On Extended General Mittag–Leffler Functions and Certain Inequalities

Abstract

In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag–Leffler function of two variables. We establish several new refinements of Hermite–Hadamard-like inequalities via co-ordinated convex functions.

1. Introduction and Preliminaries

The Hermite–Hadamard inequality states that if a function $\mathsf{\Psi }:I\subseteq \mathbb{R}\to \mathbb{R}$ is convex, then

$$\mathsf{\Psi }\left({\displaystyle \frac{a+b}{2}}\right)\le {\displaystyle \frac{1}{b-a}}{\int }_a^b\mathsf{\Psi }\left(x\right)\mathrm{d}x\le {\displaystyle \frac{\mathsf{\Psi }\left(a\right)+\mathsf{\Psi }\left(b\right)}{2}},$$

(1)

where $a,b\in I$ with $a<b$. Both inequalities hold in the reversed direction if $\mathsf{\Psi }$ is concave.

In recent years, many researchers have turned their attention to the Hermite–Hadamard inequality and have found many variations and generalizations of it via various types of convexity. Some of this research is related to functions that are convex on the coordinates (see, for instance, [1,2,3,4,5] and the references therein).

Coordinated convex functions are defined as:

 Definition 1. 

([3]) Let us consider the bidimensional interval $\Delta =[a,b]\times [c,d]$ in ${\mathbb{R}}^2$ with $a<b,c<d.$ A function $\mathsf{\Psi }:\Delta \to \mathbb{R}$ will be called convex on the coordinates if the partial mappings ${\mathsf{\Psi }}_y:[a,b]\to \mathbb{R}$, ${\mathsf{\Psi }}_y\left(u\right)=\mathsf{\Psi }(u,y)$ and ${\mathsf{\Psi }}_x:[c,d]\to \mathbb{R}$, ${\mathsf{\Psi }}_x\left(v\right)=\mathsf{\Psi }(x,v)$ are convex where defined for all $y\in [c,d]$, and $x\in [a,b].$ Recall mapping $\mathsf{\Psi }:\Delta \to \mathbb{R}$ is convex on the coordinates on Δ if the following inequality holds:

$$\begin{array}{cc}\mathsf{\Psi }\left(tx+\left(1-t\right)y,su+\left(1-s\right)w\right)\hfill & \le ts\mathsf{\Psi }(x,u)+t\left(1-s\right)\mathsf{\Psi }(x,w)\hfill \\ & +s\left(1-t\right)\mathsf{\Psi }(y,u)+\left(1-t\right)\left(1-s\right)\mathsf{\Psi }(y,w),\hfill \end{array}$$

(2)

for all $(x,u\dots y,w)\in \Delta$ and $t,s\in [0,1].$

Discuss some preliminaries of fractional calculus.

 Definition 2. 

Let $\mathsf{\Psi }\in L[a,b]$, where $a\ge 0.$ The Riemann–Liouville integrals $J_{a+}^{\nu }\mathsf{\Psi }$ and $J_{b-}^{\nu }\mathsf{\Psi },$ of order $\nu >0,$ are defined by

$$J_{a+}^{\nu }\mathsf{\Psi }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\nu \right)}}{\int }_a^x{(x-t)}^{\nu -1}\mathsf{\Psi }\left(t\right)\mathrm{d}t,\mathrm{for}x>a$$

and

$$J_{b-}^{\nu }\mathsf{\Psi }\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\nu \right)}}{\int }_x^b{(t-x)}^{\nu -1}\mathsf{\Psi }\left(t\right)\mathrm{d}t,\mathrm{for}x<b,$$

respectively. Here, $\mathsf{\Gamma }\left(\nu \right)={\int }_0^{\infty }{e}^{-t}{t}^{\nu -1}\mathrm{d}t$ is the gamma function. We also make the convention

$$J_{a+}^0\mathsf{\Psi }\left(x\right)=J_{b-}^0\mathsf{\Psi }\left(x\right)=\mathsf{\Psi }\left(x\right).$$

More details about the Riemann–Liouville fractional integrals may be found in [6].

Salim and Faraj [7] have defined the generalized fractional integral operators containing Mittag–Leffler functions:

 Definition 3. 

Let $\mu ,\nu ,k,l,\gamma$ be positive real numbers and $\omega \in \mathbb{R}$. Then the generalized fractional integral operators containing Mittag–Leffler function ${\epsilon }_{\mu ,\nu ,l,\omega ,a^{+}}^{\gamma ,\delta ,k}$ and ${\epsilon }_{\mu ,\nu ,l,\omega ,b^{-}}^{\gamma ,\delta ,k}$ for a real-valued continuous function Ψ are defined by:

$$\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)\left(x\right)={\int }_a^x{(x-t)}^{\nu -1}{E}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega {(x-t)}^{\mu }\right)\mathsf{\Psi }\left(t\right)\mathrm{d}t$$

(3)

and

$$\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)\left(x\right)={\int }_x^b{(t-x)}^{\nu -1}{E}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega {(t-x)}^{\mu }\right)\mathsf{\Psi }\left(t\right)\mathrm{d}t,$$

(4)

respectively, where the function $E_{\mu ,\nu ,l}^{\gamma ,\delta ,k}$ is a generalized Mittag–Leffler function defined as

$$E_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(t\right)=\sum _{n=0}^{\infty }\frac{{\left(\gamma \right)}_{kn}}{\mathsf{\Gamma }(\mu n+\nu )}\frac{t^n}{{\left(\delta \right)}_{ln}},$$

and ${\left(a\right)}_n$ is the Pochhammer symbol: ${\left(a\right)}_n=a(a+1)·\dots ·(a+n-1)$, ${\left(a\right)}_0=1$.

 Remark 1. 

If $k=l=1$ in (3), then the integral operator $\left({\epsilon }_{\mu ,\nu ,1,\omega ,a^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)$ reduces to an integral operator $\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+}}^{\gamma ,\delta ,1}\mathsf{\Psi }\right)$ containing generalized Mittag–Leffler function $E_{\mu ,\nu ,1}^{\gamma ,\delta ,1}$ introduced by Srivastava and Tomovski in [8]. Along with $k=l=1$, if $\delta =1$, then (3) reduces to an integral operator defined by Prabhaker in [9] containing Mittag–Leffler function $E_{\mu ,\nu }^{\gamma }$. For $\omega =0$ in (3), the integral operator $\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)$ reduces to the Riemann–Liouville fractional integral operator [7]. Note that $E_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(t\right)$ is absolutely convergent for all $t\in \mathbb{R}$, where $k<l+\mu$. Since $|E_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(t\right)|\le {\sum }_{n=0}^{\infty }\left|\frac{{\left(\gamma \right)}_{kn}}{\mathsf{\Gamma }(\mu n+\nu )}\frac{t^n}{{\left(\delta \right)}_{ln}}\right|$ with ${\sum }_{n=0}^{\infty }\left|\frac{{\left(\gamma \right)}_{kn}}{\mathsf{\Gamma }(\mu n+\nu )}\frac{t^n}{{\left(\delta \right)}_{ln}}\right|=S$, we have $|E_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(t\right)|\le S.$

Inspired by Definition 3, we give the following new definition:

 Definition 4. 

