Theory and Modern Applications

# General Raina fractional integral inequalities on coordinates of convex functions

## Abstract

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In this study, authors have established some generalized Raina fractional integral inequalities using an $$(l_{1},h_{1})$$-$$(l_{2},h_{2})$$-convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for the $$(l_{1},h_{1})$$-$$(l_{2},h_{2})$$-convex function on coordinates are given. Finally, we can say that our findings recapture some recent results as special cases.

## 1 Introduction

In the past two decades, fractional calculus has received much attention. The fast interest in the topic is due to its extensive applications in various fields such as biochemistry, physics, viscoelasticity, fluid mechanics, computer modeling, and engineering, see [13] for further detail. Most of the studies have been devoted to the existence and uniqueness of solutions for fractional differential equations (FDEs); see e.g. [49]. A fractional differential equation needs a certain inequality to be existent and unique for solution. For this reason, a huge number of mathematicians have competed to seek such inequalities; see e.g. [1029].

Always, it is important and necessary to specify which model or definition is being used because there are many different ways of defining fractional integrals and derivatives. To further facilitate the discussion of this model, we present here the definition which is most commonly used for fractional integrals and derivatives, namely the Riemann–Liouville (RL) definition.

### Definition 1.1

([1, 2])

For any $$\mathrm{L}^{1}$$ function $$f(x)$$ on an interval $$[\chi _{1},\chi _{2} ]$$ with $$x\in [\chi _{1},\chi _{2} ]$$, the ηth left-RL fractional integral of $$f(x)$$ is defined as follows:

$${}^{\mathrm{RL}}{\mathtt{J}}^{\eta }_{\chi _{1}+}f(x) :=\frac{1}{\Gamma (\eta )} \int _{\chi _{1}}^{x}(x-\xi )^{\eta -1}f( \xi ) \,{d}\xi ,\quad \chi _{1}< x,$$
(1.1)

for $$\operatorname{Re} (\eta )>0$$. Also, the ηth right-RL fractional integral of $$f(x)$$ is defined as follows:

$${}^{\mathrm{RL}}{\mathtt{J}}^{\eta }_{\chi _{2}-}f(x) :=\frac{1}{\Gamma (\eta )} \int _{x}^{\chi _{2}}(\xi -x)^{\eta -1}f( \xi ) \,{d}\xi ,\quad x< \chi _{2}.$$
(1.2)

In the recent decades, a strong modern direction of research in fractional calculus has brought the attention of interested researchers in various disciplines to investigate various possible ways to define fractional integrals and derivatives, often with different properties from the classical RL in Definition 1.1. In 2005, Raina [30] introduced the new fractional integrals, often called the Raina fractional integrals, corresponding to the classical RL integrals (1.1) and (1.2).

### Definition 1.2

([30])

For any $$\mathrm{L}^{1}$$ function $$f(x)$$ on an interval $$[\chi _{1},\chi _{2} ]$$ with $$x\in [\chi _{1},\chi _{2} ]$$, the ηth left Raina fractional integral of $$f(x)$$ is defined as follows:

$$\Im _{\rho ,\eta ,\chi _{1}^{+},\omega }^{\sigma }\varphi ( x ) = \int_{\chi _{1}}^{x} ( x-t ) ^{\eta -1} \mathfrak{F} _{\rho ,\eta }^{\sigma } \bigl[ \omega ( x-t ) ^{\rho } \bigr] \varphi ( t ) \,dt,\quad \chi _{1}< x,$$
(1.3)

and the ηth right Raina fractional integral of $$f(x)$$ is defined as follows:

$$\Im _{\rho ,\eta ,\chi _{2}^{-},\omega }^{\sigma }\varphi ( x ) = \int^{\chi _{2}}_{x} ( t-x ) ^{\eta -1}\mathfrak{F} _{\rho ,\eta }^{\sigma } \bigl[ \omega ( t-x ) ^{\rho } \bigr] \varphi ( t ) \,dt, \quad x< \chi _{2},$$
(1.4)

where $$\mathfrak{F} _{\rho ,\eta }^{\sigma } ( x )$$ is the generalization of Mittag-Leffler (ML) function defined as follows: For a bounded arbitrary sequence $$\sigma ( k )$$ of real or complex numbers, we define the function $$\mathfrak{F} _{\rho ,\eta }^{\sigma } ( x )$$ by

$$\mathfrak{F} _{\rho ,\eta }^{\sigma } ( x ) = \sum _{k=0}^{\infty} \frac{\sigma ( k ) }{\Gamma ( \rho k+\eta ) }x^{k},$$
(1.5)

where $$\rho ,\eta \in \mathbb{C}$$ with $$\operatorname{Re} (\rho )>0$$, $$x\in \mathbb{R}$$, and $$\Gamma (\cdot )$$ denotes the classical gamma function.

### Remark 1.1

By making use of $$\eta =\alpha$$, $$\sigma (0)=1$$, and $$\omega =0$$ in both (1.3) and (1.4), we obtain the classical left and right-RL fractional integrals (1.1) and (1.2), respectively.

## 2 Literature results

Before we pass to the main findings, we review and introduce some definitions, notations, theorems which will be necessary later to proceed.

### Definition 2.1

([31])

A function $$f:\mathcal{I}\subseteq \mathbb{R}\to \mathbb{R}$$ is said to be convex on $$\mathcal{I}$$ if

$$f\bigl((1-\xi )\chi _{1}+\xi \chi _{2} \bigr)\leq (1-\xi )f(\chi _{1})+{\xi }f( \chi _{2})$$
(2.1)

holds for every $$\chi _{1}, \chi _{2}\in \mathcal{I}$$ and $$\xi \in [0,1]$$.

