Some New Concepts Related to Integral Operators and Inequalities on Coordinates in Fuzzy Fractional Calculus

Abstract

In interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in convex theory because they allow us to infer more exact inequalities than convex functions. Most likely, integral operators play significant roles to define different types of inequalities. In this paper, we have successfully introduced the Riemann–Liouville fractional integrals on coordinates via fuzzy-interval-valued functions (FIVFs). Then, with the help of these integrals, some fuzzy fractional Hermite–Hadamard-type integral inequalities are also derived for the introduced coordinated convex FIVFs via a fuzzy order relation (FOR). This FOR is defined by $\phi$-cuts or level-wise by using the Kulish–Miranker order relation. Moreover, some related fuzzy fractional Hermite–Hadamard-type integral inequalities are also obtained for the product of two coordinated convex fuzzy-interval-valued functions. The main results of this paper are the generalization of several known results.

1. Introduction

The classical version of Hermite–Hadamard inequality can be put in the following manner:

Let $\mathfrak{S}:K\to ℝ$ be a convex function on a convex set $K$ and $\rho , \varsigma \in K$ with $\rho \le \varsigma$. Then:

$$\mathfrak{S}\left(\frac{\rho +\varsigma }{2}\right)\le \frac{1}{\varsigma -\rho } {{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{S}\left(\varpi \right)d\varpi \le \frac{\mathfrak{S}\left(\rho \right)+\mathfrak{S}\left(\varsigma \right)}{2}.$$

(1)

In [1], Fejér looked at the key extensions of HH inequality, which is known as Hermite–Hadamard–Fejér inequality (HH–Fejér inequality):

Let $\mathfrak{S}:K\to ℝ$ be a convex function on a convex set $K$ and $\rho , \varsigma \in K$ with $\rho \le \varsigma$. Then:

$$\mathfrak{S}\left(\frac{\rho +\varsigma }{2}\right)\le \frac{1}{{{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{D}\left(\varpi \right)d\varpi } {{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{S}\left(\varpi \right)\mathfrak{D}\left(\varpi \right)d\varpi \le \frac{\mathfrak{S}\left(\rho \right)+\mathfrak{S}\left(\varsigma \right)}{2}{{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{D}\left(\varpi \right)d\varpi .$$

(2)

If $\mathfrak{D}\left(\varpi \right)=1$, then we obtain (1) from (2).

Many authors have focused their efforts to generalize inequalities (1) and (2); see [1,2,3,4,5,6]. They play an important role in convex analysis and may be a very strong tool for monitoring and quantifying mistakes. It is worth mentioning that Sarikaya et al. [7] used the Riemann–Liouville fractional integrals to develop new Hermite–Hadamard inequalities. Since then, many papers have been published that have extended various types of fractional integrals and have provided fresh and fascinating improvements of Hermite–Hadamard-type inequalities utilizing these integrals. For the Atangana–Baleanu fractional integral, Fernandez and Mohammed [8] constructed certain Hermite–Hadamard-type inequalities. In the context of fractional calculus, Mohammed and Abdeljawad [9] demonstrated novel Hermite–Hadamard-type inequalities with regard to functions with non-singular kernels. For more similar results, refs. [7,8,9,10,11,12,13,14,15,16,17,18] is a good place to look.

On the other hand, Moore [19] established the theory of interval analysis to increase the dependability of calculation outputs and autonomous operation error analysis. Interval analysis is a robust model for dealing with interval uncertainty, and it has been widely applied and broadened in fields such as control theory [20], dynamical game theory [21], and many more. Several well-known inequalities have recently been extended to interval-valued functions. Using the Hukuhara derivative, Chalco-Cano et al. [22] derived Ostrowski-type inequalities for interval-valued functions. The Minkowski and Beckenbach-type inequalities for interval-valued functions were discovered by Román-Flores et al. [23]. For fuzzy-interval-valued functions, Khan et al. [24,25,26] derived some new versions of Hermite–Hadamard-type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. [27,28,29] discussed some novel types of Hermite–Hadamard-type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities.

For interval-valued functions, Liu et al. [30] demonstrated Hermite–Hadamard inequality using interval Riemann–Liouville-type fractional integrals. Zhao et al. [31,32] discovered Hermite–Hadamard inequalities for interval-valued coordinated functions very recently. Budak et al. [33] introduced several novel Hermite–Hadamard inequalities and defined Riemann–Liouville-type fractional integrals for interval-valued coordinated functions. Recently, Khan et al. [34] introduced the new class of convexity in fuzzy-interval calculus, which consists of coordinated convex fuzzy-interval-valued functions, and with the support of these classes, some Hermite–Hadamard-type inequalities are obtained via newly defined fuzzy-interval double integrals. For more information related to interval-valued and fuzzy-interval-valued functions, see [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65].

Inspired by ongoing research work, we provide a novel class of Hermite–Hadamard-type inequalities for coordinated convex fuzzy-interval-valued functions through fuzzy-interval Riemann–Liouville-type fractional integrals. Motivated by the work of Khan et al. [27,28,34] and Budak et al. [33], we obtain Hermite–Hadamard-type inequalities for the products of two fuzzy-interval-valued coordinated functions.

2. Preliminaries

Let ${ℝ}_I$ and ${\mathbb{F}}_0$ be the collection of all closed and bounded intervals and fuzzy intervals of $ℝ$. We use ${ℝ}_I^{+}$ to represent the set of all positive intervals. The collection of all Riemann integrable real valued functions, Riemann integrable IV-Fs and fuzzy Riemann integrable F-IV-Fs over $\left[\rho , \varsigma \right]$, is denoted by ${\mathcal{R}}_{\left[\rho , \varsigma \right]}$, $\mathit{ȷ}{\mathcal{R}}_{\left[\rho , \varsigma \right]}$, and $\mathcal{F}{\mathcal{R}}_{\left(\left[\rho , \varsigma \right]\right)}$ respectively. For more conceptions on interval-valued functions and fuzzy-interval-valued functions, see [43,44,45,46,47]. Moreover, the inclusion $“\subseteq ”$ means that:

$𝓇\subseteq \eta$ if and only if $\left[{𝓇}_{\ast }, {𝓇}^{*}\right]\subseteq \left[{\eta }_{\ast }, {\eta }^{\ast }\right]$, if and only if ${\eta }_{\ast }\le {𝓇}_{\ast }$, ${𝓇}^{\ast }\le {\eta }^{\ast },$ for all $\left[{𝓇}_{*}, {𝓇}^{\ast }\right], \left[{\eta }_{\ast }, {\eta }^{\ast }\right]\in {ℝ}_I$.

Remark 1.

Ref. [46] The relation $“{\le }_I”$ defined on ${ℝ}_I$ by $\left[{𝓇}_{\ast }, {𝓇}^{\ast }\right]{\le }_I \left[{\eta }_{\ast }, {\eta }^{\ast }\right]$ if and only if ${𝓇}_{\ast }\le {\eta }_{\ast }, {𝓇}^{\ast }\le {\eta }^{\ast },$ for all $\left[{𝓇}_{\ast }, {𝓇}^{\ast }\right], \left[{\eta }_{\ast }, {\eta }^{\ast }\right]\in {ℝ}_I,$ it is an order relation.

Proposition 1.

Ref. [47] Let ${\mathbb{F}}_0$ be a set of fuzzy numbers. If $𝓇,\eta \in {\mathbb{F}}_0$, then relation $“\preccurlyeq ”$defined on ${\mathbb{F}}_0$ by:

$$𝓇\preccurlyeq \eta if and only if {\left[𝓇\right]}^{\phi }{\le }_I{\left[\eta \right]}^{\phi }, for all \phi \in \left[0, 1\right],$$

(3)

this relation is known as a partial order relation.

Definition 1.

Ref. [16,48] Let $\mathfrak{S}:\left[\rho , \varsigma \right]\to {ℝ}_I^{+}$ be an interval-valued function and $\mathfrak{S}\in \mathit{ȷ}{\mathcal{R}}_{\left[\rho , \varsigma \right]}$. Then, interval Riemann–Liouville-type integrals of $\mathfrak{S}$ are defined as:

$${\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \mathfrak{S}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\rho }^y{\left(y-\mathrm{t}\right)}^{\alpha -1}\mathfrak{S}\left(\mathrm{t}\right)d\mathrm{t} \left(y>\rho \right),$$

(4)

$${\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \mathfrak{S}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_y^{\varsigma }{\left(\mathrm{t}-y\right)}^{\alpha -1}\mathfrak{S}\left(\mathrm{t}\right)d\mathrm{t} (y<\varsigma )$$

(5)

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

Recently, Allahviranloo et al. [49] defined the following fuzzy interval Riemann–Liouville fractional integral integrals:

Definition 2.

Let$\alpha >0$and$L\left(\left[\rho , \varsigma \right],{\mathbb{F}}_0\right)$be the collection of all Lebesgue measurable fuzzy-IVFs on$\left[\rho ,\varsigma \right]$. Then, the fuzzy interval left and right Riemann–Liouville fractional integral of$\tilde{\mathfrak{S}}\in$$L\left(\left[\rho , \varsigma \right],{\mathbb{F}}_0\right)$with order$\alpha >0$are defined by:

$${\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\rho }^y{\left(y-\mathrm{t}\right)}^{\alpha -1}\tilde{\mathfrak{S}}\left(\mathrm{t}\right)d\mathrm{t}, \left(y>\rho \right),$$

(6)

and:

$${\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(y\right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_y^{\mathit{b}}{\left(\mathrm{t}-y\right)}^{\alpha -1}\tilde{\mathfrak{S}}\left(\mathrm{t}\right)d\mathrm{t}, (y<\varsigma )$$

(7)

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

$$\begin{gathered} {\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(y\right)\right]}^{\phi }=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\rho }^y{\left(y-\mathrm{t}\right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(\mathrm{t}\right)d\mathrm{t} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\rho }^y{\left(y-\mathrm{t}\right)}^{\alpha -1}\left[{\mathfrak{S}}_{\ast }\left(\mathrm{t}, \phi \right),{\mathfrak{S}}^{\ast }\left(\mathrm{t}, \phi \right)\right]d\mathrm{t}, \left(y>\rho \right), \end{gathered}$$

(8)

where:

$${\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\ast }\left(y, \phi \right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\rho }^y{\left(y-\mathrm{t}\right)}^{\alpha -1}{\mathfrak{S}}_{*}\left(\mathrm{t}, \phi \right)d\mathrm{t}, \left(y>\rho \right),$$

(9)

and:

$${\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}^{\ast }\left(y, \phi \right)=\frac{1}{\Gamma \left(\alpha \right)}{{\displaystyle \int }}_{\rho }^y{\left(y-\mathrm{t}\right)}^{\alpha -1}{\mathfrak{S}}^{\ast }\left(\mathrm{t}, \phi \right)d\mathrm{t}, \left(y>\rho \right),$$

(10)

Similarly, we can define right Riemann–Liouville fractional integral$\tilde{\mathfrak{S}}\left(y\right)$based on left and right end point functions.