Let $\mu ,\nu ,k,l,\gamma$ be positive real numbers and $\omega \in \mathbb{R}$, then

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(x,y)\hfill \\ ={\int }_a^x{\int }_c^y{(x-t)}^{{\nu }_1-1}{(y-s)}^{{\nu }_2-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(x-t)}^{{\mu }_1}\right)E_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(y-s)}^{{\mu }_2}\right)\mathsf{\Psi }(t,s)\mathrm{d}s\mathrm{d}t,\hfill \end{array}$$

$x>a,y>c$;

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(x,y)\hfill \\ ={\int }_a^x{\int }_y^d{(x-t)}^{{\nu }_1-1}{(s-y)}^{{\nu }_2-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(x-t)}^{{\mu }_1}\right)E_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(s-y)}^{{\mu }_2}\right)\mathsf{\Psi }(t,s)\mathrm{d}s\mathrm{d}t,\hfill \end{array}$$

$x>a,y<d$;

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(x,y)\hfill \\ ={\int }_x^b{\int }_c^y{(t-x)}^{{\nu }_1-1}{(y-s)}^{{\nu }_2-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(t-x)}^{{\mu }_1}\right)E_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(y-s)}^{{\mu }_2}\right)\mathsf{\Psi }(t,s)\mathrm{d}s\mathrm{d}t,\hfill \end{array}$$

$x<b,y>d$ respectively

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(x,y)\hfill \\ ={\int }_x^b{\int }_y^d{(t-x)}^{{\nu }_1-1}{(s-y)}^{{\nu }_2-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(t-x)}^{{\mu }_1}\right)E_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(s-y)}^{{\mu }_2}\right)\mathsf{\Psi }(t,s)\mathrm{d}s\mathrm{d}t,\hfill \end{array}$$

$x<b,y<d$, where $\mu =\left({\mu }_1,{\mu }_2\right),\nu =\left({\nu }_1,{\nu }_2\right),\omega =\left({\omega }_1,{\omega }_2\right),\gamma =\left({\gamma }_1,{\gamma }_2\right),\delta =\left({\delta }_1,{\delta }_2\right),k=\left(k_1,k_2\right),\mu ,\nu ,\omega ,\gamma ,\delta ,k>(0,0)$.

Similar to Definition 4, we introduce the following fractional integrals

 Definition 5. 

Let $\mu ,\nu ,k,l,\gamma$ be positive real numbers and $\omega \in \mathbb{R}$, then

$$\begin{array}{c}\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1,a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)={\int }_a^x{(x-t)}^{{\nu }_1-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(t-x)}^{{\mu }_1}\right)\mathsf{\Psi }\left(t,\frac{c+d}{2}\right)\mathrm{d}t,\hfill \\ \left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1,b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)={\int }_x^b{(t-x)}^{{\nu }_1-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(t-x)}^{{\mu }_1}\right)\mathsf{\Psi }\left(t,\frac{c+d}{2}\right)\mathrm{d}t,\hfill \\ \left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2,c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},y\right)={\int }_c^y{(y-s)}^{{\nu }_2-1}{E}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(y-s)}^{{\mu }_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},s\right)\mathrm{d}s,\hfill \\ \left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2,d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},y\right)={\int }_y^d{(s-y)}^{{\nu }_2-1}{E}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(s-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},s\right)\mathrm{d}s.\hfill \end{array}$$

 Definition 6. 

A function $t\mathcal{G}:\Delta \to \mathbb{R}$ is said to be symmetric with respect to $\frac{a+b}{2}$ and $\frac{c+d}{2}$ on the coordinates if

$$t\mathcal{G}(x,y)=\left\{\begin{array}{c}t\mathcal{G}\left(a+b-x,c+d-y\right)\\ t\mathcal{G}\left(x,c+d-y\right)\\ t\mathcal{G}\left(a+b-x,y\right)\end{array}\right.$$

holds for all $x\in [a,b]$ and $y\in [c,d]$.

 Lemma 1. 

Let $p\in \mathbb{R}\setminus \left\{0\right\}$, and $t\mathcal{G}:[a,b]\subseteq \mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$ be integrable and p-symmetric with respect to $\frac{a^p+b^p}{2}$, then

(i)

If $p>0$,

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(b^p\right)=\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(a^p\right)\hfill \\ =\frac{1}{2}\left[\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(b^p\right)+\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(a^p\right)\right],\hfill \end{array}$$

with $h\left(x\right)=x^{1/p}$, $x\in [a^p,b^p]$.

(ii)

If $p<0$,

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(a^p\right)=\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(b^p\right)\hfill \\ =\frac{1}{2}\left[\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(a^p\right)+\left({\epsilon }_{\mu ,\nu ,l,\omega ,{\left(\frac{a^p+b^p}{2}\right)}^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\circ h\right)\left(b^p\right)\right],\hfill \end{array}$$

with $h\left(x\right)=x^{1/p}$, $x\in [b^p,a^p]$.

For the applications of and related results containing Mittag–Leffler functions, see [10,11].

For details on Hermite–Hadamard-type inequalities involving fractional integrals via different classes of convex functions, see Kunt et al. [12], Mihai [13,14], Mihai and Mitroi [15], Nisan et al. [16], Noor et al. [17], Sarikaya and Yildirim [18], and others.

2. Main Results

Now we are in a position to present our main results.

 Lemma 2. 

If the function $t\mathcal{G}:\Delta \to \mathbb{R}$ is non-negative, integrable, and symmetric with respect to $\frac{a+b}{2}$ and $\frac{c+d}{2}$ on the coordinates, then the following equalities hold:

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)=\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d)=\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)=\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\hfill \\ =\frac{1}{4}\left[\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d)+\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)\right.\hfill \\ \left.+\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\right].\hfill \end{array}$$

(5)

 Proof. 

