.

If one takes ${\pi }_1\left(ⱳ,ⱱ\right)=ⱳ-ⱱ$ and ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$, then from (39), we acquire the coming inequality, see [39]:

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{ⱱ+ⱳ}{2}, \frac{\varsigma +\sigma }{2}\right){\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{ⱳ-ⱱ}\odot {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\phi ,\frac{\varsigma +\sigma }{2}\right)d\phi \oplus \frac{1}{\sigma -\varsigma }\odot {\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\frac{ⱱ+ⱳ}{2},\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{1}{\left(ⱳ-ⱱ\right)\left(\sigma -\varsigma \right)}\odot {\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4\left(ⱳ-ⱱ\right)}\odot \left[{\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\sigma -\varsigma \right)}\odot \left[{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(46)

If ${\mathcal{Q}}_{*}\left(\phi ,\omega \right)={\mathcal{Q}}^{*}\left(\phi ,\omega \right)$ with $ɤ=1$, then from (26), we acquire the coming inequality, see [40]:

$$\begin{gathered} \mathcal{Q}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ \le \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathcal{Q}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi +\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathcal{Q}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right] \\ \le \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathcal{Q}\left(\phi ,\omega \right)d\omega d\phi \\ \le \frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathcal{Q}\left(\phi ,\varsigma \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathcal{Q}\left(\phi ,\sigma \right)d\phi \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathcal{Q}\left(ⱱ,\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathcal{Q}\left(ⱳ,\omega \right)d\omega \right] \\ \le \frac{\mathcal{Q}\left(ⱱ,\varsigma \right)+\mathcal{Q}\left(ⱳ,\varsigma \right)+\mathcal{Q}\left(ⱱ,\sigma \right)+\mathcal{Q}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(47)

If ${\mathcal{Q}}_{*}\left(\phi ,\omega \right)={\mathcal{Q}}^{*}\left(\phi ,\omega \right)$ with $ɤ=1$ and, ${\pi }_1\left(ⱳ,ⱱ\right)=ⱳ-ⱱ$ and ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$, then from (36), we acquire the coming inequality, see [37]:

$$\begin{gathered} \mathcal{Q}\left(\frac{ⱱ+ⱳ}{2}, \frac{\varsigma +\sigma }{2}\right)\le \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{ⱳ-ⱱ}{\int }_{ⱱ}^{ⱳ}\mathcal{Q}\left(\phi ,\frac{\varsigma +\sigma }{2}\right)d\phi +\frac{1}{\sigma -\varsigma }{\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(\frac{ⱱ+ⱳ}{2},\omega \right)d\omega \right] \\ \le \frac{1}{\left(ⱳ-ⱱ\right)\left(\sigma -\varsigma \right)}{\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(\phi ,\omega \right)d\omega d\phi \\ \le \frac{\begin{array}{c} \\ 1\end{array}}{4\left(ⱳ-ⱱ\right)}\left[{\int }_{ⱱ}^{ⱳ}\mathcal{Q}\left(\phi ,\varsigma \right)d\phi +{\int }_{ⱱ}^{ⱳ}\mathcal{Q}\left(\phi ,\sigma \right)d\phi \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\sigma -\varsigma \right)}\left[{\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(ⱱ,\omega \right)d\omega +{\int }_{\varsigma }^{\sigma }\mathcal{Q}\left(ⱳ,\omega \right)d\omega \right] \\ \le \frac{\mathcal{Q}\left(ⱱ,\varsigma \right)+\mathcal{Q}\left(ⱳ,\varsigma \right)+\mathcal{Q}\left(ⱱ,\sigma \right)+\mathcal{Q}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(48)

Example 2.

We consider the $F$·$N$·$V$·$M \tilde{\mathcal{Q}}:\left[0, 2\right]\times \left[0, 2\right]\to {\Omega }_0$ defined as

$$\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\left(m\right)=\left\{\begin{array}{c}\frac{m-\phi \omega }{5-\phi \omega }, \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m\in \left[\phi \omega , 5\right]\\ \frac{\left(2+\sqrt{\phi }\right)\left(2+\sqrt{\omega }\right)-m}{\left(2+\sqrt{\phi }\right)\left(2+\sqrt{\omega }\right)-5}, m\in \left(5, \left(2+\sqrt{\phi }\right)\left(2+\sqrt{\omega }\right)\right]\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} otherwise,\end{array}\right.$$

(49)

and then, for each $ɤ\in \left[0, 1\right],$ we obtain ${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[\left(1-ɤ\right)\phi \omega +5ɤ,\left(1-ɤ\right)\left(2+\sqrt{\phi }\right)\left(2+\sqrt{\omega }\right)+5ɤ\right]$. The end-point functions ${\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right),$ ${\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)$ are coordinated preinvex and preincave functions with respect to ${\pi }_1\left(ⱳ,ⱱ\right)=ⱳ-ⱱ$ and ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$ for each $ɤ\in \left[0, 1\right]$. Hence, $\tilde{\mathcal{Q}}\left(\phi ,\omega \right)$ is a coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$.