Theorem 1.

Ref. [27] Let $\tilde{\mathfrak{S}}:\left[c, d\right]\to {\mathbb{F}}_0$ be a convex fuzzy-IVF on $\left[c, d\right],$ whose $\phi$-cuts establish the series of IVFs ${\mathfrak{S}}_{\mathsf{\phi }}:\left[\mu , \nu \right]\subset ℝ\to {ℝ}_{\mathrm{C}}{}^{+}$ are given by ${\mathfrak{S}}_{\mathsf{\phi }}\left(y\right)=\left[{\mathfrak{S}}_{\ast }\left(y,\phi \right), {\mathfrak{S}}^{\ast }\left(y,\phi \right)\right]$ for all $y\in \left[\mu , \nu \right]$ and for all $\phi \in \left[0, 1\right]$. If $\tilde{\mathfrak{S}}\in L\left(\left[c, d\right],{\mathbb{F}}_0\right)$, then:

$$\tilde{\mathfrak{S}}\left(\frac{\mu +\nu }{2}\right)\preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{2{\left(\nu -\mu \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\mu \right)\right]\preccurlyeq \frac{\tilde{\mathfrak{S}}\left(\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\nu \right)}{2}.$$

(11)

Theorem 2.

Ref. [27] Let $\tilde{\mathfrak{S}},\widetilde{𝓙} :\left[\mu , \nu \right]\to {\mathbb{F}}_0$ be two convex FIVFs. Then, from $\phi$-cuts, we establish the series of IVFs ${\mathfrak{S}}_{\phi }, {𝓙}_{\phi }:\left[\mu , \nu \right]\subset ℝ\to {ℝ}_I^{+}$ are given by ${\mathfrak{S}}_{\phi }\left(x\right)=\left[{\mathfrak{S}}_{\ast }\left(x,\phi \right), {\mathfrak{S}}^{\ast }\left(x,\phi \right)\right]$ and ${𝓙}_{\phi }\left(x\right)=\left[{𝓙}_{\ast }\left(x,\phi \right), {𝓙}^{\ast }\left(x,\phi \right)\right]$ for all $x\in \left[\mu , \nu \right]$ and for all $\phi \in \left[0, 1\right]$. If $\tilde{\mathfrak{S}}\tilde{\times }\widetilde{𝓙}\in L\left(\left[\mu , \nu \right],{\mathbb{F}}_0\right)$ is fuzzy Riemann integrable, then:

$$\begin{gathered} \frac{\Gamma \left(\alpha +1\right)}{2{\left(\nu -\mu \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\nu \right)\tilde{\times }\widetilde{𝓙}\left(\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\alpha }\tilde{\mathfrak{S}}\left(\mu \right)\tilde{\times }\widetilde{𝓙}\left(\mu \right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\preccurlyeq \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\tilde{\mathcal{M}}\left(\mu ,\nu \right)\widetilde{+}\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\tilde{\mathcal{N}}\left(\mu ,\nu \right), \end{gathered}$$

(12)

and:

$$\begin{gathered} \tilde{\mathfrak{S}}\left(\frac{\mu +\nu }{2}\right)\tilde{\times }\widetilde{𝓙}\left(\frac{\mu +\nu }{2}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\nu -\mu \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\nu \right)\tilde{\times }\widetilde{𝓙}\left(\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\mu \right)\tilde{\times }\widetilde{𝓙}\left(\mu \right)\right] \\ +\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\tilde{\mathcal{M}}\left(\mu ,\nu \right)+\frac{1}{2}\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\tilde{\mathcal{N}}\left(\mu ,\nu \right), \end{gathered}$$

(13)

where $\tilde{\mathcal{M}}\left(\mu ,\nu \right)=\tilde{\mathfrak{S}}\left(\mu \right)\tilde{\times }\widetilde{𝓙}\left(\mu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\nu \right)\tilde{\times }\widetilde{𝓙}\left(\nu \right),$ $\tilde{\mathcal{N}}\left(\mu ,\nu \right)=\tilde{\mathfrak{S}}\left(\mu \right)\tilde{\times }\widetilde{𝓙}\left(\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\nu \right)\tilde{\times }\widetilde{𝓙}\left(\mu \right),$ ${\mathcal{M}}_{\phi }\left(\mu ,\nu \right)=\left[{\mathcal{M}}_{\ast }\left(\left(\mu ,\nu \right), \phi \right), {\mathcal{M}}^{\ast }\left(\left(\mu ,\nu \right), \phi \right)\right]$, and ${\mathcal{N}}_{\phi }\left(\mu ,\nu \right)=\left[{\mathcal{N}}_{\ast }\left(\left(\mu ,\nu \right), \phi \right), {\mathcal{N}}^{\ast }\left(\left(\mu ,\nu \right), \phi \right)\right].$

2.1. Fuzzy-Interval-Valued Double Integrals and Convexity

For coordinated interval-valued function $\mathfrak{S}\left(x,y\right)$ and coordinated fuzzy-interval-valued function $\tilde{\mathfrak{S}}\left(x,y\right)$, interval and fuzzy-interval Riemann–Liouville-type integrals are defined as:

Definition 3.

Ref. [34] A function $\tilde{\mathfrak{S}}:$ $\mathsf{\Delta }=\left[\rho , \varsigma \right]\times \left[\mu , \nu \right]\to {\mathbb{F}}_0$ is called a fuzzy-interval double integrable ($FD$-integrable) on $\mathsf{\Delta }$ if there exists $\tilde{B}\in {\mathbb{F}}_0$ such that, for each $ϵ$, there exists $\delta >0$ such that:

$$d\left(S\left(\tilde{\mathfrak{S}},P, \delta ,\mathsf{\Delta }\right), \tilde{B}\right)<ϵ,$$

for every Riemann sum of $\tilde{\mathfrak{S}}$ corresponding to $P\in \mathcal{P}\left(\delta , \mathsf{\Delta }\right)$ and for arbitrary choice $\left({\eta }_i,w_j\right)\in \left[x_{i-1}, x_i\right]\times \left[{\varpi }_{j-1}, {\varpi }_j\right]$ for $1\le i\le k$ and $1\le j\le n.$ Then, we say that $\tilde{B}$ is the $FR$-integral of $\tilde{\mathfrak{S}}$ on $\mathsf{\Delta }$ and is denote by $\tilde{B}=\left(FD\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,\varpi \right)d\varpi dx$ or $\tilde{B}=\left(FD\right){{\displaystyle \iint }}_{\mathsf{\Delta }}^{ }\tilde{\mathfrak{S}}dA$.

Definition 4.

Ref. [34] A fuzzy-interval-valued map $\tilde{\mathfrak{S}}:\mathsf{\Delta }=\left[\rho , \varsigma \right]\times \left[\mu , \nu \right]\to {\mathbb{F}}_0$ is called FIVF on coordinates. Then, from $\phi$-cuts, we establish the series of IVFs ${\mathfrak{S}}_{\phi }:\mathsf{\Delta }\subset {ℝ}^2\to {ℝ}_I$ on coordinates are given by ${\mathfrak{S}}_{\phi }\left(x,\varpi \right)=\left[{\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right),\phi \right), {\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right),\phi \right)\right]$ for all $\left(x,\varpi \right)\in \Delta .$ Here, for each $\phi \in \left[0, 1\right],$ the end point real valued functions ${\mathfrak{S}}_{\ast }\left(.,\phi \right), {\mathfrak{S}}^{\ast }\left(.,\phi \right):\left(x,\varpi \right)\to ℝ$ are called lower and upper functions of ${\mathfrak{S}}_{\phi }$.

Definition 5.

Ref. [34] Let $\tilde{\mathfrak{S}}:\Delta \subset {ℝ}^2\to {\mathbb{F}}_0$ be a coordinate FIVF. Then $\tilde{\mathfrak{S}}\left(x,y\right)$ is said to be continuous at $\left(x,y\right)\in \Delta =\left[\rho , \varsigma \right]\times \left[\mu , \nu \right],$ if for each $\phi \in \left[0, 1\right],$ both end point functions ${\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right)$ and ${\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)$ are continuous at $\left(x,y\right)\in \Delta .$

Definition 6.

Ref. [34] Let $\tilde{\mathfrak{S}}:\Delta \subset {ℝ}^2\to {\mathbb{F}}_0$ be a coordinate fuzzy-IVF. Then $\tilde{\mathfrak{S}}\left(x,y\right)$ is said to be continuous at $\left(x,y\right)\in \Delta =\left[\rho , \varsigma \right]\times \left[\mu , \nu \right],$ if for each $\phi \in \left[0, 1\right]$, both end point functions ${\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right)$ and ${\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)$ are continuous at $\left(x,y\right)\in \Delta .$

Definition 7.