### Definition 2.2

([32])

Denote $$\Delta := [ \chi _{1}, \chi _{2} ] \times [ \chi _{3}, \chi _{4} ]$$, where $$0<\chi _{1}<\chi _{2}$$ and $$0<\chi _{3}<\chi _{4}$$. For a function $$f:\Delta \to \mathbb{R}$$, the coordinated convex function on Δ is defined as follows:

\begin{aligned}& f \bigl(\bigl(\xi _{1}\chi _{1}+(1-\xi _{1})\chi _{2}\bigr),\bigl(\xi _{2}\chi _{3}+(1- \xi _{2})\chi _{4}\bigr) \bigr) \\& \quad \leq \xi _{1}\xi _{2}f(\chi _{1},\chi _{3})+ \xi _{2}(1-\xi _{1})f(\chi _{2},\chi _{3}) \\& \qquad {}+\xi _{1}(1-\xi _{2})f(\chi _{1},\chi _{4})+(1-\xi _{1}) (1-\xi _{2})f( \chi _{2},\chi _{4}) \end{aligned}
(2.2)

for every $$\xi _{1}, \xi _{2}\in [0,1]$$ and $$(\chi _{1}, \chi _{2}), (\chi _{3},\chi _{4})\in \Delta$$.

The well-known integral inequality of Hermite–Hadamard type (HH-type) for such a convex function (2.1) is given by

$$f \biggl( \frac{\chi _{1}+\chi _{2}}{2} \biggr) \leq \frac{1}{\chi _{2}-\chi _{1}} \int _{\chi _{1}}^{\chi _{2}}f(x)\,dx \leq \frac{f ( \chi _{1} ) +f (\chi _{2} ) }{2}.$$
(2.3)

In 2001, HH-inequality (2.3) was established on the bidimensional plane Δ for such a coordinated convex function (2.2) by Dragomir [32], his result is as follows.

### Theorem 2.1

Let $$f:\Delta \to \mathbb{R}$$ be a coordinated convex function on Δ, then we have

\begin{aligned}& f \biggl( \frac{\chi _{1}+\chi _{2}}{2},\frac{\chi _{3}+\chi _{4}}{2} \biggr) \\& \quad \leq \frac{1}{2} \biggl(\frac{1}{\chi _{2}-\chi _{1}} \int_{\chi _{1}}^{\chi _{2}} f \biggl(x , \frac{\chi _{3}+\chi _{4}}{2} \biggr)\,dx + \frac{1}{\chi _{4}-\chi _{3}} \int^{\chi _{4}}_{\chi _{3}} f \biggl( \frac{\chi _{1}+\chi _{2}}{2},y \biggr)\,dy \biggr) \\& \quad \leq \frac{1}{ ( \chi _{2}-\chi _{1} ) ( \chi _{4}-\chi _{3} )} \int_{\chi _{1}}^{\chi _{2}} \int^{\chi _{4}}_{\chi _{3}}f(x ,y)\,dy\,dx \\& \quad \leq \frac{1}{4} \biggl( \frac{1}{\chi _{2}-\chi _{1}} \int_{\chi _{1}}^{\chi _{2}} \bigl(f(x ,\chi _{3})+f(x , \chi _{4})\bigr)\,dx \frac{1}{\chi _{4}-\chi _{3}} \int^{\chi _{4}}_{\chi _{3}}\bigl(f(\chi _{1},y)+f( \chi _{2},y)\bigr)\,dy \biggr) \\& \quad \leq \frac{f(\chi _{1},\chi _{3})+f(\chi _{1},\chi _{4}) +f(\chi _{2},\chi _{3})+f(\chi _{2},\chi _{4})}{4}. \end{aligned}
(2.4)

In 2014, HH-inequality (2.4) was generalized to fractional integrals of RL type by Sarikaya [33], which is as follows.

### Theorem 2.2

Let $$f:\Delta \to \mathbb{R}$$ be a coordinated convex function on Δ, then we have

\begin{aligned} &f \biggl( \frac{\chi _{1}+\chi _{2}}{2},\frac{\chi _{3}+\chi _{4}}{2} \biggr) \\ &\quad \leq \frac{\Gamma ( \alpha +1 ) }{4 ( \chi _{2}-\chi _{1} ) ^{\alpha }} \biggl( J_{\chi _{1}^{+}}^{\alpha }f \biggl( \chi _{2}, \frac{\chi _{3}+\chi _{4}}{2} \biggr) +J_{\chi _{2}^{-}}^{\alpha }f \biggl( \chi _{1},\frac{\chi _{3}+\chi _{4}}{2} \biggr) \biggr) \\ &\qquad {}+ \frac{\Gamma ( \beta +1 ) }{4 ( \chi _{4}-\chi _{3} ) ^{\beta }} \biggl( J_{\chi _{3}^{+}}^{\alpha }f \biggl( \frac{\chi _{1}+\chi _{2}}{2},\chi _{4} \biggr) +J_{\chi _{4}^{-}}^{ \beta }f \biggl( \frac{\chi _{1}+\chi _{2}}{2},\chi _{3} \biggr) \biggr) \\ &\quad \leq \frac{\Gamma ( \alpha +1 ) \Gamma ( \beta +1 ) }{ ( \chi _{2}-\chi _{1} ) ^{\alpha } ( \chi _{4}-\chi _{3} ) ^{\beta }} \bigl(J_{\chi _{1}^{+},\chi _{3}^{+}}^{\alpha ,\beta }f ( \chi _{2}, \chi _{4} ) +J_{\chi _{1}^{+},\chi _{4}^{-}}^{\alpha ,\beta }f ( \chi _{2},\chi _{3} ) \\ &\qquad {}+J_{\chi _{2}^{-},\chi _{3}^{+}}^{ \alpha ,\beta }f ( \chi _{1},\chi _{4} ) +J_{\chi _{2}^{-}, \chi _{4}^{-}}^{\alpha ,\beta }f ( \chi _{1},\chi _{3} ) \bigr) \\ &\quad \leq \frac{\Gamma ( \alpha +1 ) }{4 ( \chi _{2}-\chi _{1} ) ^{\alpha }} \bigl( J_{\chi _{2}^{-}}^{\alpha }f ( \chi _{1},\chi _{4} ) +J_{\chi _{2}^{-}}^{\alpha } f (\chi _{1},\chi _{3} ) +J_{\chi _{1}^{+}}^{\alpha }f ( \chi _{2},\chi _{4} ) +J_{\chi _{1}^{+}}^{\alpha }f ( \chi _{2},\chi _{3} ) \bigr) \\ &\qquad {}+ \frac{\Gamma ( \beta +1 ) }{4 ( \chi _{4}-\chi _{3} ) ^{\beta }} \bigl( J_{\chi _{4}^{-}}^{\beta }f ( \chi _{2},\chi _{3} ) +J_{\chi _{4}^{-}}^{\beta } f (\chi _{1},\chi _{3} ) +J_{ \chi _{3}^{+}}^{\alpha }f ( \chi _{2},\chi _{4} ) +J_{ \chi _{3}^{+}}^{\alpha }f ( \chi _{1},\chi _{4} ) \bigr) \\ &\quad \leq \frac{f(\chi _{1},\chi _{3})+f(\chi _{1},\chi _{4})+f(\chi _{2},\chi _{3})+f(\chi _{2},\chi _{4})}{4}. \end{aligned}