$$\begin{gathered} {\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{2}, \frac{2\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{2}\right)=\left[1+4ɤ,9-4ɤ\right] \\ \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi +\frac{1}{{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \right] \\ =\left[1+4ɤ,\frac{1}{3}\left(\left(9+2\sqrt{2}\right)ɤ-2\sqrt{2}+6\right)\right], \\ \frac{1}{{\mathit{\pi }}_1\left(ⱳ,ⱱ\right){\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi =\left[1+4ɤ,\frac{1}{9}\left(\left(1+24\sqrt{2}\right)ɤ-24\sqrt{2}+44\right)\right], \\ \frac{\begin{array}{c} \\ 1\end{array}}{4{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}\left[{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)d\phi \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)d\omega \right] \\ =\left[1+4ɤ,\frac{8-5\sqrt{2}}{3}\left(1-ɤ\right)+\frac{9+2\sqrt{2}}{3}ɤ+\frac{6-2\sqrt{2}}{3}\right] \\ \frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)}{4}=\left[1+4ɤ,\frac{\left(1-ɤ\right){\left(2-\sqrt{2}\right)}^2+4\left(1-ɤ\right)\left(2-\sqrt{2}\right)+4\left(1-ɤ\right)+20ɤ}{4}\right] \end{gathered}$$

That is

$$\begin{gathered} \left[1+4ɤ,9-4ɤ\right]{\supseteq }_I\left[1+4ɤ,\frac{1}{3}\left(\left(9+2\sqrt{2}\right)ɤ-2\sqrt{2}+6\right)\right] \\ {\supseteq }_I\left[1+4ɤ,\frac{1}{9}\left(\left(1+24\sqrt{2}\right)ɤ-24\sqrt{2}+44\right)\right] \\ {\supseteq }_I\left[1+4ɤ,\frac{8-5\sqrt{2}}{3}\left(1-ɤ\right)+\frac{9+2\sqrt{2}}{3}ɤ+\frac{6-2\sqrt{2}}{3}\right] \\ {\supseteq }_I\left[1+4ɤ,\frac{\left(1-ɤ\right){\left(2-\sqrt{2}\right)}^2+4\left(1-ɤ\right)\left(2-\sqrt{2}\right)+4\left(1-ɤ\right)+20ɤ}{4}\right] \end{gathered}$$

Hence, Theorem 9 has been verified.

The Pachpatte-type inequalities that we now acquire are for the product of coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s. A few previously established inequities have been improved upon by these results.

Theorem 10.

Let $\tilde{\mathcal{Q}},\tilde{\mathcal{T}}: ∆=\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\subset {\mathfrak{R}}^2\to {\Omega }_0$ be two coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s on $∆$, for which $ɤ$-levels ${\mathcal{Q}}_{ɤ}, {\mathcal{T}}_{ɤ}:\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\to {\mathfrak{R}}_I^{+}$ are defined by ${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ and ${\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{T}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{T}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ for all $\left(\phi ,\omega \right)\in ∆$ and for all $ɤ\in \left[0, 1\right]$. If Condition 1 for ${\pi }_1$ and ${\pi }_2$ is fulfilled, then the following inequality holds:

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right){\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\odot {\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\mathit{\pi }}_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\otimes \tilde{\mathcal{T}}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{1}{9}\odot \tilde{\alpha }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{1}{18}\odot \tilde{\beta }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{1}{36}\odot \tilde{\gamma }\left(ⱱ,ⱳ,\varsigma ,\sigma \right), \end{gathered}$$

(50)

where

$$\tilde{\alpha }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)=\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\sigma \right),$$

$$\begin{gathered} \tilde{\beta }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)=\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\varsigma \right)\oplus \\ \tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\sigma \right), \end{gathered}$$

$$\tilde{\gamma }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)=\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱳ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\otimes \tilde{\mathcal{T}}\left(ⱱ,\sigma .\right)$$

and for each $\mathrm{ɤ}\in \left[0,1\right],$ $\tilde{\alpha }\left(\mathrm{ⱱ},\mathrm{ⱳ},\varsigma ,\sigma \right)$, $\tilde{\beta }\left(\mathrm{ⱱ},\mathrm{ⱳ},\varsigma ,\sigma \right)$ and $\tilde{\gamma }\left(\mathrm{ⱱ},\mathrm{ⱳ},\varsigma ,\sigma \right)$ are defined as follows:

$${\alpha }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)=\left[{\alpha }_{*}\left(\left(ⱱ,ⱳ,\varsigma ,\sigma \right), ɤ\right), {\alpha }^{*}\left(\left(ⱱ,ⱳ,\varsigma ,\sigma \right), ɤ\right)\right],$$

$${\beta }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)=\left[{\beta }_{*}\left(\left(ⱱ,ⱳ,\varsigma ,\sigma \right), ɤ\right), {\beta }^{*}\left(\left(ⱱ,ⱳ,\varsigma ,\sigma \right), ɤ\right)\right],$$

$${\gamma }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)=\left[{\gamma }_{*}\left(\left(ⱱ,ⱳ,\varsigma ,\sigma \right), ɤ\right), {\gamma }^{*}\left(\left(ⱱ,ⱳ,\varsigma ,\sigma \right), ɤ\right)\right].$$

Proof. 