Ref. [34] Let $\tilde{\mathfrak{S}}:\Delta \subset {ℝ}^2\to {\mathbb{F}}_0$ be a fuzzy-IVF on coordinates. Then fuzzy double integral of $\tilde{\mathfrak{S}}$ over $\Delta =\left[\rho , \varsigma \right]\times \left[\mu , \nu \right],$ denoted by $\left(FD\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)dydx$, it is defined level-wise by:

$$\begin{gathered} {\left[\left(FD\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)dydx\right]}^{\begin{array}{c} \\ \phi \end{array}}=\left(ID\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\mathfrak{S}}_{\phi }\left(x,y\right)dydx \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left\{{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(\left(x,y\right),\phi \right)dydx:\mathfrak{S}\left(\left(x,y\right),\phi \right)\in {\mathfrak{O}}_{\Delta }\right\}, \end{gathered}$$

(14)

for all $\phi \in \left[0, 1\right],$ where ${\mathfrak{O}}_{\Delta }$ is the collection of end point functions of IVFs on $\Delta$. $\tilde{\mathfrak{S}}$ is $FD$-integrable over $\Delta$ if $\left(FD\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)dydx\in {\mathbb{F}}_0.$ Note that, if both end point functions are Lebesgue-integrable, then $\tilde{\mathfrak{S}}$ is fuzzy double Aumann-integrable function over $\Delta$.

Theorem 3.

Ref. [34] Let $\tilde{\mathfrak{S}}:\Delta \subset {ℝ}^2\to {\mathbb{F}}_0$ be a fuzzy-IVF on coordinates, whose $\phi$-cuts establish the series of IVFs ${\mathfrak{S}}_{\phi }:\Delta \subset {ℝ}^2\to {ℝ}_I$ are given by ${\mathfrak{S}}_{\phi }\left(x,y\right)=\left[{\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right), {\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)\right]$ for all $\left(x,y\right)\in \Delta =\left[\rho , \varsigma \right]\times \left[\mu , \nu \right]$ and for all $\phi \in \left[0, 1\right].$ Then $\tilde{\mathfrak{S}}$ is $FD$-integrable over $\Delta$ if and only if ${\mathfrak{S}}_{\ast }\left(x,\phi \right)$ and ${\mathfrak{S}}^{\ast }\left(x,\phi \right)$ are both $D$-integrable over $\Delta$. Moreover, if $\tilde{\mathfrak{S}}$ is $FD$-integrable over $\Delta$, then:

$$\begin{array}{ll}{\left[\left(FD\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)dydx\right]}^{\begin{array}{c} \\ \phi \end{array}}& =\left[\left(D\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right)dydx, \left(D\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)dydx\right]\\ & =\left(ID\right){{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\mathfrak{S}}_{\phi }\left(x,y\right)dydx\end{array}$$

(15)

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

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

Definition 8.

Ref. [34] The FIVF $\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0$ is said to be coordinated convex FIVF on $\Delta$ if:

$$\begin{gathered} \tilde{\mathfrak{S}}\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , s\mu +\left(1-s\right)\nu \right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\preccurlyeq \epsilon s\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}\epsilon \left(1-s\right)\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}\left(1-\epsilon \right)s\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}\left(1-\epsilon \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right), \end{gathered}$$

(16)

for all $\left(\rho , \varsigma \right), \left(\mu ,\nu \right)\in \Delta$, and $\epsilon ,s\in \left[0, 1\right]$, where $\tilde{\mathfrak{S}}\left(x\right)\succcurlyeq \tilde{0}$. If inequality (16) is reversed, then $\tilde{\mathfrak{S}}$ is called coordinate concave FIVF on $\Delta$.

Lemma 1.

Ref. [34] Let $\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0$ be a coordinated FIVF on $\Delta$. Then, $\tilde{\mathfrak{S}}$ is coordinated convex FIVF on $\Delta$ if and only if there exist two coordinated convex FIVFs ${\tilde{\mathfrak{S}}}_x:\left[\mu ,\nu \right]\to {\mathbb{F}}_0$, ${\tilde{\mathfrak{S}}}_x\left(w\right)=\tilde{\mathfrak{S}}\left(x,w\right)$ and ${\tilde{\mathfrak{S}}}_y:\left[\rho ,\varsigma \right]\to {\mathbb{F}}_0$, ${\tilde{\mathfrak{S}}}_y\left(z\right)=\tilde{\mathfrak{S}}\left(z,y\right)$.

Theorem 4.

Ref. [34] Let $\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0$ be a FIVF on $\Delta$. Then, from $\phi$-levels, we get the collection of IVFs ${\mathfrak{S}}_{\phi }:\Delta \to {ℝ}_I^{+}\subset {ℝ}_I$ are given by:

$${\mathfrak{S}}_{\phi }\left(x,\varpi \right)=\left[{\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right),\phi \right), {\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right),\phi \right)\right],$$

(17)

for all $\left(x,\varpi \right)\in \Delta$ and for all $\phi \in \left[0, 1\right]$. Then, $\tilde{\mathfrak{S}}$ is coordinated convex FIVF on $\Delta$, if and only if, for all $\phi \in \left[0, 1\right],$ ${\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right), \phi \right)$ and ${\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right), \phi \right)$ are coordinated convex function.

Example 1.

We consider the FIVFs$\tilde{\mathfrak{S}}:\left[0, 1\right]\times \left[0, 1\right]\to {\mathbb{F}}_0$defined by:

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

Then, for each$\theta \in \left[0, 1\right],$we have${\mathfrak{S}}_{\theta }\left(x\right)=\left[2\theta \left(6+e^x\right)\left(6+e^{\varpi }\right),\left(4+2\theta \right)\left(6+e^x\right)\left(6+e^{\varpi }\right)\right]$. Since end point functions${\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right),\theta \right),$${\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right),\theta \right)$are coordinate concave functions for each$\theta \in \left[0, 1\right]$. Hence,$\tilde{\mathfrak{S}}\left(x,\varpi \right)$is coordinate concave FIVF.

From Lemma 1 and Example 1, we can easily note the each convex FIVF is coordinated convex FIVF. However, the converse is not true.

Remark 2.

If one takes${\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right),\phi \right)={\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right),\phi \right)$with$\phi =1$, then$\mathfrak{S}$is known as coordinated function if$\mathfrak{S}$satisfies the coming inequality:

$$\begin{gathered} \mathfrak{S}\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , s\mu +\left(1-s\right)\nu \right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \epsilon s\mathfrak{S}\left(\rho ,\mu \right)+\epsilon \left(1-s\right)\mathfrak{S}\left(\rho ,\nu \right)+\left(1-\epsilon \right)s\mathfrak{S}\left(\varsigma ,\mu \right)+\left(1-\epsilon \right)\left(1-s\right)\mathfrak{S}\left(\varsigma ,\nu \right) \end{gathered}$$

it is valid, which is defined by Dragomir [41].

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right),\phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right),\phi \right)$with$\phi =1$and${\mathfrak{S}}_{\ast }\left(\left(x,\varpi \right),\phi \right)$is an affine function and${\mathfrak{S}}^{\ast }\left(\left(x,\varpi \right),\phi \right)$is a concave function. If coming inequality:

$$\begin{gathered} \mathfrak{S}\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , s\mu +\left(1-s\right)\nu \right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\supseteq \epsilon s\mathfrak{S}\left(\rho ,\mu \right)+\epsilon \left(1-s\right)\mathfrak{S}\left(\rho ,\nu \right)+\left(1-\epsilon \right)s\mathfrak{S}\left(\varsigma ,\mu \right)+\left(1-\epsilon \right)\left(1-s\right)\mathfrak{S}\left(\varsigma ,\nu \right), \end{gathered}$$

is valid, then$\mathfrak{S}$is named as coordinated IVF which is defined by Zhao et al. [32], Definition 2 and Example 2.

2.2. Fuzzy-Interval-Valued Fractional Integrals on Coordinated Functions

For coordinated interval-valued function $\mathfrak{S}\left(x,y\right)$ and coordinated fuzzy-interval-valued function $\tilde{\mathfrak{S}}\left(x,y\right)$, double interval and double fuzzy-interval Riemann–Liouville-type integrals are defined as:

Definition 9.

Ref. [33] Let $\mathfrak{S}:\Delta \to {ℝ}_I^{+}$ and $\mathfrak{S}\in \mathit{ȷ}{\mathfrak{O}}_{\Delta }$. The double interval Riemann–Liouville-type integrals ${\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }, {\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-} }^{\alpha , \beta },$ ${\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta },{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-} }^{\alpha , \beta }$ of $\mathfrak{S}$ order $\alpha , \beta >0$ are defined by:

$${\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta } \mathfrak{S}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_{\mu }^y{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\mathfrak{S}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} \left(x>\rho , y>\mu \right),$$

(18)

$${\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \mathfrak{S}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_y^{\nu }{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\mathfrak{S}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} (x>\rho , y<\nu )$$

(19)

$${\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta } \mathfrak{S}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_{\mu }^y{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\mathfrak{S}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y>\mu \right),$$

(20)

$${\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \mathfrak{S}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_y^{\nu }{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\mathfrak{S}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} (x<\varsigma , y<\nu )$$

(21)

Now, with help of concepts related to fuzzy-interval-valued functions, which are given by Khan et al. [34], Allahviranloo et al. [49], and Budak et al. [33], we introduce the following fuzzy-interval-valued Riemann–Liouville double fractional integral of the function $\tilde{\mathfrak{S}}\left(x,y\right)$ by:

Definition 10.