Using $t\mathcal{G}$ symmetry with respect to $\frac{a+b}{2}$ and $\frac{c+d}{2}$ on the coordinates and substitutions $x=a+b-t$, $y=c+d-s$, we have

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\hfill \\ ={\int }_a^b{\int }_c^d{(b-t)}^{{\nu }_1-1}{(d-s)}^{{\nu }_2-1}{E}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(b-t)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(d-s)}^{{\mu }_2}\right)t\mathcal{G}(t,s)\mathrm{d}s\mathrm{d}t\hfill \\ ={\int }_a^b{\int }_c^d{(b-t)}^{{\nu }_1-1}{(d-s)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(b-t)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(d-s)}^{{\mu }_2}\right)\hfill \\ \times t\mathcal{G}(a+b-t,c+d-s)\mathrm{d}s\mathrm{d}t\hfill \\ ={\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(y-c)}^{{\mu }_2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ =\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c).\hfill \end{array}$$

(6)

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)\hfill \\ ={\int }_a^b{\int }_c^d{(b-t)}^{{\nu }_1-1}{(s-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(b-t)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(s-c)}^{{\mu }_2}\right)t\mathcal{G}(t,s)\mathrm{d}s\mathrm{d}t\hfill \\ ={\int }_a^b{\int }_c^d{(b-t)}^{{\nu }_1-1}{(s-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(b-t)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(s-c)}^{{\mu }_2}\right)\hfill \\ \times t\mathcal{G}(a+b-t,c+d-s)\mathrm{d}s\mathrm{d}t\hfill \\ ={\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(d-y)}^{{\mu }_2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ =\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d).\hfill \end{array}$$

(7)

Now, using $t\mathcal{G}$ with respect to $\frac{a+b}{2}$ and substitution $u=a+b-t$, we obtain

$$\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,\omega ,a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\hfill \\ ={\int }_a^b{\int }_c^d{(b-t)}^{{\nu }_1-1}{(d-s)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(b-t)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(d-s)}^{{\mu }_2}\right)t\mathcal{G}(t,s)\mathrm{d}s\mathrm{d}t\hfill \\ ={\int }_a^b{\int }_c^d{(b-t)}^{{\nu }_1-1}{(d-s)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(b-t)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(d-s)}^{{\mu }_2}\right)\hfill \\ \times t\mathcal{G}(a+b-t,s)\mathrm{d}s\mathrm{d}t\hfill \\ ={\int }_a^b{\int }_c^d{(u-a)}^{{\nu }_1-1}{(d-s)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{(u-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{(d-s)}^{{\mu }_2}\right)t\mathcal{G}(u,s)\mathrm{d}s\mathrm{d}u\hfill \\ =\left({\epsilon }_{\mu ,\nu ,l,\omega ,b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d).\hfill \end{array}$$

(8)

Combining Equations (6)–(8), we get Equation (5) and the proof is complete. □

 Lemma 3. 

Let $\mathsf{\Psi }:[a,b]\to \mathbb{R}$ be a convex function such that $\mathsf{\Psi }\in L[a,b]$, and let $\mu ,\nu ,l,\gamma ,\delta ,k>0$, $\omega \in \mathbb{R}$. If the function $t\mathcal{G}:[a,b]\to \mathbb{R}$ is non-negative, integrable, and symmetric with respect to $\frac{a+b}{2}$, then the following inequalities for fractional integrals hold:

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2}\right)\left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(b\right)\right]\le \left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}fg\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}fg\right)\left(b\right)\right]\hfill \\ \le \frac{\mathsf{\Psi }\left(a\right)+\mathsf{\Psi }\left(b\right)}{2}\left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(b\right)\right],\hfill \end{array}$$

(9)

where ${\omega }^{\prime }=\frac{\omega }{{(b-a)}^{\mu }}$.

 Proof. 

Since $\mathsf{\Psi }$ is a convex function on $[a,b]$, we have for all $t\in [0,1]$

$$\mathsf{\Psi }\left(\frac{a+b}{2}\right)=\mathsf{\Psi }\left(\frac{ta+(1-t)b+tb+(1-t)a}{2}\right)\le \frac{\mathsf{\Psi }(ta+(1-t\left)b\right)+\mathsf{\Psi }(tb+(1-t\left)a\right)}{2}.$$

(10)

Multiplying both sides of (10) by $2t^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)t\mathcal{G}(tb+(1-t)a)$ then integrating the resulting inequality with respect to t over $[0,1]$, we obtain

$$\begin{array}{c}2\mathsf{\Psi }\left(\frac{a+b}{2}\right){\int }_0^1{t}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)t\mathcal{G}(tb+(1-t)a)\mathrm{d}t\hfill \\ \le {\int }_0^1{t}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)\mathsf{\Psi }(ta+(1-t)b)t\mathcal{G}(tb+(1-t)a)\mathrm{d}t\hfill \\ +{\int }_0^1{t}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)\mathsf{\Psi }(tb+(1-t)a)t\mathcal{G}(tb+(1-t)a)\mathrm{d}t.\hfill \end{array}$$

Setting $x=tb+(1-t)a$ and $\mathrm{d}x=(b-a)\mathrm{d}t$ gives

$$\begin{array}{c}\frac{2\mathsf{\Psi }\left(\frac{a+b}{2}\right)}{{(b-a)}^{\nu }}{\int }_a^b{(x-a)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(x-a)}^{\mu }\right)t\mathcal{G}\left(x\right)\mathrm{d}x\hfill \\ \le \frac{1}{{(b-a)}^{\nu }}\left[{\int }_a^b{(x-a)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(x-a)}^{\mu }\right)\mathsf{\Psi }(a+b-x)t\mathcal{G}\left(x\right)\mathrm{d}x\right.\hfill \\ \left.+{\int }_a^b{(x-a)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(x-a)}^{\mu }\right)\mathsf{\Psi }\left(x\right)t\mathcal{G}\left(x\right)\mathrm{d}x\right]\hfill \\ =\frac{1}{{(b-a)}^{\nu }}\left[{\int }_a^b{(b-x)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(b-x)}^{\mu }\right)\mathsf{\Psi }\left(x\right)t\mathcal{G}(a+b-x)\mathrm{d}x\right.\hfill \\ \left.+{\int }_a^b{(x-a)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(x-a)}^{\mu }\right)\left(fg\right)\left(x\right)\mathrm{d}x\right]\hfill \\ =\frac{1}{{(b-a)}^{\nu }}\left[{\int }_a^b{(b-x)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(b-x)}^{\mu }\right)\mathsf{\Psi }\left(x\right)t\mathcal{G}\left(x\right)\mathrm{d}x\right.\hfill \\ \left.+{\int }_a^b{(x-a)}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left({\omega }^{\prime }{(x-a)}^{\mu }\right)\left(fg\right)\left(x\right)\mathrm{d}x\right].\hfill \end{array}$$

So

$$2\mathsf{\Psi }\left(\frac{a+b}{2}\right)\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(a\right)\le \left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}fg\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}fg\right)\left(b\right)\right],$$

and using Lemma 1, we have

$$\mathsf{\Psi }\left(\frac{a+b}{2}\right)\left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(b\right)\right]\le \left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}fg\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}fg\right)\left(b\right)\right].$$

The first inequality is proved.