Let $\tilde{\mathcal{Q}}$ and $\tilde{\mathcal{T}}$ both be coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s on $\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]$. Then

$$\begin{gathered} \tilde{\mathcal{Q}}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right) \\ {\supseteq }_{\mathbb{F}}\zeta \mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \zeta \left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \left(1-\zeta \right)\mathsf{ϛ}\odot \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right), \end{gathered}$$

and

$$\begin{gathered} \tilde{\mathcal{T}}\left(ⱱ+\left(1-\zeta \right){\pi }_1\left(ⱳ,ⱱ\right), \varsigma +\left(1-\mathsf{ϛ}\right){\pi }_2\left(\sigma ,\varsigma \right)\right) \\ {\supseteq }_{\mathbb{F}}\zeta \mathsf{ϛ}\odot \tilde{\mathcal{T}}\left(ⱱ,\varsigma \right)\oplus \zeta \left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{T}}\left(ⱱ,\sigma \right)\oplus \left(1-\zeta \right)\mathsf{ϛ}\odot \tilde{\mathcal{T}}\left(ⱳ,\varsigma \right)\oplus \left(1-\zeta \right)\left(1-\mathsf{ϛ}\right)\odot \tilde{\mathcal{T}}\left(ⱳ,\sigma \right). \end{gathered}$$

Since $\tilde{\mathcal{Q}}$ and $\tilde{\mathcal{T}}$ both are coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s, then by Lemma 1, there exists

$${\tilde{\mathcal{Q}}}_{\phi }:\left[\varsigma ,\sigma \right]\to {\Omega }_0, {\tilde{\mathcal{Q}}}_{\phi }\left(\omega \right)=\tilde{\mathcal{Q}}\left(\phi ,\omega \right),\hspace{1em}\hspace{1em} {\tilde{\mathcal{T}}}_{\phi }:\left[\varsigma ,\sigma \right]\to {\Omega }_0, {\tilde{\mathcal{T}}}_{\phi }\left(\omega \right)=\tilde{\mathcal{T}}\left(\phi ,\omega \right),$$

and

$${\tilde{\mathcal{Q}}}_{\omega }:\left[ⱱ,ⱳ\right]\to {\Omega }_0, {\tilde{\mathcal{Q}}}_{\omega }\left(\phi \right)=\tilde{\mathcal{Q}}\left(\phi ,\omega \right), \hspace{1em}\hspace{1em}{\tilde{\mathcal{T}}}_{\omega }:\left[ⱱ,ⱳ\right]\to {\Omega }_0, {\tilde{\mathcal{T}}}_{\omega }\left(\phi \right)=\tilde{\mathcal{T}}\left(\phi ,\omega \right).$$

Since ${\tilde{\mathcal{Q}}}_{\phi }$, ${\tilde{\mathcal{T}}}_{\phi }, {\tilde{\mathcal{Q}}}_{\omega }$, and ${\tilde{\mathcal{T}}}_{\omega }$ are $F$·$N$·$V$·$M$s, then by inequality (15), we have

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\tilde{\mathcal{Q}}}_{\omega }\left(\phi \right)\otimes {\tilde{\mathcal{T}}}_{\omega }\left(\phi \right)d\phi {\supseteq }_{\mathbb{F}}\frac{1}{3}\odot \left[{\tilde{\mathcal{Q}}}_{\omega }\left(ⱱ\right)\otimes {\tilde{\mathcal{T}}}_{\omega }\left(ⱱ\right)\oplus {\tilde{\mathcal{Q}}}_{\omega }\left(ⱳ\right)\otimes {\tilde{\mathcal{T}}}_{\omega }\left(ⱳ\right)\right] \\ \oplus \frac{1}{6}\odot \left[{\tilde{\mathcal{Q}}}_{\omega }\left(ⱱ\right)\otimes {\tilde{\mathcal{T}}}_{\omega }\left(ⱳ\right)\oplus {\tilde{\mathcal{Q}}}_{\omega }\left(ⱳ\right)\otimes {\tilde{\mathcal{T}}}_{\omega }\left(ⱱ\right)\right], \end{gathered}$$