Let$\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0$and$\tilde{\mathfrak{S}}\in \mathcal{F}{\mathfrak{O}}_{\Delta }$. The double fuzzyinterval Riemann–Liouville-type integrals${\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }, {\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-} }^{\alpha , \beta }$$,{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta },{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-} }^{\alpha , \beta }$of$\mathfrak{S}$order$\alpha , \beta >0$ are defined by:

$${\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_{\mu }^y{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\tilde{\mathfrak{S}}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} \left(x>\rho , y>\mu \right),$$

(22)

$${\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_y^{\nu }{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\tilde{\mathfrak{S}}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} (x>\rho , y<\nu )$$

(23)

$${\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_{\mu }^y{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\tilde{\mathfrak{S}}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y\mu ,>\right),$$

(24)

$${\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_y^{\nu }{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\tilde{\mathfrak{S}}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} (x<\varsigma , y<\nu )$$

(25)

The fuzzy interval left and right Riemann–Liouville fractional integral of$\tilde{\mathfrak{S}}$based on left and right end point functions can be defined level-wise, that is, for all$\phi \in \left[0, 1\right]$, Equation (22), we have:

$$\begin{array}{l}{\left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)\right]}^{\phi }\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_{\mu }^y{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)d\mathrm{s}d\mathrm{t}, \left(x>\rho , y>\mu \right),\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_{\mu }^y{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\left[{\mathfrak{S}}_{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)\right]d\mathrm{s}d\mathrm{t} \left(x>\rho , y>\mu \right)\end{array}$$

(26)

where:

$${\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta } {\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_{\mu }^y{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}{\mathfrak{S}}_{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)d\mathrm{s}d\mathrm{t}, \left(x>\rho , y>\mu \right),$$

(27)

and:

$${\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta } {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_{\mu }^y{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}{\mathfrak{S}}^{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)d\mathrm{s}d\mathrm{t}, \left(x>\rho , y>\mu \right),$$

(28)

Now, for all$\phi \in \left[0, 1\right]$, Equation (23) we have:

$$\begin{array}{l}{\left[{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)\right]}^{\phi }\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_y^{\nu }{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)d\mathrm{s}d\mathrm{t} (x>\rho , y<\nu )\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_y^{\nu }{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\left[{\mathfrak{S}}_{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)\right]d\mathrm{s}d\mathrm{t} (x>\rho , y<\nu )\end{array}$$

where:

$${\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_y^{\nu }{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}{\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)d\mathrm{s}d\mathrm{t} (x>\rho , y<\nu )$$

(29)

and:

$${\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_{\rho }^x{{\displaystyle \int }}_y^{\nu }{\left(x-\mathrm{t}\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}{\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)d\mathrm{s}d\mathrm{t} (x>\rho , y<\nu )$$

(30)

Similarly, for all$\phi \in \left[0, 1\right]$, Equation (24) we have:

$$\begin{array}{l}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_{\mu }^y{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\tilde{\mathfrak{S}}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y>\mu \right),\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_{\mu }^y{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}\left[{\mathfrak{S}}_{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)\right]d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y>\mu \right)\end{array}$$

(31)

where:

$${\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_{\mu }^y{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}{\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y>\mu \right),$$

and:

$${\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_{\mu }^y{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(y-\mathrm{s}\right)}^{\beta -1}{\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y>\mu \right),$$

Similarly, for all$\phi \in \left[0, 1\right]$, Equation (25) we have:

$$\begin{array}{l}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(x,y\right)\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_y^{\nu }{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\tilde{\mathfrak{S}}\left(\mathrm{t},\mathrm{s}\right)d\mathrm{s}d\mathrm{t} (x<\varsigma , y<\nu )\\ =\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_y^{\nu }{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}\left[{\mathfrak{S}}_{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\mathrm{t},\mathrm{s}\right), \phi \right)\right]d\mathrm{s}d\mathrm{t} \left(x<\varsigma , y<\nu \right),\end{array}$$

(32)

where:

$${\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_y^{\nu }{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}{\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)d\mathrm{s}d\mathrm{t} (x<\varsigma , y<\nu )$$

(33)

and:

$${\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)=\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{{\displaystyle \int }}_x^{\varsigma }{{\displaystyle \int }}_y^{\nu }{\left(\mathrm{t}-x\right)}^{\alpha -1}{\left(\mathrm{s}-y\right)}^{\beta -1}{\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)d\mathrm{s}d\mathrm{t} (x<\varsigma , y<\nu )$$

(34)

Remark 3.

It can be easily noted that if${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then from Definition 10, we achieve Definition 9.

3. Fuzzy-Interval Fractional Hermite–Hadamard Inequalities

In this section, we shall continue with the following fractional $HH$ inequality for convex fuzzy-IVFs, and we also give some fractional inequalities for the product of two coordinated convex fuzzy-IVF through FOR.

Theorem 5.

Let$\tilde{\mathfrak{S}}:\Delta \to {\mathbb{F}}_0$be a coordinate convex FIVF on$\Delta$. Then, from$\phi$-cuts, we establish that the series of IVFs${\mathfrak{S}}_{\phi }:\Delta \to {ℝ}_I^{+}$are given by${\mathfrak{S}}_{\phi }\left(x,y\right)=\left[{\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right), {\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)\right]$for all$\left(x,y\right)\in \Delta$and for all$\phi \in \left[0, 1\right]$. If$\tilde{\mathfrak{S}}\in \mathcal{F}{\mathfrak{O}}_{\Delta }$, and then the following inequalities hold:

$$\begin{array}{ll}\tilde{\mathfrak{S}}& \left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\frac{\mu +\nu }{2}\right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha }\tilde{\mathfrak{S}}\left(\rho ,\frac{\mu +\nu }{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\mu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{8{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\right]\\ & +\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+} }^{ \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\right]\\ & \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)}{4}.\end{array}$$

(35)

If$\mathfrak{S}\left(x\right)$coordinated concave FIVF then:

$$\begin{array}{ll}\tilde{\mathfrak{S}}& \left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ & \succcurlyeq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\frac{\mu +\nu }{2}\right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\frac{\mu +\nu }{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\mu \right)\right]\\ & \succcurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\right]\\ & \succcurlyeq \frac{\Gamma \left(\alpha +1\right)}{8{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\right]\\ & +\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+} }^{ \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\right]\\ & \succcurlyeq \frac{\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)}{4}.\end{array}$$

(36)

Proof. 

Let$\tilde{\mathfrak{S}}:\left[\rho , \varsigma \right]\to {\mathbb{F}}_0$be a coordinated convex FIVF. Then, by hypothesis, we have

$$4\tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\preccurlyeq \tilde{\mathfrak{S}}\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , \epsilon \mu +\left(1-\epsilon \right)\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\left(1-\epsilon \right)\rho +\epsilon \varsigma , \left(1-\epsilon \right)\mu +\epsilon \nu \right).$$

By using Theorem 4, for every$\phi \in \left[0, 1\right]$, we have:

$$\begin{array}{ll}4{\mathfrak{S}}_{\ast }& \left(\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right), \phi \right)\\ & \le {\mathfrak{S}}_{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , \epsilon \mu +\left(1-\epsilon \right)\nu \right), \phi \right)+{\mathfrak{S}}_{\ast }\left(\left(\left(1-\epsilon \right)\rho +\epsilon \varsigma , \left(1-\epsilon \right)\mu +\epsilon \nu \right), \phi \right), \\ 4{\mathfrak{S}}^{\ast }& \left(\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right), \phi \right)\\ & \le {\mathfrak{S}}^{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , \epsilon \mu +\left(1-\epsilon \right)\nu \right), \phi \right)+{\mathfrak{S}}^{\ast }\left(\left(\left(1-\epsilon \right)\rho +\epsilon \varsigma , \left(1-\epsilon \right)\mu +\epsilon \nu \right),\phi \right).\end{array}$$

By using Lemma 1, we have:

$$\begin{array}{c}2{\mathfrak{S}}_{\ast }\left(\left(x,\frac{\mu +\nu }{2}\right), \phi \right)\le {\mathfrak{S}}_{\ast }\left(\left(x, \epsilon \mu +\left(1-\epsilon \right)\nu \right), \phi \right)+{\mathfrak{S}}_{\ast }\left(\left(x, \left(1-\epsilon \right)\mu +\epsilon \nu \right), \phi \right),\\ 2{\mathfrak{S}}^{\ast }\left(\left(x,\frac{\mu +\nu }{2}\right), \phi \right)\le {\mathfrak{S}}^{\ast }\left(\left(x, \epsilon \mu +\left(1-\epsilon \right)\nu \right), \phi \right)+{\mathfrak{S}}^{\ast }\left(\left(x, \left(1-\epsilon \right)\mu +\epsilon \nu \right),\phi \right),\end{array}$$

(37)

and:

$$\begin{array}{c}2{\mathfrak{S}}_{\ast }\left(\left(\frac{\rho +\varsigma }{2},y\right), \phi \right)\le {\mathfrak{S}}_{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right), \phi \right)+{\mathfrak{S}}_{\ast }\left(\left(\left(1-\epsilon \right)\rho +t\varsigma , y\right), \phi \right),\\ 2{\mathfrak{S}}^{\ast }\left(\left(\frac{\rho +\varsigma }{2},y\right), \phi \right)\le {\mathfrak{S}}^{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right), \phi \right)+{\mathfrak{S}}^{\ast }\left(\left(\left(1-\epsilon \right)\rho +t\varsigma , y\right),\phi \right).\end{array}$$

(38)

From (37) and (38), we have:

$$\begin{array}{ll}2[{\mathfrak{S}}_{\ast }& \left(\left(x,\frac{\mu +\nu }{2}\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(x,\frac{\mu +\nu }{2}\right), \phi \right)]\\ & {\le }_I\left[{\mathfrak{S}}_{\ast }\left(\left(x, \epsilon \mu +\left(1-\epsilon \right)\nu \right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(x, \epsilon \mu +\left(1-\epsilon \right)\nu \right), \phi \right)\right]\\ & +\left[{\mathfrak{S}}_{\ast }\left(\left(x, \left(1-\epsilon \right)\mu +\epsilon \nu \right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(x, \left(1-\epsilon \right)\mu +\epsilon \nu \right),\phi \right)\right],\end{array}$$

and:

$$\begin{array}{ll}2[{\mathfrak{S}}_{\ast }& \left(\left(\frac{\rho +\varsigma }{2},y\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\frac{\rho +\varsigma }{2},y\right), \phi \right)]\\ & {\le }_I\left[{\mathfrak{S}}_{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right), \phi \right)\right]\\ & +\left[{\mathfrak{S}}_{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right), \phi \right),{\mathfrak{S}}^{\ast }\left(\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right),\phi \right)\right],\end{array}$$

It follows that:

$${\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right){\le }_I{\mathfrak{S}}_{\phi }\left(x, \epsilon \mu +\left(1-\epsilon \right)\nu \right)+{\mathfrak{S}}_{\phi }\left(x, \left(1-\epsilon \right)\mu +\epsilon \nu \right),$$

(39)

and:

$${\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right){\le }_I{\mathfrak{S}}_{\phi }\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right)+{\mathfrak{S}}_{\phi }\left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , y\right)$$

(40)