For the proof of the second inequality of (9), we first note that if $\mathsf{\Psi }$ is a convex function, then for all $t\in [0,1]$, it yields

$$\mathsf{\Psi }(ta+(1-t\left)b\right)+\mathsf{\Psi }\left(\right(1-t)a+tb)\le \mathsf{\Psi }\left(a\right)+\mathsf{\Psi }\left(b\right).$$

(11)

Then, multiplying both sides of (11) by $t^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)t\mathcal{G}(tb+(1-t)a)$ and integrating the resulting inequality with respect to t over $[0,1]$, we obtain

$$\begin{array}{c}{\int }_0^1{t}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)\mathsf{\Psi }(ta+(1-t)b)t\mathcal{G}(tb+(1-t)a)\mathrm{d}t\hfill \\ +{\int }_0^1{t}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)\mathsf{\Psi }(tb+(1-t)a)t\mathcal{G}(tb+(1-t)a)\mathrm{d}t\hfill \\ \le (\mathsf{\Psi }\left(a\right)+\mathsf{\Psi }\left(b\right)){\int }_0^1{t}^{\nu -1}{\mathcal{E}}_{\mu ,\nu ,l}^{\gamma ,\delta ,k}\left(\omega t^{\mu }\right)t\mathcal{G}(tb+(1-t)a)\mathrm{d}t.\hfill \end{array}$$

That is,

$$\begin{array}{c}\left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}fg\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}fg\right)\left(b\right)\right]\hfill \\ \le \frac{\mathsf{\Psi }\left(a\right)+\mathsf{\Psi }\left(b\right)}{2}\left[\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(a\right)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)\left(b\right)\right].\hfill \end{array}$$

The proof is completed. □

 Remark 2. 

If in Lemma 3 we put $\omega =(0,0)$, we obtain [19] (Theorem 2.2).

The next result is the Hermite–Hadamard-type inequality for coordinated convex functions containing the generalized Mittag–Leffler function.

 Theorem1. 

Let $\mathsf{\Psi }:\Delta \to \mathbb{R}$ be coordinated convex on $\Delta =[a,b]\times [c,d]$ in ${\mathbb{R}}^2$ with $a<b,c<d$ and $\mathsf{\Psi }\in L[\Delta ]$. Then, one has the inequalities

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}1\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}1\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}1\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}1\right)(b,d)\end{array}\right]\hfill \\ \le \left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,d)\end{array}\right]\hfill \\ \le \frac{\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d)}{4}\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}1\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}1\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}1\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}1\right)(b,d)\end{array}\right],\hfill \end{array}$$

(12)

where $\mu =\left({\mu }_1,{\mu }_2\right),\nu =\left({\nu }_1,{\nu }_2\right),{\omega }^{\prime }=\left({\omega }_1^{\prime },{\omega }_2^{\prime }\right),\gamma =\left({\gamma }_1,{\gamma }_2\right),\delta =\left({\delta }_1,{\delta }_2\right),k=\left(k_1,k_2\right)$, $\mu ,\nu ,{\omega }^{\prime },\gamma ,\delta ,k>(0,0)$ with ${\omega }_1^{\prime }=\frac{{\omega }_1}{{(b-a)}^{{\mu }_1}},{\omega }_2^{\prime }=\frac{{\omega }_2}{{(d-c)}^{{\mu }_2}}$.

 Proof. 

Using (2) with $x=ta+(1-t)b,y=(1-t)a+tb$, $u=sc+(1-s)d,w=(1-s)c+td$ and $t=s=\frac{1}{2}$, we find that

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\hfill \\ \le \left[\mathsf{\Psi }(ta+(1-t)b,sc+(1-s\left)d\right)+\mathsf{\Psi }(ta+(1-t)b,(1-s)c+sd)\right.\hfill \\ \left.+\mathsf{\Psi }\left(\right(1-t)a+tb,sc+(1-s\left)d\right)+\mathsf{\Psi }\left(\right(1-t)a+tb,(1-s)c+s)d)\right].\hfill \end{array}$$

(13)

Thus, multiplying both sides of (13) by $t^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)$ and by integrating with respect to $(t,s)$ on $[0,1]\times [0,1]$, we obtain

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right){\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathrm{d}s\mathrm{d}t\hfill \\ \le \frac{1}{4}\left[\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,sc+(1-s)d)\mathrm{d}s\mathrm{d}t\\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,sc+(1-s)d)\mathrm{d}s\mathrm{d}t\\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\end{array}\right].\hfill \end{array}$$

(14)

Using substitutions $u+ta+(1-t)b,v=sc+(1-s)d$, we have

$$\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(b-u)}^{{\nu }_1-1}{(d-v)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(b-u)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(d-v)}^{{\mu }_2}\right)\mathrm{d}v\mathrm{d}u\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}1\right)(b,d),\hfill \end{array}$$

(15)

with ${\omega }_1^{\prime }=\frac{{\omega }_1}{{(b-a)}^{{\nu }_1-1}}$, ${\omega }_2^{\prime }=\frac{{\omega }_2}{{(d-c)}^{{\nu }_2-1}}$, and

$$\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,sc+(1-s)d)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(b-u)}^{{\nu }_1-1}{(d-v)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(b-u)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(d-v)}^{{\mu }_2}\right)\mathsf{\Psi }(u,v)\mathrm{d}v\mathrm{d}u\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,d).\hfill \end{array}$$

(16)

Analogously, we obtain

$$\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,c),\hfill \end{array}$$

(17)

$$\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,sc+(1-s)d)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,d),\hfill \end{array}$$

(18)

$$\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,c).\hfill \end{array}$$

(19)

Using Lemma 2, introducing relationships (15)–(19) in (14) and after multiplying with ${(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}$, we get

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}1\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}1\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}1\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}1\right)(b,d)\end{array}\right]\hfill \\ \le \left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,d)\end{array}\right],\hfill \end{array}$$

(20)

by which the first inequality of (12) is proved.

For the proof of the second inequality in (12) using (2), we have

$$\begin{array}{c}\mathsf{\Psi }\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)+\mathsf{\Psi }\left(ta+\left(1-t\right)b,(1-s)c+sd\right)\hfill \\ +\mathsf{\Psi }\left((1-t)a+tb,sc+\left(1-s\right)d\right)+\mathsf{\Psi }\left((1-t)a+tb,(1-s)c+sd\right)\hfill \\ \le \mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d).\hfill \end{array}$$

(21)

Then, multiplying both sides of (21) by $t^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)$ and integrating with respect to $(t,s)$ on $[0,1]\times [0,1]$, we get

$$\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,sc+(1-s)d)\mathrm{d}s\mathrm{d}t\hfill \\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,sc+(1-s)d)\mathrm{d}s\mathrm{d}t\hfill \\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ \le {\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\left[\begin{array}{c}\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)\\ +\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d)\end{array}\right]\mathrm{d}s\mathrm{d}t.\hfill \end{array}$$

So, after multiplying with ${(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}$ and using Lemma 2, we have

$$\begin{array}{c}\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}\mathsf{\Psi }\right)(b,d)\end{array}\right]\hfill \\ \le \frac{\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d)}{4}\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}1\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}1\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}1\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}1\right)(b,d)\end{array}\right],\hfill \end{array}$$

which finishes the proof. □

The following theorem establishes Hermite–Hadamard–Fejér-type inequalities for coordinated convex functions containing the generalized Mittag–Leffler function.

 Theorem 2. 