and

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\tilde{\mathcal{Q}}}_{\phi }\left(\omega \right)\otimes {\tilde{\mathcal{T}}}_{\phi }\left(\omega \right)d\omega {\supseteq }_{\mathbb{F}}\frac{1}{3}\odot \left[{\tilde{\mathcal{Q}}}_{\phi }\left(\varsigma \right)\otimes {\tilde{\mathcal{T}}}_{\phi }\left(\varsigma \right)\oplus {\tilde{\mathcal{Q}}}_{\phi }\left(\sigma \right)\otimes {\tilde{\mathcal{T}}}_{\phi }\left(\sigma \right)\right] \\ \oplus \frac{1}{6}\odot \left[{\tilde{\mathcal{Q}}}_{\phi }\left(\varsigma \right)\otimes {\tilde{\mathcal{T}}}_{\phi }\left(\sigma \right)\oplus {\tilde{\mathcal{Q}}}_{\phi }\left(\varsigma \right)\otimes {\tilde{\mathcal{T}}}_{\phi }\left(\sigma \right)\right]. \end{gathered}$$

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

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{{\mathcal{Q}}_{ɤ}}_{\omega }\left(\phi \right)\times {{\mathcal{T}}_{ɤ}}_{\omega }\left(\phi \right)d\phi \\ {\supseteq }_I\frac{1}{3}\left[{{\mathcal{Q}}_{ɤ}}_{\omega }\left(ⱱ\right)\times {{\mathcal{T}}_{ɤ}}_{\omega }\left(ⱱ\right)+{{\mathcal{Q}}_{ɤ}}_{\omega }\left(ⱳ\right)\times {{\mathcal{T}}_{ɤ}}_{\omega }\left(ⱳ\right)\right]+\frac{1}{6}\left[{{\mathcal{Q}}_{ɤ}}_{\omega }\left(ⱱ\right)\times {{\mathcal{T}}_{ɤ}}_{\omega }\left(ⱳ\right)+{{\mathcal{Q}}_{ɤ}}_{\omega }\left(ⱳ\right)\times {{\mathcal{T}}_{ɤ}}_{\omega }\left(ⱱ\right)\right], \end{gathered}$$

and

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\omega \right)\times {{\mathcal{T}}_{ɤ}}_{\phi }\left(\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}\left[{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\varsigma \right)\times {{\mathcal{T}}_{ɤ}}_{\phi }\left(\varsigma \right)+{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\sigma \right)\times {{\mathcal{T}}_{ɤ}}_{\phi }\left(\sigma \right)\right]+\frac{1}{6}\left[{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\varsigma \right)\times {{\mathcal{T}}_{ɤ}}_{\phi }\left(\sigma \right)+{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\varsigma \right)\times {{\mathcal{T}}_{ɤ}}_{\phi }\left(\sigma \right)\right]. \end{gathered}$$

The above inequalities can be written as

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\phi \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)\right] \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)\right], \end{gathered}$$

(51)

and

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)\right] \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)\right]. \end{gathered}$$

(52)

Initially, we resolve inequality (51) by considering integration on both sides of the inequality about $\omega$ throughout the interval $\left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]$ and taking the division of both sides with ${\pi }_2\left(\sigma ,\varsigma \right)$ to obtain

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{1}{3{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)\right]d\omega \\ +\frac{1}{6{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)\right]d\omega . \end{gathered}$$

(53)

Now, again by inequality (15), for each $ɤ\in \left[0, 1\right],$ we have

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right]d\omega \\ +\frac{1}{6}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right]d\omega . \end{gathered}$$

(54)

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right]d\omega \\ +\frac{1}{6}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right]d\omega . \end{gathered}$$

(55)

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right]d\omega \\ +\frac{1}{6}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)\right]d\omega . \end{gathered}$$

(56)

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right]d\omega \\ +\frac{1}{6}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)\right]d\omega . \end{gathered}$$

(57)

From (54)–(57), inequality (53), we have

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_1\left(ⱳ,ⱱ\right){\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{1}{9}{\alpha }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{1}{18}{\beta }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{1}{36}{\gamma }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

That is

$$\begin{gathered} \frac{1}{{\mathit{\pi }}_1\left(ⱳ,ⱱ\right){\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\odot {\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\otimes \tilde{\mathcal{T}}\left(\phi ,\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{1}{9}\odot \tilde{\alpha }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{1}{18}\odot \tilde{\beta }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{1}{36}\odot \tilde{\gamma }\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

Hence, this concludes the proof of the theorem.  □

Theorem 11.