Since${\mathfrak{S}}_{\phi }\left(x,.\right)$and${\mathfrak{S}}_{\phi }\left(.,y\right)$are both coordinated convex-IVFs, then from inequality (11), for every$\phi \in \left[0, 1\right]$, inequalities (39) and (40) we have:

$${\mathfrak{S}}_{{\phi }_x}\left(\frac{\mu +\nu }{2}\right){\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{{\phi }_x}\left(\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{{\phi }_x}\left(\mu \right)\right]{\le }_I\frac{{\mathfrak{S}}_{{\phi }_x}\left(\mu \right)+{\mathfrak{S}}_{{\phi }_x}\left(\nu \right)}{2}.$$

(41)

and:

$${\mathfrak{S}}_{{\phi }_y}\left(\frac{\rho +\varsigma }{2}\right){\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{{\phi }_y}\left(\varsigma \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{{\phi }_y}\left(\rho \right)\right]{\le }_I\frac{{\mathfrak{S}}_{{\phi }_y}\left(\rho \right)+{\mathfrak{S}}_{{\phi }_y}\left(\varsigma \right)}{2}$$

(42)

Since${\mathfrak{S}}_{{\phi }_x}\left(w\right)={\mathfrak{S}}_{\phi }\left(x,w\right)$, then (41) can be written as:

$${\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right){\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(x,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(x,\mu \right)\right]{\le }_I\frac{{\mathfrak{S}}_{\phi }\left(x,\mu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)}{2}.$$

(43)

That is:

$$\begin{array}{l}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\beta }{2{\left(\nu -\mu \right)}^{\beta }}\left[{{\displaystyle \int }}_{\mu }^{\nu }{\left(\nu -s\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,s\right)ds+{{\displaystyle \int }}_{\mu }^{\nu }{\left(s-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,s\right)ds\right]{\le }_I\frac{{\mathfrak{S}}_{\phi }\left(x,\mu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)}{2}.\end{array}$$

Multiplying double inequality (43) by$\frac{\alpha {\left(\varsigma -x\right)}^{\alpha -1}}{2{\left(\varsigma -\rho \right)}^{\alpha }}$and integrating it with respect to$x$over$\left[\rho , \varsigma \right],$we have:

$$\begin{array}{l}\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right){\left(\varsigma -x\right)}^{\alpha -1}dx\\ {\le }_I{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(\varsigma -x\right)}^{\alpha -1}{\left(\nu -s\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,s\right)dsdx+{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(\varsigma -x\right)}^{\alpha -1}{\left(s-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,s\right)dsdx\\ {\le }_I\frac{\alpha }{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(\varsigma -x\right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\mu \right)dx+{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(\varsigma -x\right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\nu \right)dx\right]\end{array}$$

(44)

Again, multiplying double inequality (43) by$\frac{\alpha {\left(x-\rho \right)}^{\alpha -1}}{2{\left(\varsigma -\rho \right)}^{\alpha }}$and integrating it with respect to$x$over$\left[\rho , \varsigma \right],$we have:

$$\begin{array}{ll}\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }& {\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right){\left(\varsigma -x\right)}^{\alpha -1}dx\\ & {\le }_I\frac{\alpha \beta }{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }}{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(x-\rho \right)}^{\alpha -1}{\left(\nu -s\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,s\right)dsdx\\ & +\frac{\alpha \beta }{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }}{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(x-\rho \right)}^{\alpha -1}{\left(s-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,s\right)dsdx\\ & {\le }_I\frac{\alpha }{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(x-\rho \right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\mu \right)dx+{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(x-\rho \right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\nu \right)dx\right]\end{array}$$

(45)

From (44), we have:

$$\begin{array}{ll}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}& \left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\right]\\ & {\le }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\right]\\ & {\le }_I\frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\right]\end{array}$$

(46)

From (45), we have:

$$\begin{array}{ll}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}& \left[{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\right]\\ & {\le }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\right]\\ & {\le }_I\frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\right]\end{array}$$

(47)

Since from$\phi$-cuts, we obtain the collection of IVFs${\mathfrak{S}}_{\phi }:\Delta \to {ℝ}_I^{+}$, then we have:

$$\begin{array}{ll}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}& \left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\frac{\mu +\nu }{2}\right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\right]\end{array}$$

(48)

and:

$$\begin{array}{ll}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}& \left[{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\frac{\mu +\nu }{2}\right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\right]\end{array}$$

(49)

Similarly, since${\tilde{\mathfrak{S}}}_y\left(z\right)=\tilde{\mathfrak{S}}\left(z,y\right)$then, from the (42), (48), and (49), we have:

$$\begin{array}{ll}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}& \left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2}, \nu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\right]\end{array}$$

(50)

and:

$$\begin{array}{ll}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\alpha }}& \left[{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\mu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\right]\\ & \preccurlyeq \frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\right]\end{array}$$

(51)

After adding the inequalities (48), (49), (50), and (51), we will obtain as a result the second, third, and fourth inequalities of (35).

Now, from the left part of inequality (11), for all$\phi \in \left[0, 1\right]$we have:

$${\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right){\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\right]$$

(52)

and:

$${\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right){\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\right]$$

(53)

Summing the inequalities (52) and (53), we obtain the following inequality:

$$\begin{gathered} {\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right){\le }_I\frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\right] \\ +\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\right] \end{gathered}$$

Similarly, since for$\phi \in \left[0, 1\right]$, we obtain the collection of IVFs${\mathfrak{S}}_{\phi }:\Delta \to {ℝ}_I^{+}$, then the above inequality can be as follows:

$$\begin{array}{l}\tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ \preccurlyeq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \tilde{\mathfrak{S}}\left(\varsigma ,\frac{\mu +\nu }{2}\right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\frac{\mu +\nu }{2}\right)\right]\widetilde{+}\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\mu \right)\right].\end{array}$$

(54)

This is the first inequality of (35).

Now, from the right part of inequality (11), for all$\phi \in \left[0, 1\right]$we have:

$$\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\right]{\le }_I\frac{{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)}{2}.$$

(55)

$$\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\right]{\le }_I\frac{{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)}{2}.$$

(56)

$$\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\right]{\le }_I\frac{{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)}{2}$$

(57)

$$\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\right]{\le }_I\frac{{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)}{2}$$

(58)

Summing inequalities (55), (56), (57), and (58), and then taking multiplication of the resultant with$\frac{1}{4}$, we have:

$$\begin{array}{ll}\frac{\Gamma \left(\alpha +1\right)}{8{\left(\varsigma -\rho \right)}^{\alpha }}& \left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\right]\\ & +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\mu }^{+}}^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\right]\\ & {\le }_I\frac{{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)}{4}.\end{array}$$

Since from$\phi$-cuts, we obtain the collection of IVFs${\mathfrak{S}}_{\phi }:\Delta \to {ℝ}_I^{+}$, then we have:

$$\begin{array}{ll}\frac{\Gamma \left(\alpha +1\right)}{8{\left(\varsigma -\rho \right)}^{\alpha }}[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha }\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha }\tilde{\mathfrak{S}}\left(\rho ,\mu \right)& \widetilde{+}{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \tilde{\mathfrak{S}}\left(\rho ,\nu \right)]\\ & \widetilde{+}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}{\mathit{ȷ}}_{{\mu }^{+}}^{\beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\right]\\ & \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)}{4}.\end{array}$$

(59)

This is last inequality of (35) and the result has been proven. □

Remark 4.

If one is to take$\alpha =1$and$\beta =1$, then from (35), we achieve the coming inequality, see [34]:

$$\begin{array}{ll}\tilde{\mathfrak{S}}(\frac{\rho +\varsigma }{2},& \frac{\mu +\nu }{2})\\ & \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{\varsigma -\rho }{{\displaystyle \int }}_{\rho }^{\varsigma }\tilde{\mathfrak{S}}\left(x,\frac{\mu +\nu }{2}\right)dx\widetilde{+}\frac{1}{\nu -\mu }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},y\right)dy\right]\preccurlyeq \frac{1}{\left(\varsigma -\rho \right)\left(\nu -\mu \right)} {{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)dydx\\ & \preccurlyeq \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\varsigma -\rho \right)}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }\tilde{\mathfrak{S}}\left(x,\mu \right)dx\widetilde{+} {{\displaystyle \int }}_{\rho }^{\varsigma }\tilde{\mathfrak{S}}\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\nu -\mu \right)}\left[{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(\rho ,y\right)dy\widetilde{+} {{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(\varsigma ,y\right)dy\right]\\ & \preccurlyeq \frac{\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)}{4}.\end{array}$$

(60)

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)$, which is an affine function, and${\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$, which is concave function. If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then Remark 2 is correct, and from (36), we acquire the following inequality, see [33]:

$$\begin{array}{l}\mathfrak{S}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ \supseteq \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \mathfrak{S}\left(\varsigma ,\frac{\mu +\nu }{2}\right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha }\mathfrak{S}\left(\rho ,\frac{\mu +\nu }{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \mathfrak{S}\left(\frac{\rho +\varsigma }{2},\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \mathfrak{S}\left(\frac{\rho +\varsigma }{2},\mu \right)\right]\\ \supseteq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\mathfrak{S}\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \mathfrak{S}\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\mathfrak{S}\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \mathfrak{S}\left(\rho ,\mu \right)\right]\\ \supseteq \frac{\Gamma \left(\alpha +1\right)}{8{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }\mathfrak{S}\left(\varsigma ,\mu \right)\mathfrak{S}{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \mathfrak{S}\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }\mathfrak{S}\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \mathfrak{S}\left(\rho ,\nu \right)\right]\\ +\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+} }^{ \beta }\mathfrak{S}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \mathfrak{S}\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }\mathfrak{S}\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \mathfrak{S}\left(\varsigma ,\mu \right)\right]\\ \supseteq \frac{\mathfrak{S}\left(\rho ,\mu \right)+\mathfrak{S}\left(\varsigma ,\mu \right)+\mathfrak{S}\left(\rho ,\nu \right)+\mathfrak{S}\left(\varsigma ,\nu \right)}{4}.\end{array}$$

(61)