The above inequalities have attracted many researchers in the recent years, see e.g. [3438].

### Definition 2.3

([39])

Let $$f\in L ( \Delta )$$. The fractional integral operators for two variable functions, where $$(\rho ,\eta ,\omega )\in [0,+\infty )^{2}\times [0,+\infty )^{2} \times \mathbb{R}^{2}$$ with $$\rho =(\rho _{1},\rho _{2})$$, $$\eta =(\eta _{1},\eta _{2})$$, $$\omega =( \omega _{1},\omega _{2})$$, and $$\sigma =(\sigma _{1},\sigma _{2})$$, are given as follows:

\begin{aligned}& \begin{aligned} \Im _{\rho ,\eta ,\chi _{1}^{+},\chi _{3}^{+},\omega }^{\sigma } \varphi ( x,y ) ={}& \int_{\chi _{1}}^{x} \int_{\chi _{3}}^{y} ( x-\xi _{1} ) ^{ \eta _{1}-1} ( y-\xi _{2} ) ^{\eta _{2}-1}\mathfrak{F} _{ \rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} ( x-\xi _{1} ) ^{\rho _{1}} \bigr] \\ &{}\times \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} ( y-\xi _{2} ) ^{\rho _{1}} \bigr] \varphi ( \xi _{1},\xi _{2} ) \,d\xi _{2}\,d\xi _{1}, \end{aligned} \\& \begin{aligned} \Im _{\rho ,\eta ,\chi _{1}^{+},\chi _{4}^{-},\omega }^{\sigma } \varphi ( x,y ) ={}& \int_{\chi _{1}}^{x} \int^{\chi _{4}}_{y} ( x-\xi _{1} ) ^{ \eta _{1}-1} ( \xi _{2}-y ) ^{\eta _{2}-1}\mathfrak{F} _{ \rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} ( x-\xi _{1} ) ^{\rho _{1}} \bigr] \\ &{}\times \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} ( \xi _{2}-y ) ^{\rho _{1}} \bigr] \varphi ( \xi _{1},\xi _{2} ) \,d\xi _{2}\,d\xi _{1}, \end{aligned} \\& \begin{aligned} \Im _{\rho ,\eta ,\chi _{2}^{-},\chi _{3}^{+},\omega }^{\sigma } \varphi ( x,y ) ={}& \int^{\chi _{2}}_{x} \int_{\chi _{3}}^{y} ( \xi _{1}-x ) ^{ \eta _{1}-1} ( y-\xi _{2} ) ^{\eta _{2}-1}\mathfrak{F} _{ \rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} ( \xi _{1}-x ) ^{\rho _{1}} \bigr] \\ &{}\times \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} ( y-\xi _{2} ) ^{\rho _{1}} \bigr] \varphi ( \xi _{1},\xi _{2} ) \,d\xi _{2}\,d\xi _{1}, \end{aligned} \end{aligned}

and

\begin{aligned} \Im _{\rho ,\eta ,\chi _{2}^{-},\chi _{4}^{-},\omega }^{\sigma } \varphi ( x,y ) ={}& \int^{\chi _{2}}_{x} \int^{\chi _{4}}_{y} ( \xi _{1}-x ) ^{ \eta _{1}-1} ( \xi _{2}-y ) ^{\eta _{2}-1}\mathfrak{F} _{ \rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} ( \xi _{1}-x ) ^{\rho _{1}} \bigr] \\ &{}\times \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} ( \xi _{2}-y ) ^{\rho _{1}} \bigr] \varphi ( \xi _{1},\xi _{2} ) \,d\xi _{2}\,d\xi _{1}. \end{aligned}