Let $\tilde{\mathcal{Q}},\tilde{\mathcal{T}}: ∆=\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\subset {\mathfrak{R}}^2\to {\Omega }_0$ be two coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s. Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}, {\mathcal{T}}_{ɤ}:∆\subset {\mathfrak{R}}^2\to {\mathfrak{R}}_I^{+}$and offered by ${\mathcal{Q}}_{ɤ}\left(\phi \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ and ${\mathcal{T}}_{ɤ}\left(\phi \right)=\left[{\mathcal{T}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{T}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ for all $\left(\phi ,\omega \right)\in ∆$ and for all $ɤ\in \left[0, 1\right]$. If Condition 1 for ${\pi }_1$ and ${\pi }_2$ is fulfilled, then the following inequality holds:

$$\begin{gathered} 4\odot \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{2}\right)\otimes \tilde{\mathcal{T}}\left(\frac{2ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\mathit{\pi }}_1\left(ⱳ,ⱱ\right){\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\odot {\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{{\varsigma +\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\otimes \tilde{\mathcal{T}}\left(\phi ,\omega \right)d\omega d\phi \\ \oplus \frac{5}{36}\odot \tilde{\alpha }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{7}{36}\odot \tilde{\beta }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{2}{9}\odot \tilde{\gamma }\left(ⱱ,ⱳ,\varsigma ,\sigma \right), \end{gathered}$$

(58)

where $\tilde{\alpha }\left(\mathrm{ⱱ},\mathrm{ⱳ},\varsigma ,\sigma \right)$, $\tilde{\beta }\left(\mathrm{ⱱ},\mathrm{ⱳ},\varsigma ,\sigma \right)$ and $\tilde{\gamma }\left(\mathrm{ⱱ},\mathrm{ⱳ},\varsigma ,\sigma \right)$ are given in Theorem 10.

Proof. 

Since $\tilde{\mathcal{Q}},\tilde{\mathcal{T}}: ∆\to {\Omega }_0$ is two coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s, and then, from inequality (16) and for each $ɤ\in \left[0, 1\right],$ we have

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\times \mathcal{T}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\times \mathcal{T}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\times \mathcal{T}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\times \mathcal{T}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\times \mathcal{T}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\right], \end{gathered}$$

(59)

and

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\omega \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)+{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\right]. \end{gathered}$$

(60)

 □

By them adding together, the inequalities (59) and (60), and then multiplying the resulting number by two, we arrive at

$$\begin{gathered} 8{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{2}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \\ +\frac{2}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\phi \\ +\frac{1}{6}\left[2{\mathcal{Q}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)+2{\mathcal{Q}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\right] \\ +\frac{1}{6}\left[2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)+2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\right] \\ +\frac{1}{3}\left[2{\mathcal{Q}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)+2{\mathcal{Q}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\right] \\ +\frac{1}{3}\left[2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)+2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\right]. \end{gathered}$$

(61)

At this point, we may use integral inequality (16) to obtain the value of each integral on the right side of (61):

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)\right]. \end{gathered}$$

(62)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)\right]. \end{gathered}$$

(63)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)\right]. \end{gathered}$$

(64)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(ⱳ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)\right]. \end{gathered}$$

(65)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\right]. \end{gathered}$$

(66)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)+{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\right]. \end{gathered}$$

(67)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)+{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\right]. \end{gathered}$$

(68)

$$\begin{gathered} 2{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)\right] \\ +\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\varsigma \right)\right]. \end{gathered}$$

(69)

From (62)–(69), we have

$$\begin{gathered} 8{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{2}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \\ +\frac{2}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\phi \\ +\frac{1}{6{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega +\frac{1}{6{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ +\frac{1}{6{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi +\frac{1}{6{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi , \\ +\frac{1}{18}{\alpha }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{1}{9}{\beta }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{2}{9}{\gamma }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

(70)

Once more, we may obtain the following relation by using integral inequality (16) for the first two integrals on the right-hand side of (70):

$$\begin{gathered} \frac{2}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)d\phi \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)\right]d\phi \\ +\frac{1}{6{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)\right]d\phi , \end{gathered}$$

(71)

$$\begin{gathered} \frac{2}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)d\phi \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)\right]d\omega \\ +\frac{1}{6{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)\right]d\omega . \end{gathered}$$

(72)

From (71) and (72), we have

$$\begin{gathered} 8{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)\right]d\phi \\ +\frac{1}{6{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)\right]d\phi \\ +\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)\right]d\omega \\ +\frac{1}{6{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)\right]d\omega \\ +\frac{1}{6{\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ +\frac{1}{6{\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ +\frac{1}{6{\pi }_1\left(ⱳ,ⱱ\right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi \\ +\frac{1}{6{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi \\ +\frac{1}{18}{\alpha }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{1}{9}{\beta }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{2}{9}{\gamma }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

It follows that

$$\begin{gathered} 8{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{2}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ +\frac{2}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)\right]d\phi \\ +\frac{1}{3{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\left[{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)\right]d\phi \\ +\frac{2}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)\right]d\omega \\ +\frac{1}{3{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\omega \right)\right]d\omega \\ +\frac{1}{18}{\alpha }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{1}{9}{\beta }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{2}{9}{\gamma }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