Let $\mathsf{\Psi }:\Delta \to \mathbb{R}$ be coordinated convex on $\Delta =[a,b]\times [c,d]$ in ${\mathbb{R}}^2$ with $a<b,c<d$ and $\mathsf{\Psi }\in L[\Delta ]$. If $t\mathcal{G}:\Delta \to \mathbb{R}$ is non-negative, integrable, and symmetric with respect to $\frac{a+b}{2},\frac{c+d}{2}$ on the coordinates, then the following integral inequalities hold:

$$\begin{array}{l}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\end{array}\right]\\ \le \left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}fg\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}fg\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}fg\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}fg\right)(b,d)\end{array}\right]\hfill \\ \le \frac{\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d)}{4}\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\end{array}\right],\hfill \end{array}$$

(22)

where $\mu =\left({\mu }_1,{\mu }_2\right),\nu =\left({\nu }_1,{\nu }_2\right),{\omega }^{\prime }=\left({\omega }_1^{\prime },{\omega }_2^{\prime }\right),\gamma =\left({\gamma }_1,{\gamma }_2\right),\delta =\left({\delta }_1,{\delta }_2\right),k=\left(k_1,k_2\right)$, $\mu ,\nu ,{\omega }^{\prime },\gamma ,\delta ,k>(0,0)$ with ${\omega }_1^{\prime }=\frac{{\omega }_1}{{(b-a)}^{{\mu }_1}},{\omega }_2^{\prime }=\frac{{\omega }_2}{{(d-c)}^{{\mu }_2}}$.

 Proof. 

Since $\mathsf{\Psi }$ is a convex function on $\Delta$, then for all $t,s\in [0,1]\times [0,1]$, we can write

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)=\mathsf{\Psi }\left(\frac{ta+(1-t)b+(1-t)a+tb}{2},\frac{sc+(1-s)d+(1-s)c+sd}{2}\right)\hfill \\ \le \frac{1}{4}\left[\begin{array}{c}\mathsf{\Psi }\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)+\mathsf{\Psi }\left(ta+\left(1-t\right)b,(1-s)c+sd\right)\\ +\mathsf{\Psi }\left((1-t)a+tb,sc+\left(1-s\right)d\right)+\mathsf{\Psi }\left((1-t)a+tb,(1-s)c+sd\right)\end{array}\right].\hfill \end{array}$$

(23)

Then, multiplying both sides of (23) by $t^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)t\mathcal{G}((1-t)a+tb$, $(1-s)c+sd)$ and integrating with respect to $(t,s)$ on $[0,1]\times [0,1]$, we obtain

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\hfill \\ \times {\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)t\mathcal{G}((1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ \le \frac{1}{4}\left[\begin{array}{c}{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,sc+(1-s)d)\\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,(1-s)c+sd)\\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,sc+(1-s)d)\\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\\ +{\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,(1-s)c+sd)\\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\end{array}\right].\hfill \end{array}$$

(24)

Setting $x=(1-t)a+tb$ and $y=(1-s)c+sd$, we obtain

$$\begin{array}{c}{I}_1={\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)t\mathcal{G}((1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)\hfill \end{array}$$

(25)

and

$$\begin{array}{c}{I}_2={\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,sc+(1-s)d)\hfill \\ t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(a+b-x,c+d-y)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x.\hfill \end{array}$$

Now, using substitutions $u=a+b-x$, $v=c+d-y$ and the symmetry of function $t\mathcal{G}$, we have

$$\begin{array}{ll}{I}_2& =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\\ & \times {\int }_a^b{\int }_c^d{(b-u)}^{{\nu }_1-1}{(d-v)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(b-u)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(d-v)}^{{\mu }_2}\right)\hfill \\ & \times \mathsf{\Psi }(u,v)t\mathcal{G}(a+b-u,c+d-v)\mathrm{d}v\mathrm{d}u\hfill \\ & =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ & \times {\int }_a^b{\int }_c^d{(b-u)}^{{\nu }_1-1}{(d-v)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(b-u)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(d-v)}^{{\mu }_2}\right)\hfill \\ & \times \mathsf{\Psi }(u,v)t\mathcal{G}(u,v)\mathrm{d}v\mathrm{d}u\hfill \\ & =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}fg\right)(b,d).\hfill \end{array}$$

(26)

Analogously, we get

$$\begin{array}{c}{I}_3={\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }(ta+(1-t)b,(1-s)c+sd)\hfill \\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(a+b-x,y)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(b-u)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(b-u)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(u,y)t\mathcal{G}(a+b-u,y)\mathrm{d}y\mathrm{d}u\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(b-u)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(b-u)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(u,y)t\mathcal{G}(u,y)\mathrm{d}y\mathrm{d}u\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}fg\right)(b,c),\hfill \end{array}$$

(27)

$$\begin{array}{c}{I}_4={\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,sc+(1-s)d)\hfill \\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(x,c+d-y)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-v)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(d-v)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(x,v)t\mathcal{G}(x,c+d-v)\mathrm{d}v\mathrm{d}x\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-v)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(d-v)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(x,v)t\mathcal{G}(x,v)\mathrm{d}v\mathrm{d}x\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}fg\right)(a,d),\hfill \end{array}$$

(28)

and

$$\begin{array}{c}{I}_5={\int }_0^1{\int }_0^1{t}^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)\mathsf{\Psi }((1-t)a+tb,(1-s)c+sd)\hfill \\ \times t\mathcal{G}\left(\right(1-t)a+tb,(1-s)c+sd)\mathrm{d}s\mathrm{d}t\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\hfill \\ \times {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\hfill \\ \times \mathsf{\Psi }(x,y)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ =\frac{1}{{(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}fg\right)(a,c).\hfill \end{array}$$

(29)

Introducing (25)–(29) in (24), multiplying the inequality with ${(b-a)}^{{\nu }_1-1}{(d-c)}^{{\nu }_2-1}$, and using Lemma 2, we have the first inequality of (22).

We shall prove the second inequality of (22). Since $\mathsf{\Psi }$ is a convex function on $\Delta$, for all $(t,s)\in [0,1]\times [0,1]$, it yields

$$\begin{array}{c}\mathsf{\Psi }\left(ta+\left(1-t\right)b,sc+\left(1-s\right)d\right)+\mathsf{\Psi }\left(ta+\left(1-t\right)b,(1-s)c+sd\right)\hfill \\ +\mathsf{\Psi }\left((1-t)a+tb,sc+\left(1-s\right)d\right)+\mathsf{\Psi }\left((1-t)a+tb,(1-s)c+sd\right)\hfill \\ \le \mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d).\hfill \end{array}$$

(30)

Multiplying both sides of (30) by $t^{{\nu }_1-1}{s}^{{\nu }_2-1}{\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}\left({\omega }_1{t}^{{\mu }_1}\right){\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}\left({\omega }_2{s}^{{\mu }_2}\right)t\mathcal{G}((1-t)a+tb,(1-s)c+sd)$ and integrating with respect to $(t,s)$ on $[0,1]\times [0,1]$, we obtain

$$\begin{array}{c}\hfill I_2+I_3+I_4+I_5\le (\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d))I_1.\end{array}$$

That is, using (25)–(29),

$$\begin{array}{c}\frac{1}{4}\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}fg\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}fg\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}fg\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}fg\right)(b,d)\end{array}\right]\hfill \\ \le \frac{\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d)}{4}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c).\hfill \end{array}$$

Finally, by using Lemma 2 we get the second part of inequality (22), and the proof is complete. □

 Remark 3. 