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)$, which is an affine function, and${\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$, which is a concave function. If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then Remark 2 is correct, and from (36), we acquire the following inequality, see [32]:

$$\begin{array}{l}\mathfrak{S}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{2}\left[\frac{1}{\varsigma -\rho }{{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{S}\left(x,\frac{\mu +\nu }{2}\right)dx+\frac{1}{\nu -\mu }{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(\frac{\rho +\varsigma }{2},y\right)dy\right]\subseteq \frac{1}{\left(\varsigma -\rho \right)\left(\nu -\mu \right)} {{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(x,y\right)dydx.\\ \supseteq \frac{\begin{array}{c} \\ 1\end{array}}{4\left(\varsigma -\rho \right)}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{S}\left(x,\mu \right)dx+{{\displaystyle \int }}_{\rho }^{\varsigma }\mathfrak{S}\left(x,\nu \right)dx\right]+\frac{\begin{array}{c} \\ 1\end{array}}{4\left(\nu -\mu \right)}\left[{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(\rho ,y\right)dy+{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(\varsigma ,y\right)dy\right]\\ \supseteq \frac{\mathfrak{S}\left(\rho ,\mu \right)+\mathfrak{S}\left(\varsigma ,\mu \right)+\mathfrak{S}\left(\rho ,\nu \right)+\mathfrak{S}\left(\varsigma ,\nu \right)}{4}\end{array}$$

(62)

If ${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)={\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$ with $\phi =1$, then from (35) we acquire the following iquality, see [40]:

$$\begin{array}{l}\mathfrak{S}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ \le \frac{\Gamma \left(\alpha +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \mathfrak{S}\left(\varsigma ,\frac{\mu +\nu }{2}\right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha }\mathfrak{S}\left(\rho ,\frac{\mu +\nu }{2}\right)\right]+\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } \mathfrak{S}\left(\frac{\rho +\varsigma }{2},\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \mathfrak{S}\left(\frac{\rho +\varsigma }{2},\mu \right)\right]\\ \le \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\mathfrak{S}\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \mathfrak{S}\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\mathfrak{S}\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \mathfrak{S}\left(\rho ,\mu \right)\right]\\ \le \frac{\Gamma \left(\alpha +1\right)}{8{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }\mathfrak{S}\left(\varsigma ,\mu \right)\mathfrak{S}{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } \mathfrak{S}\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }\mathfrak{S}\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } \mathfrak{S}\left(\rho ,\nu \right)\right]\\ +\frac{\Gamma \left(\beta +1\right)}{4{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+} }^{ \beta }\mathfrak{S}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \mathfrak{S}\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }\mathfrak{S}\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } \mathfrak{S}\left(\varsigma ,\mu \right)\right]\\ \le \frac{\mathfrak{S}\left(\rho ,\mu \right)+\mathfrak{S}\left(\varsigma ,\mu \right)+\mathfrak{S}\left(\rho ,\nu \right)+\mathfrak{S}\left(\varsigma ,\nu \right)}{4}.\end{array}$$

(63)

Theorem 6.

Let$\tilde{\mathfrak{S}}, \widetilde{𝓙}:\Delta \to {\mathbb{F}}_0$be a coordinate convex FIVFs on$\Delta$. Then, from$\phi$-cuts, we establish that the series of IVFs${\mathfrak{S}}_{\phi },{𝓙}_{\phi }:\Delta \to {ℝ}_I^{+}$are given by${\mathfrak{S}}_{\phi }\left(x,y\right)=\left[{\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right), {\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)\right]$and${𝓙}_{\phi }\left(x,y\right)=\left[{𝓙}_{\ast }\left(\left(x,y\right),\phi \right), {𝓙}^{\ast }\left(\left(x,y\right),\phi \right)\right]$for all$\left(x,y\right)\in \Delta$and for all$\phi \in \left[0, 1\right]$. If$\tilde{\mathfrak{S}}\tilde{\times }\widetilde{𝓙}\in \mathcal{F}{\mathfrak{O}}_{\Delta }$, then the following inequalities hold:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\right]\\ \widetilde{+}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right)\right].\\ \preccurlyeq \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)\widetilde{+}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ \widetilde{+}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)\widetilde{+}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(64)

If$\tilde{\mathfrak{S}}$and$\widetilde{𝓙}$are both coordinate concave FIVFs on$\Delta$, then the above inequality can be written as:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\right]\\ \widetilde{+}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right)\right]\\ \succcurlyeq \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)\widetilde{+}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ \widetilde{+}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)\widetilde{+}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(65)

where:

$$\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)=\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right) \widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\widetilde{+} \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right),$$

$$\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)=\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right),$$

$$\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)=\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right) \widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right)\widetilde{+} \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right),$$

$$\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right)=\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right) \widetilde{+} \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+} \tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\widetilde{+}\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right) ,$$

and for each$\phi \in \left[0, 1\right]$,$\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)$,$\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)$,$\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)$, and$\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right)$are defined as follows:

$$K_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)=\left[K_{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right), K^{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right)\right],$$

$$L_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)=\left[L_{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right), L^{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right)\right],$$

$${\mathcal{M}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)=\left[{\mathcal{M}}_{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right), {\mathcal{M}}^{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right)\right],$$

$${\mathcal{N}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)=\left[{\mathcal{N}}_{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right), {\mathcal{N}}^{\ast }\left(\left(\rho ,\varsigma ,\mu ,\nu \right), \phi \right)\right].$$

Proof. 

Let$\tilde{\mathfrak{S}}$and$\widetilde{𝓙}$both be coordinated convex FIVFs on$\left[\rho , \varsigma \right]\times \left[\mu , \nu \right]$. Then:

$$\begin{array}{ll}\tilde{\mathfrak{S}}& \left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , s\mu +\left(1-s\right)\nu \right)\\ & \preccurlyeq \epsilon s\tilde{\mathfrak{S}}\left(\rho ,\mu \right)\widetilde{+}\epsilon \left(1-s\right)\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\widetilde{+}\left(1-\epsilon \right)s\tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\widetilde{+}\left(1-\epsilon \right)\left(1-s\right)\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right),\end{array}$$

and:

$$\begin{array}{ll}\widetilde{𝓙}& \left(\epsilon \rho +\left(1-\epsilon \right)\varsigma , s\mu +\left(1-s\right)\nu \right)\\ & \preccurlyeq \epsilon s\widetilde{𝓙}\left(\rho ,\mu \right)\widetilde{+}\epsilon \left(1-s\right)\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}\left(1-\epsilon \right)s\widetilde{𝓙}\left(\varsigma ,\mu \right)\widetilde{+}\left(1-\epsilon \right)\left(1-s\right)\widetilde{𝓙}\left(\varsigma ,\nu \right).\end{array}$$

Since$\tilde{\mathfrak{S}}$and$\widetilde{𝓙}$are both coordinated convex FIVFs, then by Lemma 1, there exist:

$${\tilde{\mathfrak{S}}}_x:\left[\mu ,\nu \right]\to {\mathbb{F}}_0,{\tilde{\mathfrak{S}}}_x\left(y\right)=\tilde{\mathfrak{S}}\left(x,y\right), {\widetilde{𝓙}}_x:\left[\mu ,\nu \right]\to {\mathbb{F}}_0,{\widetilde{𝓙}}_x\left(y\right)=\widetilde{𝓙}\left(x,y\right),$$

Since${\tilde{\mathfrak{S}}}_x$, and${\widetilde{𝓙}}_x$are FIVFs, then by inequality (12), we have:

$$\begin{array}{l}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\tilde{\mathfrak{S}}}_x\left(\nu \right)\tilde{\times }{\widetilde{𝓙}}_x\left(\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\nu }^{-}}^{\beta }{\tilde{\mathfrak{S}}}_x\left(\mu \right)\tilde{\times }{\widetilde{𝓙}}_x\left(\mu \right)\right]\\ \preccurlyeq \left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\tilde{\mathfrak{S}}}_x\left(\mu \right)\tilde{\times }{\widetilde{𝓙}}_x\left(\mu \right) \widetilde{+} {\tilde{\mathfrak{S}}}_x\left(\nu \right)\tilde{\times }{\widetilde{𝓙}}_x\left(\nu \right)\right)\\ \widetilde{+}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\tilde{\mathfrak{S}}}_x\left(\mu \right)\tilde{\times }{\widetilde{𝓙}}_x\left(\nu \right) \widetilde{+} {\tilde{\mathfrak{S}}}_x\left(\nu \right)\tilde{\times }{\widetilde{𝓙}}_x\left(\mu \right)\right).\end{array}$$

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

$$\begin{array}{l}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{{\phi }_x}\left(\nu \right)\times {𝓙}_{{\phi }_x}\left(\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta }{\mathfrak{S}}_{{\phi }_x}\left(\mu \right)\times {𝓙}_{{\phi }_x}\left(\mu \right)\right]\\ {\le }_I\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{{\phi }_x}\left(\mu \right)\times {𝓙}_{{\phi }_x}\left(\mu \right)+{\mathfrak{S}}_{{\phi }_x}\left(\nu \right)\times {𝓙}_{{\phi }_x}\left(\nu \right)\right)\\ +\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{{\phi }_x}\left(\mu \right)\times {𝓙}_{{\phi }_x}\left(\nu \right)+{\mathfrak{S}}_{{\phi }_x}\left(\nu \right)\times {𝓙}_{{\phi }_x}\left(\mu \right)\right).\end{array}$$

That is:

$$\begin{array}{l}\frac{\beta }{2{\left(\nu -\mu \right)}^{\beta }}\left[{{\displaystyle \int }}_{\mu }^{\nu }{\left(\nu -y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,y\right)\times {𝓙}_{\phi }\left(x,y\right)dy+{{\displaystyle \int }}_{\mu }^{\nu }{\left(y-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,y\right)\times {𝓙}_{\phi }\left(x,y\right)dy\right]\\ {\le }_I\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(x,\mu \right)\times {𝓙}_{\phi }\left(x,\mu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)\times {𝓙}_{\phi }\left(x,\nu \right)\right)\\ +\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(x,\mu \right)\times {𝓙}_{\phi }\left(x,\nu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)\times {𝓙}_{\phi }\left(x,\mu \right)\right).\end{array}$$

(66)

Multiplying double inequality (66) by$\frac{\alpha {\left(\varsigma -x\right)}^{\alpha -1}}{2{\left(\varsigma -\rho \right)}^{\alpha }}$and integrating it with respect to$x$over$\left[\rho , \varsigma \right],$we get:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\end{array}\right]\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right).\end{array}$$

(67)

Again, multiplying double inequality (66) by$\frac{\alpha {\left(x-\rho \right)}^{\alpha -1}}{2{\left(\varsigma -\rho \right)}^{\alpha }}$and integrating it with respect to$x$over$\left[\rho , \varsigma \right],$we gain:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\end{array}\right]\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right).\end{array}$$

(68)

Summing (67) and (68), we have:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\\ +\begin{array}{c}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\end{array}\end{array}\right]\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right) \right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\end{array}$$

(69)

Now, again with the help of integral inequality (12) for the first two integrals on the right-hand side of (69), we have the following relation:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ +\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right).\end{array}$$

(70)

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right).\end{array}$$

(71)

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right) \right)\\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right).\end{array}$$

(72)

and

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ +\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right).\end{array}$$

(73)

From (70)–(73) and inequality (69), we have

$$\begin{gathered} \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\\ +{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\end{array}\right] \\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)L_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right) \\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}{\mathcal{M}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}{\mathcal{N}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right) \end{gathered}$$

Since from$\phi$-cuts, we obtain the collection of IVFs${\mathfrak{S}}_{\phi }, {𝓙}_{\phi }:\Delta \to {ℝ}_I^{+}$, then the above inequality can be written in the form of inequality (64). Hence, the result has been proven. □

Remark 5.