(73)

The relationship that follows is now obtained by applying integral inequality (15) to the integrals on the right side of (73):

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\varsigma \right)d\phi \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\varsigma \right)\right], \end{gathered}$$

(74)

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right] \\ +\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right], \end{gathered}$$

(75)

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\sigma \right)d\phi \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{T}}_{ɤ}\left(ⱱ,\sigma \right)\right], \end{gathered}$$

(76)

$$\begin{gathered} \frac{1}{{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(\phi ,\varsigma \right)d\phi \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\varsigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\varsigma \right)\right], \end{gathered}$$

(77)

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\sigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\varsigma \right)\right], \end{gathered}$$

(78)

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\sigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\varsigma \right)\right], \end{gathered}$$

(79)

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\sigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱳ,\varsigma \right)\right], \end{gathered}$$

(80)

$$\begin{gathered} \frac{1}{{\pi }_2\left(\sigma ,\varsigma \right)} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\omega \right)d\omega \\ {\supseteq }_I\frac{1}{3}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\sigma \right)\right]+\frac{1}{6}\left[{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)\times {\mathcal{J}}_{ɤ}\left(ⱱ,\varsigma \right)\right]. \end{gathered}$$

(81)

From (74)–(81), inequality (95), we have

$$\begin{gathered} 4{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\times {\mathcal{T}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_I\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\times {\mathcal{T}}_{ɤ}\left(\phi ,\omega \right)d\omega d\phi \\ +\frac{5}{36}{\alpha }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{7}{36}{\beta }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right)+\frac{2}{9}{\gamma }_{ɤ}\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

That is

$$\begin{gathered} 4\odot \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\otimes \tilde{\mathcal{T}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\pi }_1\left(ⱳ,ⱱ\right){\pi }_2\left(\sigma ,\varsigma \right)}\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\otimes \tilde{\mathcal{T}}\left(\phi ,\omega \right)d\omega d\phi \\ \oplus \frac{5}{36}\odot \tilde{\alpha }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{7}{36}\odot \tilde{\beta }\left(ⱱ,ⱳ,\varsigma ,\sigma \right)\oplus \frac{2}{9}\odot \tilde{\gamma }\left(ⱱ,ⱳ,\varsigma ,\sigma \right). \end{gathered}$$

We now give the HH-Fejér inequality for coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ s by means of FOR in the following result.

Theorem 12.

Let $\tilde{\mathcal{Q}}:∆=\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\times \left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\to {\Omega }_0$ be a coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ with $ⱱ<ⱳ$ and $\varsigma <\sigma .$ Then, from $ɤ$-levels, we receive the set of $I$·$V$·$M$s ${\mathcal{Q}}_{ɤ}:∆\to {\mathfrak{R}}_I^{+}$ and offered by ${\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)=\left[{\mathcal{Q}}_{*}\left(\left(\phi ,\omega \right),ɤ\right), {\mathcal{Q}}^{*}\left(\left(\phi ,\omega \right),ɤ\right)\right]$ for all $\left(\phi ,\omega \right)\in ∆$ and for all $ɤ\in \left[0, 1\right]$. Let $\mathsf{ƕ}:\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]\to \mathfrak{R}$ with $\mathsf{ƕ}\left(\phi \right)\ge 0,$  ${\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi >0$and $\mathsf{Ԃ}:\left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]\to \mathfrak{R}$ with $\mathsf{Ԃ}\left(\omega \right)\ge 0,$  ${\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega >0,$be two symmetric functions with respect to $\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}$ and $\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}$, respectively. If Condition 1 for ${\pi }_1$ and ${\pi }_2$ holds, then the following inequality holds:

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}{2}\right) \\ {\supseteq }_{\mathbb{F}}\frac{1}{2}\left[\begin{array}{c}\frac{1}{{\int }_{ⱱ}^{ⱳ}\mathsf{ƕ}\left(\phi \right)d\phi }\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi \\ \oplus \frac{1}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \end{array}\right] \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi {\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot \\ {\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }\odot \left[{\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱱ+{\mathit{\pi }}_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot \left[{\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\varsigma +{\mathit{\pi }}_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(82)

Proof. 