If we take $\omega =(0,0)$ in Theorem 2, we have [20] (Theorem 7).

 Theorem 3. 

Let $\mathsf{\Psi }:\Delta \to \mathbb{R}$ be coordinated convex on $\Delta =[a,b]\times [c,d]$ in ${\mathbb{R}}^2$ with $a<b,c<d$ and $\mathsf{\Psi }\in L[\Delta ]$. If $t\mathcal{G}:\Delta \to \mathbb{R}$ is non-negative, integrable, and symmetric with respect to $\frac{a+b}{2},\frac{c+d}{2}$ on the coordinates, then the following integral inequalities hold:

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\end{array}\right]\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(b,\frac{c+d}{2}\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(b,d\right)\right)\\ +\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(b,\frac{c+d}{2}\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(b,c\right)\right)\\ +\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(a,\frac{c+d}{2}\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(a,d\right)\right)\\ +\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(a,\frac{c+d}{2}\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(a,c\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(\frac{a+b}{2},d\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(b,d\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(\frac{a+b}{2},d\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(a,d\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(\frac{a+b}{2},c\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(b,c\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(\frac{a+b}{2},c\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(a,c\right)\right)\end{array}\right]\hfill \\ \le \left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}fg\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}fg\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}fg\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}fg\right)(b,d)\end{array}\right]\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(b,c\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(b,d\right)\right)\\ +\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(b,d\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(b,c\right)\right)\\ +\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(a,c\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(a,d\right)\right)\\ +\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)\left(\mathsf{\Psi }\left(a,d\right)\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)t\mathcal{G}\left(a,c\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(a,d\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(b,d\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(b,d\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(a,d\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(a,c\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(b,c\right)\right)\\ +\left({\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}\right)\left(\mathsf{\Psi }\left(b,c\right)\left({\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}\right)t\mathcal{G}\left(a,c\right)\right)\end{array}\right]\hfill \\ \le \frac{\mathsf{\Psi }(a,c)+\mathsf{\Psi }(b,c)+\mathsf{\Psi }(a,d)+\mathsf{\Psi }(b,d)}{4}\left[\begin{array}{c}\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(a,d)\\ +\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,c)+\left({\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}t\mathcal{G}\right)(b,d)\end{array}\right].\hfill \end{array}$$

(31)

 Proof. 

For an easier proof, we will use the following notations: ${\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },a^{+}}^{{\gamma }_1,{\delta }_1,k_1}={\epsilon }_{a^{+}}$, ${\epsilon }_{{\mu }_1,{\nu }_1,l_1,{\omega }_1^{\prime },b^{-}}^{{\gamma }_1,{\delta }_1,k_1}={\epsilon }_{b^{-}}$, ${\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },c^{+}}^{{\gamma }_2,{\delta }_2,k_2}={\epsilon }_{c^{+}}$, ${\epsilon }_{{\mu }_2,{\nu }_2,l_2,{\omega }_2^{\prime },d^{-}}^{{\gamma }_2,{\delta }_2,k_2}={\epsilon }_{d^{-}}$, ${\mathcal{E}}_{{\mu }_1,{\nu }_1,l_1}^{{\gamma }_1,{\delta }_1,k_1}={\mathcal{E}}_1$, ${\mathcal{E}}_{{\mu }_2,{\nu }_2,l_2}^{{\gamma }_2,{\delta }_2,k_2}={\mathcal{E}}_2$, ${\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},d^{-}}^{\gamma ,\delta ,k}={\epsilon }_{b^{-},d^{-}}$, ${\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },b^{-},c^{+}}^{\gamma ,\delta ,k}={\epsilon }_{b^{-},c^{+}}$, ${\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},d^{-}}^{\gamma ,\delta ,k}={\epsilon }_{a^{+},d^{-}}$, and ${\epsilon }_{\mu ,\nu ,l,{\omega }^{\prime },a^{+},c^{+}}^{\gamma ,\delta ,k}={\epsilon }_{a^{+},c^{+}}$.

Since $\mathsf{\Psi }:\Delta \to \mathbb{R}$ is convex on the coordinates, it follows that the mapping ${\mathsf{\Psi }}_x:[c,d]\to \mathbb{R},{\mathsf{\Psi }}_x\left(y\right)=\mathsf{\Psi }(x,y)$ is convex on $[c,d]$ and $t{\mathcal{G}}_x:[c,d]\to \mathbb{R},t{\mathcal{G}}_x=t\mathcal{G}(x,y)$ is non-negative, integrable, and symmetric with respect to $\frac{c+d}{2}$, for all $x\in [a,b]$. Then, thanks to inequalities (9) we have

$$\begin{array}{c}{\mathsf{\Psi }}_x\left(\frac{c+d}{2}\right)\left[\left({\epsilon }_{c^{+}}t{\mathcal{G}}_x\right)\left(d\right)+\left({\epsilon }_{d^{-}}t{\mathcal{G}}_x\right)\left(c\right)\right]\le \left[\left({\epsilon }_{c^{+}}{\mathsf{\Psi }}_{x}t{\mathcal{G}}_x\right)\left(d\right)+\left({\epsilon }_{d^{-}}{\mathsf{\Psi }}_{x}t{\mathcal{G}}_x\right)\left(c\right)\right]\hfill \\ \le \frac{{\mathsf{\Psi }}_x\left(c\right)+{\mathsf{\Psi }}_x\left(d\right)}{2}\left[\left({\epsilon }_{c^{+}}t{\mathcal{G}}_x\right)\left(d\right)+\left({\epsilon }_{d^{-}}t{\mathcal{G}}_x\right)\left(c\right)\right].\hfill \end{array}$$

That is,

$$\begin{array}{c}\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)\left[\begin{array}{c}{\int }_c^d{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)t\mathcal{G}(x,y)\mathrm{d}y\\ +{\int }_c^d{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)t\mathcal{G}(x,y)\mathrm{d}y\end{array}\right]\hfill \\ \le \left[\begin{array}{c}{\int }_c^d{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }(x,y)t\mathcal{G}(x,y)\mathrm{d}y\\ +{\int }_c^d{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }(x,y)t\mathcal{G}(x,y)\mathrm{d}y\end{array}\right]\hfill \\ \le \frac{\mathsf{\Psi }(x,c)+\mathsf{\Psi }(x,d)}{2}\left[\begin{array}{c}{\int }_c^d{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)t\mathcal{G}(x,y)\mathrm{d}y\\ +{\int }_c^d{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)t\mathcal{G}(x,y)\mathrm{d}y\end{array}\right].\hfill \end{array}$$

(32)