If one is to take$\alpha =1$and$\beta =1$, then from (64), we achieve the coming inequality, see [34]:

$$\begin{array}{l}\frac{1}{\left(\varsigma -\rho \right)\left(\nu -\mu \right)} {{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)\tilde{\times }\widetilde{𝓙}\left(x,y\right)dydx\\ \preccurlyeq \frac{1}{9}\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)\widetilde{+}\frac{1}{18}\left[\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)+\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)\right]\widetilde{+}\frac{1}{36}\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(74)

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)$as an affine function and${\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$as a concave function. If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$then, by Remark 2 and (65), we acquire the following inequality, see [33]:

$$\begin{gathered} \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }S\left(\varsigma ,\nu \right)\times 𝓙\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } S\left(\varsigma ,\mu \right)\times 𝓙\left(\varsigma ,\mu \right)\end{array}\right] \\ +\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }S\left(\rho ,\nu \right)\times 𝓙\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } S\left(\rho ,\mu \right)\times 𝓙\left(\rho ,\mu \right)\end{array}\right] \\ \supseteq \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)L\left(\rho ,\varsigma ,\mu ,\nu \right) \\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\mathcal{M}\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\mathcal{N}\left(\rho ,\varsigma ,\mu ,\nu \right) \end{gathered}$$

(75)

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)$as an affine function and${\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$as a concave function. If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then by Remark 2 and (65), we acquire the following inequality, see [32]:

$$\begin{array}{l}\frac{1}{\left(\varsigma -\rho \right)\left(\nu -\mu \right)} {{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(x,y\right)\times 𝓙\left(x,y\right)dydx\\ \supseteq \frac{1}{9}K\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{1}{18}\left[L\left(\rho ,\varsigma ,\mu ,\nu \right)+\mathcal{M}\left(\rho ,\varsigma ,\mu ,\nu \right)\right]+\frac{1}{36}\mathcal{N}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(76)

If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)={\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$and${𝓙}_{\ast }\left(\left(x,y\right), \phi \right)={𝓙}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then from (64), we acquire the following inequality, see [50]:

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }S\left(\varsigma ,\nu \right)\times 𝓙\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } S\left(\varsigma ,\mu \right)\times 𝓙\left(\varsigma ,\mu \right)\end{array}\right]\\ +\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}+{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }S\left(\rho ,\nu \right)\times 𝓙\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } S\left(\rho ,\mu \right)\times 𝓙\left(\rho ,\mu \right)\end{array}\right]\\ \le \left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)L\left(\rho ,\varsigma ,\mu ,\nu \right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\mathcal{M}\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\mathcal{N}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(77)

Theorem 7.

Let$\tilde{\mathfrak{S}}, \widetilde{𝓙}:\Delta \to {\mathbb{F}}_0$be a coordinate convex FIVFs on$\Delta$. Then, from$\phi$-cuts, we establish that the series of IVFs${\mathfrak{S}}_{\phi },{𝓙}_{\phi }:\Delta \to {ℝ}_I^{+}$are given by${\mathfrak{S}}_{\phi }\left(x,y\right)=\left[{\mathfrak{S}}_{\ast }\left(\left(x,y\right),\phi \right), {\mathfrak{S}}^{\ast }\left(\left(x,y\right),\phi \right)\right]$and${𝓙}_{\phi }\left(x,y\right)=\left[{𝓙}_{\ast }\left(\left(x,y\right),\phi \right), {𝓙}^{\ast }\left(\left(x,y\right),\phi \right)\right]$for all$\left(x,y\right)\in \Delta$and for all$\phi \in \left[0, 1\right]$. If$\tilde{\mathfrak{S}}\tilde{\times }\widetilde{𝓙}\in \mathcal{F}{\mathfrak{O}}_{\Delta }$, then the following inequalities hold:

$$\begin{gathered} 4\tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\tilde{\times }\widetilde{𝓙}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right) \\ \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\end{array}\right] \\ \widetilde{+}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right)\end{array}\right] \\ \widetilde{+}\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right) \\ \widetilde{+}\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right) \\ \widetilde{+}\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right) \\ \widetilde{+}\left[\frac{1}{4}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right) \end{gathered}$$

(78)

If$\tilde{\mathfrak{S}}$and$\widetilde{𝓙}$are both coordinate concave FIVFs on$\Delta$, then the above inequality can be written as:

$$\begin{gathered} 4\tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\tilde{\times }\widetilde{𝓙}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right) \\ \succcurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\end{array}\right] \\ \widetilde{+}\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right)\end{array}\right] \\ \widetilde{+}\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right) \\ \widetilde{+}\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right) \\ \widetilde{+}\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right) \\ \widetilde{+}\left[\frac{1}{4}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right) \end{gathered}$$

(79)

where$\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)$,$\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)$,$\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)$, and$\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right)$are given in Theorem 6.

Proof. 

Since$\tilde{\mathfrak{S}},\widetilde{𝓙} :\Delta \to {\mathbb{F}}_0$are two convex FIVFs, from inequality (13) and for each$\phi \in \left[0, 1\right]$, we have:

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }} \left[\begin{array}{c}{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(\varsigma -x\right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)dx\\ +{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(x-\rho \right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)dx\end{array}\right]\\ +\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\right)\end{array}$$

(80)

and:

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\beta }{2{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{{\displaystyle \int }}_{\mu }^{\nu }{\left(\nu -y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)dy \\ +{{\displaystyle \int }}_{\mu }^{\nu }{\left(y-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)dy\end{array}\right]\\ +\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)+{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)+{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\right)\end{array}$$

(81)

Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain:

$$\begin{array}{l}8{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }} \left[\begin{array}{c}{{\displaystyle \int }}_{\rho }^{\varsigma }2{\left(\varsigma -x\right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)dx\\ +{{\displaystyle \int }}_{\rho }^{\varsigma }2{\left(x-\rho \right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)dx\end{array}\right]\\ +\frac{\beta }{2{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{{\displaystyle \int }}_{\mu }^{\nu }2{\left(\nu -y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)dy \\ +{{\displaystyle \int }}_{\mu }^{\nu }2{\left(y-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)dy\end{array}\right]\\ +\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(2{\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)+2{\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(2{\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)+2{\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\right)\\ +\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left(2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)+2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left(2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)+2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\right)\end{array}$$

(82)

Again, with the help of integral inequality (13) and Lemma 1 for each integral on the right-hand side of (82), we have:

$$\begin{array}{l}\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }2{\left(\varsigma -x\right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)dx\\ {\le }_I\frac{\alpha \beta }{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(\varsigma -x\right)}^{\alpha -1}{\left(\nu -y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,y\right)dydx\right]\\ +\frac{\alpha \beta }{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(\varsigma -x\right)}^{\alpha -1}{\left(y-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,y\right)dydx\right]\\ +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(\varsigma -x\right)}^{\alpha -1}\left({\mathfrak{S}}_{\phi }\left(x,\mu \right)\times {𝓙}_{\phi }\left(x,\mu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)\times {𝓙}_{\phi }\left(x,\nu \right)\right)dx\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(\varsigma -x\right)}^{\alpha -1}\left({\mathfrak{S}}_{\phi }\left(x,\mu \right)\times {𝓙}_{\phi }\left(x,\nu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)\times {𝓙}_{\phi }\left(x,\mu \right)\right)dx\\ =\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\end{array}\right]\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right).\end{array}$$

(83)

$$\begin{array}{l}\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }2{\left(x-\rho \right)}^{\alpha -1}{\mathfrak{S}}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(x,\frac{\mu +\nu }{2}\right)dx\\ {\le }_I\frac{\alpha \beta }{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(x-\rho \right)}^{\alpha -1}{\left(\nu -y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,y\right)dydx\right]\\ +\frac{\alpha \beta }{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }}\left[{{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }{\left(x-\rho \right)}^{\alpha -1}{\left(y-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(x,y\right)dydx\right]\\ +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(x-\rho \right)}^{\alpha -1}\left({\mathfrak{S}}_{\phi }\left(x,\mu \right)\times {𝓙}_{\phi }\left(x,\mu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)\times {𝓙}_{\phi }\left(x,\nu \right)\right)dx\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\frac{\alpha }{2{\left(\varsigma -\rho \right)}^{\alpha }}{{\displaystyle \int }}_{\rho }^{\varsigma }{\left(x-\rho \right)}^{\alpha -1}\left({\mathfrak{S}}_{\phi }\left(x,\mu \right)\times {𝓙}_{\phi }\left(x,\nu \right)+{\mathfrak{S}}_{\phi }\left(x,\nu \right)\times {𝓙}_{\phi }\left(x,\mu \right)\right)dx\\ =\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\end{array}\right]\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right).\end{array}$$

(84)

$$\begin{array}{l}\frac{\beta }{2{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{{\displaystyle \int }}_{\mu }^{\nu }2{\left(\nu -y\right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)dy\end{array}\right]\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\end{array}\right]\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right).\end{array}$$