Since $\tilde{\mathcal{Q}}$ is both a coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$ on $∆$, it follows those functions, then by Lemma 1, there exists

$${\tilde{\mathcal{Q}}}_{\phi }:\left[\varsigma ,\sigma \right]\to {\Omega }_0, {\tilde{\mathcal{Q}}}_{\phi }\left(\omega \right)=\tilde{\mathcal{Q}}\left(\phi ,\omega \right), \hspace{1em}{\tilde{\mathcal{Q}}}_{\omega }:\left[ⱱ,ⱳ\right]\to {\Omega }_0, {\tilde{\mathcal{Q}}}_{\omega }\left(\phi \right)=\tilde{\mathcal{Q}}\left(\phi ,\omega \right).$$

Thus, from inequality (17), for each $ɤ\in \left[0, 1\right],$ we have

$${{\mathcal{Q}}_{ɤ}}_{\phi }\left(\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega {\supseteq }_I\frac{{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\varsigma \right)+{{\mathcal{Q}}_{ɤ}}_{\phi }\left(\sigma \right)}{2},$$

and

$${{\mathcal{Q}}_{ɤ}}_{\omega }\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2}\right){\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{{\mathcal{Q}}_{ɤ}}_{\omega }\left(\phi \right)\mathsf{ƕ}\left(\phi \right)d\phi {\supseteq }_I\frac{{{\mathcal{Q}}_{ɤ}}_{\omega }\left(ⱱ\right)+{{\mathcal{Q}}_{ɤ}}_{\omega }\left(ⱳ\right)}{2}.$$

The above inequalities can be written as

$${\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi , \sigma \right)}{2},$$

(83)

and

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right){\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)d\phi {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)}{2}$$

(84)

After multiplying (83) by $\mathsf{ƕ}\left(\phi \right)$ and integrating the product over $\left[ⱱ, ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)\right]$ pertaining to $\phi$, we obtain the following:

$$\begin{gathered} {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\omega d\phi \\ {\supseteq }_I{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\frac{{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi , \sigma \right)}{2}\mathsf{ƕ}\left(\phi \right)d\phi . \end{gathered}$$

(85)

Upon multiplying (84) by $\mathsf{Ԃ}\left(\omega \right)$ and subsequently integrating the resulting value pertaining to $\omega$ across $\left[\varsigma , \varsigma +{\pi }_2\left(\sigma ,\varsigma \right)\right]$, we obtain the following:

$$\begin{gathered} {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\phi d\omega \\ {\supseteq }_I{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)}{2}\mathsf{Ԃ}\left(\omega \right)d\omega . \end{gathered}$$

(86)

Since ${\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\pi }_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\mathrm{ƕ}\left(\phi \right)d\phi >0$ and ${\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathrm{Ԃ}\left(\omega \right)d\omega >0,$ then dividing (85) and (86) by ${\int }_{\mathrm{ⱱ}}^{\mathrm{ⱱ}+{\pi }_1\left(\mathrm{ⱳ},\mathrm{ⱱ}\right)}\mathrm{ƕ}\left(\phi \right)d\phi >0$ and ${\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathrm{Ԃ}\left(\omega \right)d\omega >0$, respectively, we obtain

$$\begin{gathered} \frac{1}{2}[\frac{1}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi + \\ \frac{1}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega ] \\ {\supseteq }_I\frac{1}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{1}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\frac{{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(\phi , \sigma \right)}{4}\mathsf{ƕ}\left(\phi \right)d\phi \\ +\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)}{4}\mathsf{Ԃ}\left(\omega \right)d\omega . \end{gathered}$$

(87)

Now, we obtain the following from the left half of double inequalities (83) and (84):

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega ,$$

(88)

and

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi .$$

(89)

The result of adding the inequalities (88) and (89) is

$${\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{1}{2}\left[\begin{array}{c}\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi \\ +\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \end{array}\right].$$

(90)

Similarly, we can derive, from the right portion of (89) and (90),

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ, \sigma \right)}{2},$$

(91)

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ, \sigma \right)}{2},$$

(92)

and

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\mathsf{ƕ}\left(\phi \right)d\phi {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)}{2}.$$

(93)

$$\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\mathsf{ƕ}\left(\phi \right)d\phi {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)}{2}.$$

(94)

Adding (91)–(94) and dividing by 4, we obtain

$$\begin{gathered} \frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }\left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\mathsf{ƕ}\left(\phi \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\mathsf{ƕ}\left(\phi \right)d\phi \right] \\ {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ, \sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ, \sigma \right)}{4}. \end{gathered}$$

(95)

When we combine inequalities (90), (95), and (87), we obtain

$$\begin{gathered} {\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_I\frac{1}{2}\left[\begin{array}{c}\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi \\ +\frac{\begin{array}{c} \\ 1\end{array}}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \end{array}\right] \\ {\supseteq }_I\frac{1}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathcal{Q}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\omega d\phi \\ {\supseteq }_I\frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱱ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega +{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{\mathcal{Q}}_{ɤ}\left(ⱳ,\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \right] \\ +\frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }\left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\varsigma \right)\mathsf{ƕ}\left(\phi \right)d\phi +{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\mathcal{Q}}_{ɤ}\left(\phi ,\sigma \right)\mathsf{ƕ}\left(\phi \right)d\phi \right] \\ {\supseteq }_I\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱱ, \sigma \right)}{2}+\frac{{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ, \sigma \right)}{2}+\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\varsigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\varsigma \right)}{2}+\frac{{\mathcal{Q}}_{ɤ}\left(ⱱ,\sigma \right)+{\mathcal{Q}}_{ɤ}\left(ⱳ,\sigma \right)}{2}. \end{gathered}$$