Also, we have

\begin{aligned}& \Im _{\rho _{1},\eta _{1},\chi _{1}^{+},\omega _{1}}^{\sigma _{1}} \varphi \biggl( x,\frac{\chi _{3}+\chi _{4}}{2} \biggr) = \int_{\chi _{1}}^{x} ( x-\xi _{1} ) ^{ \eta _{1}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} ( x-\xi _{1} ) ^{\rho _{1}} \bigr] \varphi \biggl( \xi _{1},\frac{\chi _{3}+\chi _{4} }{2} \biggr) \,d\xi _{1}, \\& \Im _{\rho _{1},\eta _{1},\chi _{2}^{-},\omega _{1}}^{\sigma _{1}} \varphi \biggl( x,\frac{\chi _{3}+\chi _{4}}{2} \biggr) = \int^{\chi _{2}}_{x} ( \xi _{1}-x ) ^{ \eta _{1}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} ( \xi _{1}-x ) ^{\rho _{1}} \bigr] \varphi \biggl( \xi _{1},\frac{\chi _{3}+\chi _{4} }{2} \biggr) \,d\xi _{1}, \\& \Im _{\rho _{2},\eta _{2},\chi _{3}^{+},\omega _{2}}^{\sigma _{2}} \varphi \biggl( \frac{\chi _{1}+\chi _{2}}{2},y \biggr) = \int_{\chi _{3}}^{y} ( y-\xi _{2} ) ^{ \eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} ( y-\xi _{2} ) ^{\rho _{1}} \bigr] \varphi \biggl( \frac{\chi _{1}+\chi _{2}}{2 },\xi _{2} \biggr) \,d\xi _{2}, \end{aligned}

and

$$\Im _{\rho _{2},\eta _{2},\chi _{4}^{-},\omega _{2}}^{\sigma _{2}} \varphi \biggl( \frac{\chi _{1}+\chi _{2}}{2},y \biggr) = \int^{\chi _{4}}_{y} ( \xi _{2}-y ) ^{ \eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} ( \xi _{2}-y ) ^{\rho _{1}} \bigr] \varphi \biggl( \frac{\chi _{1}+\chi _{2}}{2 },\xi _{2} \biggr) \,d\xi _{2}.$$

In [39], Tunç and Sarikaya investigate the following Hermite–Hadamard for coordinated convex functions:

\begin{aligned}& f \biggl( \frac{\chi _{1}+\chi _{2}}{2},\frac{\chi _{3}+\chi _{4}}{2} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{2}-\chi _{1} ) ^{\eta _{1}} ( \chi _{4}-\chi _{3} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}-\chi _{1} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}-\chi _{3} ) ^{\rho _{2}} ) } \\& \qquad {} \times \bigl\{ \Im _{\rho ,\eta ,\chi _{1}^{+},\chi _{3}^{+}, \omega }^{\sigma }f ( \chi _{2},\chi _{4} ) +\Im _{\rho , \eta ,\chi _{1}^{+},\chi _{4}^{-},\omega }^{\sigma }f ( \chi _{2}, \chi _{3} ) \\& \qquad {} + \Im _{\rho ,\eta ,\chi _{2}^{-},\chi _{3}^{+},\omega }^{ \sigma }f ( \chi _{1},\chi _{4} ) +\Im _{\rho ,\eta ,\chi _{2}^{-}, \chi _{4}^{-},\omega }^{\sigma }f ( \chi _{1},\chi _{3} ) \bigr\} \\& \quad \leq \frac{ ( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1},\chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) ) }{4}. \end{aligned}

### Definition 2.4

([40])

Let $$h_{1},h_{2}:J\to \mathbb{R}$$ be two nonnegative and nonzero functions. A function $$f:\Delta \to \mathbb{R}$$ is said to be $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on the coordinates on Δ if

\begin{aligned}& f \bigl( \bigl[ \xi _{1}x^{l_{1}}+ ( 1-\xi _{1} ) u^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ \xi _{2}y^{l_{2}}+ ( 1-\xi _{2} ) v^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\& \quad \leq h_{1} ( \xi _{1} ) h_{2} ( \xi _{2} ) f ( x,y ) +h_{1} ( \xi _{1} ) h_{2} ( 1- \xi _{2} ) f ( x,v ) +h_{1} ( 1- \xi _{1} ) h_{2} ( \xi _{2} ) f ( u,y ) \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( u,v ) \end{aligned}

holds for all $$( x,y ) , ( u,v ) \in \Delta$$ and $$\xi _{1}, \xi _{2}\in ( 0,1 )$$.

As we know, there are many results on coordinates of convex functions via other types of fractional operators and other types of convex functions, see e.g. [4144]. Therefore, integral inequalities on coordinates of convex functions via general Raina fractional integrals open a new door in the field of mathematical analysis and theory of convexity.

Motivated by the above results, in this paper we establish some generalized integral inequalities using an $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates. Also, we obtain an integral identity for partial differentiable functions. As an effect of this result, two interesting integral inequalities for an $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates are given. At the end, a brief conclusion is provided as well.

## 3 Main results

In what follows, we assume that $$h_{1},h_{2}:J\to \mathbb{R}$$ are two nonnegative and nonzero functions, with $$h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) \neq 0$$, $$\sigma = ( \sigma _{1},\sigma _{2} )$$, $$\rho = ( \rho _{1},\rho _{2} )$$, $$\eta = ( \eta _{1},\eta _{2} )$$, and $$\omega = ( \omega _{1},\omega _{2} )$$ with $$\rho _{1},\rho _{2},\eta _{1},\eta _{2}\in [0,+\infty )$$ and $$\omega _{1},\omega _{2}\in \mathbb{R}$$.

### Theorem 3.1

Let $$f:\Delta \to \mathbb{R}$$ be an integrable and $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates on Δ. Then we have

\begin{aligned}& \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad = \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f_{g} \biggl( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2}, \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \bigl\{ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \bigr\} \\& \quad \leq \frac{ ( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1},\chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) ) }{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {} \times \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}\,d\xi _{1}, \end{aligned}

where $$f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) )$$ with $$g_{1} ( x ) =x^{\frac{1}{l_{1}}}$$ and $$g_{2} ( y ) =y^{\frac{1}{l_{2}}}$$.