(85)

$$\begin{array}{l}\frac{\beta }{2{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{{\displaystyle \int }}_{\mu }^{\nu }2{\left(y-\mu \right)}^{\beta -1}{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},y\right)dy\end{array}\right]\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\end{array}\right]\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left(\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right).\end{array}$$

(86)

and:

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho , \mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right]\\ +\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\end{array}$$

(87)

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho , \nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right]\\ +\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\end{array}$$

(88)

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho , \mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right]\\ +\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\end{array}$$

(89)

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2},\nu \right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\mu \right)\\ {\le }_I\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left[{\mathit{ȷ}}_{{\rho }^{+}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-}}^{\alpha } {\mathfrak{S}}_{\phi }\left(\rho , \nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right]\\ +\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ +\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\end{array}$$

(90)

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\rho , \nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right]\\ +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right),\end{array}$$

(91)

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma , \nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right]\\ +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\end{array}$$

(92)

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\rho , \nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right]\\ +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\end{array}$$

(93)

and:

$$\begin{array}{l}2{\mathfrak{S}}_{\phi }\left(\varsigma ,\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\rho ,\frac{\mu +\nu }{2}\right)\\ {\le }_I\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left[{\mathit{ȷ}}_{{\mu }^{+}}^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\nu }^{-}}^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma , \nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right]\\ +\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left({\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ +\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)\left({\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\end{array}$$

(94)

From inequalities (83) to (94) and inequality (82), we have:

$$\begin{array}{ll}8{\mathfrak{S}}_{\phi }& \left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\times {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ & {\le }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\end{array}\\ +{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\end{array}\right]\\ & +\left(\frac{2\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right) \left[\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\end{array}\right]\\ & +2\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right) \left[\begin{array}{c}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\end{array}\right]\\ & +2\left(\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right) \left[\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\end{array}\right]\\ & +2\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right) \left[\begin{array}{c}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\end{array}\right]\\ & +\frac{2\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}{K}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)++\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\frac{2\beta }{\left(\beta +1\right)\left(\beta +2\right)}{L}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\frac{2\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathcal{M}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +2\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right){\mathcal{N}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(95)

Again, with the help of integral inequality (12) and Lemma 1, for each integral on the right-hand side of (95), we have:

$$\begin{array}{l}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)K_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}{\mathcal{M}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(96)

$$\begin{array}{l}\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\mu }^{+} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\beta +1\right)}{2{\left(\nu -\mu \right)}^{\beta }}\left({\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\nu }^{-} }^{\beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)L_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}{\mathcal{N}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(97)

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\rho }^{+} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)+{\mathit{ȷ}}_{{\rho }^{+} }^{\alpha } {\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)K_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}{L}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}\begin{array}{}\end{array}$$

(98)

$$\begin{array}{l}\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\\ +\frac{\Gamma \left(\alpha +1\right)}{2{\left(\varsigma -\rho \right)}^{\alpha }}\left({\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-} }^{\alpha }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\right)\\ {\le }_I\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right){\mathcal{M}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}{\mathcal{N}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(99)

From (92) to (99) we have:

$$\begin{array}{ll}4{\mathfrak{S}}_{\phi }\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\times & {𝓙}_{\phi }\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ & {\le }_I\frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\varsigma ,\nu \right)\times {𝓙}_{\phi }\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\varsigma ,\mu \right)\times {𝓙}_{\phi }\left(\varsigma ,\mu \right)\\ +{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }{\mathfrak{S}}_{\phi }\left(\rho ,\nu \right)\times {𝓙}_{\phi }\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } {\mathfrak{S}}_{\phi }\left(\rho ,\mu \right)\times {𝓙}_{\phi }\left(\rho ,\mu \right)\end{array}\right]\\ & +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]K_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]L_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]{\mathcal{M}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{4}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]{\mathcal{N}}_{\phi }\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(100)

That is:

$$\begin{array}{ll}4\tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\tilde{\times }& \widetilde{𝓙}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ & \preccurlyeq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\varsigma ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\varsigma ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\varsigma ,\mu \right)\\ \widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }\tilde{\mathfrak{S}}\left(\rho ,\nu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\nu \right)\widetilde{+}{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } \tilde{\mathfrak{S}}\left(\rho ,\mu \right)\tilde{\times }\widetilde{𝓙}\left(\rho ,\mu \right)\end{array}\right]\\ & \widetilde{+}\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & \widetilde{+}\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & \widetilde{+}\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & \widetilde{+}\left[\frac{1}{4}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right) \end{array}$$

Hence, the result has been proven. □

Remark 6.

If one is to take$\alpha =1$and$\beta =1$, then from (78), we achieve the coming inequality, see [34]:

$$\begin{array}{l}4 \tilde{\mathfrak{S}}\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\tilde{\times }\widetilde{𝓙}\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ \preccurlyeq \frac{1}{\left(\varsigma -\rho \right)\left(\nu -\mu \right)} {{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\tilde{\mathfrak{S}}\left(x,y\right)\tilde{\times }\widetilde{𝓙}\left(x,y\right)dydx\widetilde{+} \frac{5}{36}\tilde{K}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ \widetilde{+}\frac{7}{36}\left[\tilde{L}\left(\rho ,\varsigma ,\mu ,\nu \right)\widetilde{+}\tilde{\mathcal{M}}\left(\rho ,\varsigma ,\mu ,\nu \right)\right]\widetilde{+}\frac{2}{9}\tilde{\mathcal{N}}\left(\rho ,\varsigma ,\mu ,\nu \right).\end{array}$$

(101)

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)$as an affine function and${\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$as a convex function. If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then from Remark 2 and (79), we acquire the following inequality, see [32]:

$$\begin{array}{l}4 \mathfrak{S}\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\times 𝓙\left(\frac{\rho +\varsigma }{2},\frac{\mu +\nu }{2}\right)\\ \supseteq \frac{1}{\left(\varsigma -\rho \right)\left(\nu -\mu \right)} {{\displaystyle \int }}_{\rho }^{\varsigma }{{\displaystyle \int }}_{\mu }^{\nu }\mathfrak{S}\left(x,y\right)\times 𝓙\left(x,y\right)dydx+\frac{5}{36}K\left(\rho ,\varsigma ,\mu ,\nu \right)\\ +\frac{7}{36}\left[L\left(\rho ,\varsigma ,\mu ,\nu \right)+\mathcal{M}\left(\rho ,\varsigma ,\mu ,\nu \right)\right]+\frac{2}{9}\mathcal{N}\left(\rho ,\varsigma ,\mu ,\nu \right).\end{array}$$

(102)

Let one take${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)$as an affine function and${\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$as a convex function. If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)\ne {\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then from Remark 2 and (79), we acquire the following inequality, see [33]:

$$\begin{array}{ll}4\mathfrak{S}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\times & 𝓙\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ & \supseteq \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }S\left(\varsigma ,\nu \right)\times 𝓙\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } S\left(\varsigma ,\mu \right)\times 𝓙\left(\varsigma ,\mu \right)\\ +{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }S\left(\rho ,\nu \right)\times 𝓙\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } S\left(\rho ,\mu \right)\times 𝓙\left(\rho ,\mu \right)\end{array}\right]\\ & +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]K\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]L\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{M}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{4}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(103)

If${\mathfrak{S}}_{\ast }\left(\left(x,y\right), \phi \right)={\mathfrak{S}}^{\ast }\left(\left(x,y\right), \phi \right)$and${𝓙}_{\ast }\left(\left(x,y\right), \phi \right)={𝓙}^{\ast }\left(\left(x,y\right), \phi \right)$with$\phi =1$, then from (78), we acquire the following inequality, see [50]:

$$\begin{array}{ll}4\mathfrak{S}\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)& \times 𝓙\left(\frac{\rho +\varsigma }{2}, \frac{\mu +\nu }{2}\right)\\ & \le \frac{\Gamma \left(\alpha +1\right)\Gamma \left(\beta +1\right)}{4{\left(\varsigma -\rho \right)}^{\alpha }{\left(\nu -\mu \right)}^{\beta }} \left[\begin{array}{c}{\mathit{ȷ}}_{{\rho }^{+},{\mu }^{+} }^{\alpha , \beta }S\left(\varsigma ,\nu \right)\times 𝓙\left(\varsigma ,\nu \right)+{\mathit{ȷ}}_{{\rho }^{+},{\nu }^{-}}^{\alpha , \beta } S\left(\varsigma ,\mu \right)\times 𝓙\left(\varsigma ,\mu \right)\\ +{\mathit{ȷ}}_{{\varsigma }^{-},{\mu }^{+} }^{\alpha , \beta }S\left(\rho ,\nu \right)\times 𝓙\left(\rho ,\nu \right)+{\mathit{ȷ}}_{{\varsigma }^{-},{\nu }^{-}}^{\alpha , \beta } S\left(\rho ,\mu \right)\times 𝓙\left(\rho ,\mu \right)\end{array}\right]\\ & +\left[\frac{\alpha }{2\left(\alpha +1\right)\left(\alpha +2\right)}+\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)\right]K\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]L\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{2}\left(\frac{1}{2}-\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right)+\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{M}\left(\rho ,\varsigma ,\mu ,\nu \right)\\ & +\left[\frac{1}{4}-\frac{\alpha }{\left(\alpha +1\right)\left(\alpha +2\right)}\frac{\beta }{\left(\beta +1\right)\left(\beta +2\right)}\right]\mathcal{N}\left(\rho ,\varsigma ,\mu ,\nu \right)\end{array}$$

(104)

4. Conclusions and Future Plans

The major goal for this study was to introduce new fuzzy fractional operators and to show some Hermite–Hadamard-type integral inequalities with the help of Lemma 1 for introduced coordinated convex fuzzy-interval-valued functions via newly defined fuzzy fractional integrals. We also demonstrated that the newly established inequalities could be transformed into Hermite–Hadamard-type inequalities for classical convex functions, for fuzzy-interval-valued and interval-valued functions [24,27,30], and for coordinated convex fuzzy-interval-valued and interval-valued functions [32,33,34,41,42,50] via Riemann integrals and fuzzy Riemann integrals, respectively, without having to prove each one separately. This is a novel and intriguing topic, and future research will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.