Multiplying both sides of (32) by ${(b-x)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right)$ and ${(x-a)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right)$, and integrating with respect to x over $[a,b]$, respectively, we have

$$\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right)E_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \le {\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,c\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,d\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right],\hfill \end{array}$$

(33)

$$\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \hfill +& {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \le {\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,c\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(x,d\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right].\hfill \end{array}$$

(34)

For the mappings ${\mathsf{\Psi }}_y:[a,b]\to \mathbb{R},{\mathsf{\Psi }}_y=\mathsf{\Psi }(x,y)$ and $t{\mathcal{G}}_y:[a,b]\to \mathbb{R},t{\mathcal{G}}_y=t\mathcal{G}(x,y)$, we use the same arguments as before. So we can state that

$$\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(a,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\mathsf{\Psi }\left(b,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right]\hfill \end{array}$$

(35)

and

$$\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(a,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\mathsf{\Psi }\left(b,y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right].\hfill \end{array}$$

(36)

Adding the inequalities (33)–(36), we can write

$$\begin{array}{c}{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,\frac{c+d}{2}\right){\epsilon }_{c^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,\frac{c+d}{2}\right){\epsilon }_{d^{-}}t\mathcal{G}(b,c)\right]\hfill \\ +{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,\frac{c+d}{2}\right){\epsilon }_{c^{+}}t\mathcal{G}(a,d)\right]+{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,\frac{c+d}{2}\right){\epsilon }_{d^{-}}t\mathcal{G}(a,c)\right]\hfill \\ +{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},d\right){\epsilon }_{b^{-}}t\mathcal{G}(a,d)\right]\hfill \\ +{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,c)\right]+{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},c\right){\epsilon }_{b^{-}}t\mathcal{G}(a,c)\right]\hfill \\ \le 2\left[\left({\epsilon }_{a^{+}{c}^{+}}fg\right)(b,d)+\left({\epsilon }_{a^{+}{d}^{-}}fg\right)(b,c)+\left({\epsilon }_{b^{-}{c}^{+}}fg\right)(a,d)+\left({\epsilon }_{b^{-}{d}^{-}}fg\right)(a,c)\right]\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,c\right){\epsilon }_{c^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,d\right){\epsilon }_{d^{-}}t\mathcal{G}(b,c)\right]\\ +{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,c\right){\epsilon }_{c^{+}}t\mathcal{G}(a,d)\right]+{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,d\right){\epsilon }_{d^{-}}t\mathcal{G}(a,c)\right]\\ +{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(a,d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(b,d\right){\epsilon }_{b^{-}}t\mathcal{G}(a,d)\right]\\ +{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(a,c\right){\epsilon }_{a^{+}}t\mathcal{G}(b,c)\right]+{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(b,c\right){\epsilon }_{b^{-}}t\mathcal{G}(a,c)\right]\end{array}\right].\hfill \end{array}$$

So,

$$\begin{array}{c}\frac{1}{2}\left\{\begin{array}{c}{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,\frac{c+d}{2}\right){\epsilon }_{c^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,\frac{c+d}{2}\right){\epsilon }_{d^{-}}t\mathcal{G}(b,c)\right]\\ +{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,\frac{c+d}{2}\right){\epsilon }_{c^{+}}t\mathcal{G}(a,d)\right]+{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,\frac{c+d}{2}\right){\epsilon }_{d^{-}}t\mathcal{G}(a,c)\right]\\ +{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},d\right){\epsilon }_{b^{-}}t\mathcal{G}(a,d)\right]\\ +{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,c)\right]+{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(\frac{a+b}{2},c\right){\epsilon }_{b^{-}}t\mathcal{G}(a,c)\right]\end{array}\right\}\hfill \end{array}$$

$$\begin{array}{c}\le \left[\left({\epsilon }_{a^{+}{c}^{+}}fg\right)(b,d)+\left({\epsilon }_{a^{+}{d}^{-}}fg\right)(b,c)+\left({\epsilon }_{b^{-}{c}^{+}}fg\right)(a,d)+\left({\epsilon }_{b^{-}{d}^{-}}fg\right)(a,c)\right]\hfill \\ \le \frac{1}{4}\left[\begin{array}{c}{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,c\right){\epsilon }_{c^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,d\right){\epsilon }_{d^{-}}t\mathcal{G}(b,c)\right]\\ +{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,c\right){\epsilon }_{c^{+}}t\mathcal{G}(a,d)\right]+{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,d\right){\epsilon }_{d^{-}}t\mathcal{G}(a,c)\right]\\ +{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(a,d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(b,d\right){\epsilon }_{b^{-}}t\mathcal{G}(a,d)\right]\\ +{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(a,c\right){\epsilon }_{a^{+}}t\mathcal{G}(b,c)\right]+{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(b,c\right){\epsilon }_{b^{-}}t\mathcal{G}(a,c)\right]\end{array}\right]\hfill \\ \le \frac{1}{2}\left[\begin{array}{c}{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,c\right){\epsilon }_{c^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{a^{+}}\left[\mathsf{\Psi }\left(b,d\right){\epsilon }_{d^{-}}t\mathcal{G}(b,c)\right]\\ +{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,c\right){\epsilon }_{c^{+}}t\mathcal{G}(a,d)\right]+{\epsilon }_{b^{-}}\left[\mathsf{\Psi }\left(a,d\right){\epsilon }_{d^{-}}t\mathcal{G}(a,c)\right]\\ +{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(a,d\right){\epsilon }_{a^{+}}t\mathcal{G}(b,d)\right]+{\epsilon }_{c^{+}}\left[\mathsf{\Psi }\left(b,d\right){\epsilon }_{b^{-}}t\mathcal{G}(a,d)\right]\\ +{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(a,c\right){\epsilon }_{a^{+}}t\mathcal{G}(b,c)\right]+{\epsilon }_{d^{-}}\left[\mathsf{\Psi }\left(b,c\right){\epsilon }_{b^{-}}t\mathcal{G}(a,c)\right]\end{array}\right].\hfill \end{array}$$

This gives the second and third inequalities in (31).

Now we will show the first inequality in (31).