### Proof

It is easy to see that

$$\biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{ \frac{1}{l_{1}}}= \biggl[ \frac{ ( (\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} ) ^{\frac{1 }{l_{1}}} ) ^{l_{1}}+ ( ( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1}\chi _{2}^{l_{1}} ) ^{\frac{1}{l_{1}}} ) ^{l_{1}}}{2} \biggr] ^{ \frac{1}{l_{1}}}$$
(3.1)

and

$$\biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}}= \biggl[ \frac{ ( (\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} ) ^{\frac{1 }{l_{2}}} ) ^{l_{2}}+ ( ( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2}\chi _{4}^{l_{2}} ) ^{\frac{1}{l_{2}}} ) ^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}}.$$
(3.2)

Making use of (3.1) and (3.2), and the fact that f is $$(l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex on the coordinates, we have

\begin{aligned}& f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq h_{1} \biggl( \frac{1}{2} \biggr) h_{2} \biggl( \frac{1}{2} \biggr) \bigl\{ f \bigl( \bigl(\xi _{1} \chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{ 1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{ l_{2}}} \bigr) \\& \qquad {}+f \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+ f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \bigr\} . \end{aligned}
(3.3)

Multiplying on both sides of (3.3) by

$$\xi _{1}^{\eta _{1}-1}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl(\chi _{4}^{l_{2}} -\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr],$$

and then integrating the resulting inequality with respect to $$(\xi _{1},\xi _{2} )$$ on $$[ 0,1 ] ^{2}$$, we get

\begin{aligned}& \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \qquad {}\times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \,d\xi _{2}\,d\xi _{1} \\& \quad \leq \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ \xi _{2} \chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2}^{ \rho _{2}} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) \xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ \xi _{2} \chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\& \qquad {}\times f \bigl( \bigl[ ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr] ^{ \frac{1}{l_{1}}}, \bigl[ ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}
(3.4)

By making a change of variables in (3.4), we obtain

\begin{aligned}& \frac{1}{h_{1} ( \frac{1}{2} ) h_{2} ( \frac{1}{2} ) } f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} )} \\& \qquad {}\times \biggl\{ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\rho _{2}} \bigr] f \bigl( x^{ \frac{1}{l_{1}} },y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \\& \qquad {}+ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}} \bigr] f \bigl( x^{ \frac{1}{l_{1}} },y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \\& \qquad {}+ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\eta _{1}-1} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2}^{\rho _{2}} \bigl( \chi _{4}^{l_{2}}-y \bigr) \bigr] f \bigl( x^{ \frac{1}{l_{1}}},y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \\& \qquad {}+ \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\eta _{1}-1} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( x-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}} \bigr] \\& \qquad {}\times\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( y-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}} \bigr] f \bigl( x^{ \frac{1}{ l_{1}}},y^{\frac{1}{l_{2}}} \bigr) \,dy\,dx \biggr\} \\& \quad = \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} )} \\& \qquad {}\times \bigl\{ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {}+\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}}, \chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{ \sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \bigr\} . \end{aligned}
(3.5)

Since f is $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex on the coordinates, we have

\begin{aligned}& \begin{aligned}[b] &f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ \xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{4} ) +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2},\chi _{3} ) \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{4} ) , \end{aligned} \end{aligned}
(3.6)
\begin{aligned}& \begin{aligned}[b] &f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{4} ) +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{3} ) \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2},\chi _{4} ) , \end{aligned} \end{aligned}
(3.7)
\begin{aligned}& \begin{aligned}[b] &f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{4} ) +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2}, \chi _{3} ) \\ &\qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{4} ), \end{aligned} \end{aligned}
(3.8)

and

\begin{aligned} &f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\ &\quad \leq h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{1},\chi _{3} ) +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{1},\chi _{4} ) \\ &\qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) f ( \chi _{2},\chi _{3} )+h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) f ( \chi _{2},\chi _{4} ) . \end{aligned}
(3.9)

Adding inequalities (3.6)–(3.9), multiplying the resulting inequality by

$$\xi _{1}^{\eta _{1}-1}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr],$$

and then integrating the result with respect to $$(\xi _{1},\xi _{2} )$$ on $$[ 0,1 ] ^{2}$$, we get

\begin{aligned}& \int^{1}_{0} \int^{1}_{0}\xi _{1}^{ \eta _{1}-1} \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{ \sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl\{ f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1- \xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{l_{1}}}, \bigl[ \xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr] ^{ \frac{1}{l_{2}}} \bigr) \\& \qquad {}+f \bigl( \bigl[ \xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr] ^{\frac{1}{ l_{1}}}, \bigl[ ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2} \chi _{4}^{l_{2}} \bigr] ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{ l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2} }} \bigr) \\& \qquad {}+ f \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \bigr\} \,d\xi _{2}\,d\xi _{1} \\& \quad \leq \bigl( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1}, \chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) \bigr) \\& \qquad {}\times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl( h_{1} (\xi _{1} ) h_{2} ( \xi _{2} ) +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}

Making use of the change of variables and multiplying the result by

$$\frac{1}{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) },$$

we obtain

\begin{aligned}& \frac{1}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1}, \eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \bigl\{ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega }^{\sigma }f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \bigr\} \\& \quad \leq \frac{ ( f ( \chi _{1},\chi _{3} ) +f ( \chi _{1},\chi _{4} ) +f ( \chi _{2},\chi _{3} ) +f ( \chi _{2},\chi _{4} ) ) }{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {} \times \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}-1} \xi _{2}^{ \eta _{2}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {} \times \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}

This rearranges to the proof of Theorem 3.1. □

### Remark 3.1

Theorem 3.1 with $$l_{1}=l_{2}=1$$ and $$h_{1} (\xi _{1} ) =h_{2} (\xi _{1} ) =\xi _{1}$$ becomes Theorem 2.1 in [39].