That is

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{2}\left[\begin{array}{c}\frac{1}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }\odot {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\frac{2\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi \\ \oplus \frac{1}{{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\frac{2ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega \end{array}\right] \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot \\ {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\mathsf{ƕ}\left(\phi \right)d\phi }\odot \left[{\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱱ+{\pi }_1\left(ⱳ,ⱱ\right)}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot \left[{\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\varsigma +{\pi }_2\left(\sigma ,\varsigma \right)}\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

Hence, this concludes the proof.  □

Remark 4.

If one takes $\mathsf{Ԃ}\left(\omega \right)=1=\mathsf{ƕ}\left(\phi \right)$, then from (82), we achieve (26).

If one takes ${\pi }_1\left(ⱳ,ⱱ\right)=ⱳ-ⱱ$ and ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$, then from (82), we acquire the coming inequality, see [37]:

$$\begin{gathered} \tilde{\mathcal{Q}}\left(\frac{ⱱ+ⱳ}{2},\frac{\varsigma +\sigma }{2}\right){\supseteq }_{\mathbb{F}}\frac{1}{2}[\frac{1}{{\int }_{ⱱ}^{ⱳ}\mathsf{ƕ}\left(\phi \right)d\phi }\odot {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\phi ,\frac{\varsigma +\sigma }{2}\right)\mathsf{ƕ}\left(\phi \right)d\phi \oplus \frac{1}{{\int }_{ⱱ}^{ⱳ}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot \\ {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\frac{ⱱ+ⱳ}{2},\omega \right)\mathsf{Ԃ}\left(\omega \right)d\omega ] \\ {\supseteq }_{\mathbb{F}}\frac{1}{{\int }_{ⱱ}^{ⱳ}\mathsf{ƕ}\left(\phi \right)d\phi {\int }_{ⱱ}^{ⱳ}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot {\int }_{ⱱ}^{ⱳ}{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(\phi ,\omega \right)\mathsf{ƕ}\left(\phi \right)\mathsf{Ԃ}\left(\omega \right)d\omega d\phi \\ {\supseteq }_{\mathbb{F}}\frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{ⱱ}^{ⱳ}\mathsf{ƕ}\left(\phi \right)d\phi }\odot \left[{\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\phi ,\varsigma \right)d\phi \oplus {\int }_{ⱱ}^{ⱳ}\tilde{\mathcal{Q}}\left(\phi ,\sigma \right)d\phi \right] \\ \oplus \frac{\begin{array}{c} \\ 1\end{array}}{4{\int }_{ⱱ}^{ⱳ}\mathsf{Ԃ}\left(\omega \right)d\omega }\odot \left[{\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(ⱱ,\omega \right)d\omega \oplus {\int }_{\varsigma }^{\sigma }\tilde{\mathcal{Q}}\left(ⱳ,\omega \right)d\omega \right] \\ {\supseteq }_{\mathbb{F}}\frac{\tilde{\mathcal{Q}}\left(ⱱ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\varsigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱱ,\sigma \right)\oplus \tilde{\mathcal{Q}}\left(ⱳ,\sigma \right)}{4}. \end{gathered}$$

(96)

If one takes ${\pi }_1\left(ⱳ,ⱱ\right)=ⱳ-ⱱ$, ${\pi }_2\left(\sigma ,\varsigma \right)=\sigma -\varsigma$ and $\mathsf{Ԃ}\left(\omega \right)=1=\mathsf{ƕ}\left(\phi \right)$, then from (82), we acquire the inequality (46), see [37].

4. Conclusions

We present new coordinated $F$·$N$·$V$·$M$s called the coordinated $U$·$D$ preinvex $F$·$N$·$V$·$M$s, which is a generalization of the previously defined $U$·$D$ convex $F$·$N$·$V$·$M$s that Khan et al. provided. Numerous new disparities emerged as a result of the generalizations. By combining noninteger operators with the H·H inequality, we were able to obtain additional variations that expanded upon the results of the earlier H·H type. Previous results from the $U$·$D$ preinvex were covered due to the $F$·$N$·$V$·$M$ and $I$·$V$·$M$ environments by keeping the upper and lower bounds equal to the classical convexity. Moreover, some nontrivial examples are also provided to discuss the validity of our main outcomes, but some of the interesting remarks are provided to acquire the classical and new exceptional cases that can considered as applications of the main outcomes. This concept can be extended to new inequalities via fuzzy fractional integrals.