Apply Lemma 3 for the functions $F_1:[a,b]\to \mathbb{R},F_1\left(x\right)=\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)$ and $t{\mathcal{G}}_y:[a,b]\to \mathbb{R},t{\mathcal{G}}_y\left(x\right)=t\mathcal{G}(x,y)$:

$$\begin{array}{c}{F}_1\left(\frac{a+b}{2}\right)\left[\left({\epsilon }_{b^{-}}t{\mathcal{G}}_y\right)\left(a\right)+\left({\epsilon }_{a^{+}}t{\mathcal{G}}_y\right)\left(b\right)\right]\le \left[\left({\epsilon }_{b^{-}}{F}_{1}t{\mathcal{G}}_y\right)\left(a\right)+\left({\epsilon }_{a^{+}}{F}_{1}t{\mathcal{G}}_y\right)\left(b\right)\right].\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}{\int }_a^b{(x-a)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right)t\mathcal{G}(x,y)\mathrm{d}x\\ +{\int }_a^b{(b-x)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right)t\mathcal{G}(x,y)\mathrm{d}x\end{array}\right]\hfill \\ \le \left[\begin{array}{c}{\int }_a^b{(x-a)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}x\\ +{\int }_a^b{(b-x)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right)\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}x\end{array}\right].\hfill \end{array}$$

(37)

Multiplying both sides of (37) by ${\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right){(d-y)}^{{\nu }_2-1}$ and ${\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right){(y-c)}^{{\nu }_2-1}$ and integrating with respect to y over $[c,d]$, respectively, we have

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right]\hfill \\ \le \left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right],\hfill \end{array}$$

(38)

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right]\hfill \\ \le \left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(x,\frac{c+d}{2}\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right].\hfill \end{array}$$

(39)

The same way, we apply Lemma 3 for the functions $F_2:[c,d]\to \mathbb{R},F_1\left(x\right)=\mathsf{\Psi }\left(\frac{a+b}{2},y\right)$ and $t{\mathcal{G}}_x:[a,b]\to \mathbb{R},t{\mathcal{G}}_x\left(y\right)=t\mathcal{G}(x,y)$, and we get:

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right]\hfill \\ \le \left[\begin{array}{c}{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(x-a)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right],\hfill \end{array}$$

(40)

$$\begin{array}{c}\mathsf{\Psi }\left(\frac{a+b}{2},\frac{c+d}{2}\right)\left[\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right]\hfill \\ \le \left[\begin{array}{c}{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(y-c)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\\ +{\int }_a^b{\int }_c^d{(b-x)}^{{\nu }_1-1}{(d-y)}^{{\nu }_2-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right){\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)\\ ·\mathsf{\Psi }\left(\frac{a+b}{2},y\right)t\mathcal{G}(x,y)\mathrm{d}y\mathrm{d}x\end{array}\right].\hfill \end{array}$$

(41)

Adding (38)–(41), we get the first inequality of (31).

For the last inequality of (31), we apply Lemma 3 to the functions

1.

$F_3:[a,b]\to \mathbb{R},F_3\left(x\right)=\mathsf{\Psi }(x,c)$ and $t{\mathcal{G}}_y:[a,b]\to \mathbb{R},t{\mathcal{G}}_y\left(x\right)=t\mathcal{G}(x,y)$,

2.

$F_4:[a,b]\to \mathbb{R},F_4\left(x\right)=\mathsf{\Psi }(x,d)$ and $t{\mathcal{G}}_y:[a,b]\to \mathbb{R},t{\mathcal{G}}_y\left(x\right)=t\mathcal{G}(x,y)$,

3.

$F_5:[c,d]\to \mathbb{R},F_5\left(x\right)=\mathsf{\Psi }(a,y)$ and $t{\mathcal{G}}_x:[c,d]\to \mathbb{R},t{\mathcal{G}}_x\left(y\right)=t\mathcal{G}(x,y)$

4.

$F_6:[c,d]\to \mathbb{R},F_6\left(x\right)=\mathsf{\Psi }(b,y)$ and $t{\mathcal{G}}_x:[c,d]\to \mathbb{R},t{\mathcal{G}}_x\left(y\right)=t\mathcal{G}(x,y)$,

and we obtain

$$\begin{array}{c}\left({\epsilon }_{b^{-}}{F}_{3}t{\mathcal{G}}_y\right)\left(a\right)+\left({\epsilon }_{a^{+}}{F}_{3}t{\mathcal{G}}_y\right)\left(b\right)\le \frac{F_3\left(a\right)+F_3\left(b\right)}{2}\left[\left({\epsilon }_{b^{-}}t{\mathcal{G}}_y\right)\left(a\right)+\left({\epsilon }_{a^{+}}t{\mathcal{G}}_y\right)\left(b\right)\right],\hfill \end{array}$$

(42)

$$\begin{array}{c}\left({\epsilon }_{b^{-}}{F}_{4}t{\mathcal{G}}_y\right)\left(a\right)+\left({\epsilon }_{a^{+}}{F}_{4}t{\mathcal{G}}_y\right)\left(b\right)\le \frac{F_4\left(a\right)+F_4\left(b\right)}{2}\left[\left({\epsilon }_{b^{-}}t{\mathcal{G}}_y\right)\left(a\right)+\left({\epsilon }_{a^{+}}t{\mathcal{G}}_y\right)\left(b\right)\right],\hfill \end{array}$$

(43)

$$\begin{array}{c}\left({\epsilon }_{d^{-}}{F}_{5}t{\mathcal{G}}_x\right)\left(c\right)+\left({\epsilon }_{c^{+}}{F}_{5}t{\mathcal{G}}_x\right)\left(d\right)\le \frac{F_5\left(c\right)+F_5\left(d\right)}{2}\left[\left({\epsilon }_{d^{-}}t{\mathcal{G}}_x\right)\left(c\right)+\left({\epsilon }_{c^{+}}t{\mathcal{G}}_x\right)\left(d\right)\right],\hfill \end{array}$$

(44)

and

$$\begin{array}{c}\left({\epsilon }_{d^{-}}{F}_{6}t{\mathcal{G}}_x\right)\left(c\right)+\left({\epsilon }_{c^{+}}{F}_{6}t{\mathcal{G}}_x\right)\left(d\right)\le \frac{F_6\left(c\right)+F_6\left(d\right)}{2}\left[\left({\epsilon }_{d^{-}}t{\mathcal{G}}_x\right)\left(c\right)+\left({\epsilon }_{c^{+}}t{\mathcal{G}}_x\right)\left(d\right)\right].\hfill \end{array}$$

(45)

Then we have inequality

• (42) and (43) by ${(y-c)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(y-c)}^{{\mu }_2}\right)$, respectively ${(d-y)}^{{\nu }_2-1}{\mathcal{E}}_2\left({\omega }_2^{\prime }{(d-y)}^{{\mu }_2}\right)$ and integrating with respect to y over $[c,d]$,
• (44) and (45) by ${(x-a)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(x-a)}^{{\mu }_1}\right)$, respectively ${(b-x)}^{{\nu }_1-1}{\mathcal{E}}_1\left({\omega }_1^{\prime }{(b-x)}^{{\mu }_1}\right)$ and integrating with respect to x over $[a,b]$.

We get four inequalities that we are adding, and taking into account the symmetry of $t\mathcal{G}$, we obtain the last inequality of (31). The proof is complete. □

 Remark 4. 

If in Theorem 3 we put $\omega =(0,0)$, we obtain [20] (Theorem 8).

3. Conclusions

First, we introduced and studied generalized fractional integral operators containing the new extended general Mittag–Leffler function of two variables. We then obtained several new two-dimensional versions of trapezium-like inequalities via coordinated convex functions. We also discussed the linkage of the obtained results with previously known results by considering some special cases. It is expected that the ideas and techniques of this paper may stimulate further research.