### Remark 3.2

Theorem 3.1 with $$l_{1}=l_{2}=1$$, $$\eta _{1}=\eta _{2}=\alpha$$, $$\sigma _{1}(0) =\sigma _{2} ( 0 ) =1$$, $$\omega _{1}=\omega _{2}=0$$, and $$h_{1} (\xi _{1} )=h_{2} (\xi _{1} ) =\xi _{1}$$ becomes Theorem 3 in [33].

### Remark 3.3

Theorem 3.1 with $$\eta _{1}=\eta _{2}=1$$, $$\sigma _{1} ( 0 ) =\sigma _{2} ( 0 ) =1$$, and $$\omega _{1}=\omega _{2}=0$$ becomes Theorem 2.1 in [40].

### Remark 3.4

Theorem 3.1 with $$l_{1}=l_{2}=1$$, $$\eta _{1}=\eta _{2}=1$$, $$\sigma _{1} ( 0 ) = \sigma _{2} ( 0 ) =1$$, $$\omega _{1}=\omega _{2}=0$$, and $$h_{1} (\xi _{1} ) =h_{2} (\xi _{1} )= h(\xi _{1})$$ becomes Theorem 7 in [35].

### Theorem 3.2

Let $$f:\Delta \to \mathbb{R}$$ be an integrable and $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates on Δ. Then we have

\begin{aligned} &f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\ &\quad \leq \frac{h_{1} ( \frac{1}{2} ) ( \Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}f_{g} ( \chi _{2}^{l_{1}},\frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ) +\Im _{\rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{\sigma _{1}}f_{{g}} ( \chi _{1}^{l_{1}},\frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ) ) }{2 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} )^{\rho _{1}} ) } \\ &\qquad {}+ \frac{h_{2} ( \frac{1}{2} ) ( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}f_{g} ( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2},\chi _{4}^{l_{2}} ) +\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}f_{{g}} ( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2},\chi _{3}^{l_{2}} ) ) }{2 ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\ &\quad \leq h_{1} \biggl( \frac{1}{2} \biggr) \frac{f ( \chi _{1}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}}} ) +f ( \chi _{2}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}} } ) }{2\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) } \\ &\qquad {}\times \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho 1} \bigr] \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \,d\xi _{1} \\ &\qquad {}+h_{2} \biggl( \frac{1}{2} \biggr) \frac{f ( [ \frac{ \chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{3} ) +f ( [ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{4} ) }{ 2\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\ &\qquad {}\times \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}, \end{aligned}

where $$f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) )$$ with $$g_{1} ( x ) =x^{\frac{1}{l_{1}}}$$ and $$g_{2} ( y ) =y^{\frac{1}{l_{2}}}$$.

### Proof

Since f is an $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates on Δ, the partial mapping $$f_{x}: [ \chi _{3},\chi _{4} ] \to \mathbb{R}$$ defined by $$f_{x} ( v ) =f ( x,v )$$ is $$(l_{2},h_{2} )$$-convex with respect to v on $$[ \chi _{3},\chi _{4} ]$$, and $$f_{y}: [ \chi _{1},\chi _{2} ] \to \mathbb{R}$$ defined by $$f_{y} ( u ) =f ( u,y )$$ is $$(l_{1},h_{1} )$$-convex with respect to u on $$[ \chi _{1},\chi _{2} ]$$. So, we have

\begin{aligned}& f_{x} \biggl( \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \\& \quad =f_{x} \biggl( \biggl[ \frac{ ( (\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} ) ^{\frac{1}{l_{2}}} ) ^{l_{2}}+ ( ( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2}\chi _{4}^{l_{2}} ) ^{\frac{1}{l_{2}}} ) ^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \\& \quad \leq h_{2} \biggl( \frac{1}{2} \biggr) \bigl( f_{x} \bigl( \bigl( \xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) +f_{x} \bigl( \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{ l_{2}}} \bigr) \bigr) \\& \quad \leq h_{2} \biggl( \frac{1}{2} \biggr) \bigl( h_{2} (\xi _{2} ) f_{x} ( \chi _{3} ) +h_{2} ( 1-\xi _{2} ) f_{x} ( \chi _{4} ) + h_{2} ( 1-\xi _{2} ) f_{x} ( \chi _{3} ) +h_{2} (\xi _{2} ) f_{x} ( \chi _{4} ) \bigr) \\& \quad =h_{2} \biggl( \frac{1}{2} \biggr) \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1-\xi _{2} ) \bigr) \bigl( f_{x} ( \chi _{3} ) +f_{x} ( \chi _{4} ) \bigr). \end{aligned}
(3.10)

From (3.10), we get

\begin{aligned} \frac{1}{h_{2} ( \frac{1}{2} ) }f_{x} \biggl( \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \leq& f_{x} \bigl( \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) +f_{x} \bigl( \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \\ \leq& \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \bigl( f_{x} ( \chi _{3} ) +f_{x} ( \chi _{4} ) \bigr). \end{aligned}
(3.11)

Multiplying (3.11) by $$\xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}}\xi _{2}^{\rho _{2}} ]$$ and then integrating the resulting inequalities with respect to $$\xi _{2}$$ on $$[ 0,1 ]$$, we obtain

\begin{aligned}& \frac{1}{h_{2} ( \frac{1}{2} ) }f_{x} \biggl( \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2}, \eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \,d\xi _{2} \\& \quad =\frac{1}{h_{2} ( \frac{1}{2} ) }f_{x} \biggl( \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl( \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{ \rho _{2}} \bigr) \\& \quad \leq \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] f_{x} \bigl( \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}} } \bigr) \,d\xi _{2} \\& \qquad {}+ \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] f_{x} \bigl( \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \,d\xi _{2} \\& \quad = \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{4}^{l_{2}}-w \bigr) ^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-w \bigr) ^{\rho _{2}} \bigr] f_{x} ( w ) \,dw \\& \qquad {}+ \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( w-\chi _{3}^{l_{2}} \bigr) ^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( w-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}} \bigr] f_{x} ( w ) \,dw \\& \quad = \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}} \bigl( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+}, \omega _{2}}^{\sigma _{2}}f_{x} \bigl( \chi _{4}^{l_{2}} \bigr) + \Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega _{2}}^{\sigma _{2}}f_{x} \bigl( \chi _{3}^{l_{2}} \bigr) \bigr) \\& \quad \leq \bigl( f_{x} ( \chi _{3} ) +f_{x} ( \chi _{4} ) \bigr) \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2}, \eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1-\xi _{2} ) \bigr) \,d\xi _{2}. \end{aligned}
(3.12)

This implies that

\begin{aligned}& \frac{1}{h_{2} ( \frac{1}{2} ) }f \biggl( x, \biggl[ \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{\frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{1}{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {}\times \bigl( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}f_{g} \bigl( x^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega _{2}}^{\sigma _{2}}f_{{g}} \bigl( x^{l_{1}}, \chi _{3}^{l_{2}} \bigr) \bigr) \\& \quad \leq \frac{f ( x,\chi _{3} ) +f ( x,\chi _{4} ) }{\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\& \qquad {}\times \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1-\xi _{2} ) \bigr) \,d\xi _{2}. \end{aligned}
(3.13)

Put $$x= [ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{ \frac{1}{l_{1}}}$$ into (3.13) to get

\begin{aligned}& f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{h_{2} ( \frac{1}{2} ) }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {}\times \biggl( \Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}f_{g} \biggl( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ,\chi _{4}^{l_{2}} \biggr) + \Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-}, \omega _{2}}^{\sigma _{2}}f_{{g}} \biggl( \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ,\chi _{3}^{l_{2}} \biggr) \biggr) \\& \quad \leq h_{2} \biggl( \frac{1}{2} \biggr) \frac{f ( [ \frac{ \chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{3} ) +f ( [ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} ] ^{\frac{1}{l_{1}}},\chi _{4} ) }{ \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} ( \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ) } \\& \qquad {}\times \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \bigl( h_{2} (\xi _{2} ) +h_{2} ( 1- \xi _{2} ) \bigr) \,d\xi _{2}. \end{aligned}
(3.14)

Similarly, we can deduce

\begin{aligned}& f \biggl( \biggl[ \frac{\chi _{1}^{l_{1}}+\chi _{2}^{l_{1}}}{2} \biggr] ^{\frac{1}{l_{1}}}, \biggl[ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr] ^{ \frac{1}{l_{2}}} \biggr) \\& \quad \leq \frac{h_{1} ( \frac{1}{2} ) }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) } \\& \qquad {}\times \biggl( \Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}f_{g} \biggl( \chi _{2}^{l_{1}}, \frac{ \chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr) + \Im _{\rho _{1}, \eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{ \sigma _{1}}f_{{g}} \biggl( \chi _{1}^{l_{1}}, \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} \biggr) \biggr) \\& \quad \leq h_{1} \biggl( \frac{1}{2} \biggr) \frac{f ( \chi _{1}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}}} ) +f ( \chi _{2}, [ \frac{\chi _{3}^{l_{2}}+\chi _{4}^{l_{2}}}{2} ] ^{\frac{1}{l_{2}} } ) }{\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} ( \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ) } \\& \qquad {}\times \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho 1} \bigr] \bigl( h_{1} (\xi _{1} ) +h_{1} ( 1- \xi _{1} ) \bigr) \,d\xi _{1}. \end{aligned}
(3.15)

By adding (3.14) and (3.15) together, and then multiplying the result by $$\frac{1}{2}$$, we get the desired result. Thus we get the proof of Theorem 3.2. □

### Remark 3.5

Theorem 3.2 with $$l_{1}=l_{2}=1$$ and $$h_{1} (\xi _{1} ) =h_{2} (\xi _{1} ) =\xi _{1}$$ becomes Theorem 2.2 in [39].

### Lemma 3.1

Let $$f:\Delta \to \mathbb{R}$$ be a partial differentiable function on Δ. If $$\frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}}\in L ( \Delta )$$, then we have

\begin{aligned}& \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}} \chi _{4}^{1-l_{2}}}-A \\& \qquad {}+ \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times \bigl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\& \qquad {}+ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \bigr) \\& \quad =\frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times \biggl( \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1} \chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl( ( 1- \xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1- \xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {}+ \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}} } \bigr) \,d\xi _{1}\,d\xi _{2} \biggr) , \end{aligned}

where

$$\mathcal{B} (\xi _{1},\xi _{2} ) =\xi _{1}^{\eta _{1}}\xi _{2}^{ \eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{ \rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr]$$
(3.16)

and

\begin{aligned} A =& \frac{1}{4 ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \biggl( \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}} \\ &{}+ \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}} + \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}} \\ &{}+ \frac{\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}} \biggr) + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] } \\ &{}\times \biggl( \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{\sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{3}^{1-l_{2}}}+ \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{\sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{4}^{1-l_{2}}} \\ &{}+ \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{3}^{l_{2}-1}}+ \frac{\Im _{\rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{\sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{4}^{1-l_{2}}} \biggr), \end{aligned}
(3.17)

and $$f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) )$$ with $$g_{1} ( x ) =x^{\frac{1}{l_{1}}}$$ and $$g_{2} ( y ) =y^{\frac{1}{l_{2}}}$$.

### Proof

Set

$$\hbar :=\hbar _{1}-\hbar _{2}- \hbar _{3}+\hbar _{4},$$
(3.18)

where

\begin{aligned}& \begin{aligned} \hbar _{1}:={} & \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}} + ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}; \end{aligned} \\& \begin{aligned} \hbar _{2}:={}& \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2} \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}; \end{aligned} \\& \begin{aligned} \hbar _{3}:={}& \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}; \end{aligned} \\& \begin{aligned} \hbar _{4}:={}& \int^{1}_{0} \int^{1}_{0} \xi _{1}^{\eta _{1}} \xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+\xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{1}\,d\xi _{2}. \end{aligned} \end{aligned}

Integrating by parts $$\hbar _{1}$$, we have

\begin{aligned} \hbar _{1}={}& \int^{1}_{0}\xi _{2}^{\eta _{2}} \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \biggl( \int^{1}_{0}\xi _{1}^{\eta _{1}} \mathfrak{F} _{\rho _{1}, \eta _{1}+1}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \\ &{}\times \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{1} \biggr) \,d\xi _{2} \\ ={}& \frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [\omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ (\chi _{1}^{l_{1}} -\chi _{2}^{l_{1}} ) \chi _{1}^{1-l_{1}}} \int^{1}_{0}\xi _{2}^{\eta _{2}} \mathfrak{F} _{ \rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[\omega _{2} \bigl( \chi _{4}^{l_{2}} -\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times\frac{\partial f}{\partial \xi _{2}} \bigl( \chi _{1}, \bigl(\xi _{2} \chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{2} \\ &{}- \int^{1}_{0} \frac{l_{1}\xi _{1}^{\eta _{1}-1}\mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}}\xi _{1}^{\rho _{1}} ] }{ ( \chi _{1}^{l_{1}}-\chi _{2}^{l_{1}} ) (\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} ) ^{\frac{1}{l_{1}}-1}} \biggl( \int^{1}_{0} \xi _{2}^{\eta _{2}} \mathfrak{F} _{ \rho _{2},\eta _{2}+1}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \frac{\partial f}{\partial \xi _{2}} \bigl( \bigl( \xi _{1}\chi _{1}^{l_{1}} + ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}}}, \bigl(\xi _{2} \chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \,d\xi _{2} \biggr) \,d\xi _{1} \\ ={}& \frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{3} ) }{\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) \chi _{1}^{1-l_{1}}} \int^{1}_{0}\xi _{2}^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{ \rho _{2}} \bigr] \\ &{}\times \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{1-\frac{1}{ l_{2}}}f \bigl( \chi _{1}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \,d\xi _{2} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) \chi _{3}^{1-l_{2}}} \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{ \rho _{1}} \bigr] \\ &{}\times \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1- \xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{1-\frac{1}{ l_{1}}}f \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1 }{l_{1}}},\chi _{3} \bigr) \,d\xi _{1} \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \int^{1}_{0} \int^{1}_{0}\xi _{1}^{\eta _{1}-1} \mathfrak{F} _{\rho _{1}, \eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}- \chi _{1}^{l_{1}} \bigr) ^{\rho _{1}}\xi _{1}^{\rho _{1}} \bigr] \\ &{}\times \xi _{2}^{\eta _{2}-1}\mathfrak{F} _{\rho _{2},\eta _{2}}^{ \sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}- \chi _{3}^{l_{2}} \bigr) ^{\rho _{2}}\xi _{2}^{\rho _{2}} \bigr] \\ &{}\times \bigl(\xi _{1} \chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{1-\frac{1 }{l_{1}}}\bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{1-\frac{1}{ l_{2}}} \\ &{}\times f \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1 }{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{ l_{2}}} \bigr) \,d\xi _{2}\,d\xi _{1}. \end{aligned}

By the change of variables, we get

\begin{aligned} \hbar _{1} ={}&\frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{3} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}\chi _{1}^{1-l_{1}}} \\ &{}\times \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1}\mathfrak{F} _{ \rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\rho _{2}} \bigr] y^{1-\frac{1}{l_{2}}}f_{g} \bigl( \chi _{1}^{l_{1}},y \bigr) \,dy \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) \chi _{3}^{1-l_{2}}} \\ &{}\times\int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] x^{1- \frac{1}{l_{1}}}f_{g} \bigl( x,\chi _{3}^{l_{2}} \bigr) \,dx \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \int^{\chi _{2}^{l_{1}}}_{\chi _{1}^{l_{1}}} \int^{\chi _{4}^{l_{2}}}_{\chi _{3}^{l_{2}}} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\eta _{1}-1} \mathfrak{F} _{\rho _{1},\eta _{1}}^{\sigma _{1}} \bigl[ \omega _{1} \bigl( \chi _{2}^{l_{1}}-x \bigr) ^{\rho _{1}} \bigr] \\ &{}\times \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\eta _{2}-1} \mathfrak{F} _{\rho _{2},\eta _{2}}^{\sigma _{2}} \bigl[ \omega _{2} \bigl( \chi _{4}^{l_{2}}-y \bigr) ^{\rho _{2}} \bigr] x^{1- \frac{1}{l_{1}}}y^{1-\frac{1}{l_{2}}}f_{g} ( x,y ) \,dy\,dx. \end{aligned}
(3.19)

Making use of Definition 2.3 in (3.19), we get

\begin{aligned} \hbar _{1} ={}&\frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{3} ) }{\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{1}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{3}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1}, ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{ \sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr). \end{aligned}
(3.20)

Likewise, we can deduce

\begin{aligned}& \begin{aligned}[b] \hbar _{2} ={}&{-} \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{1},\chi _{4} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{1}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{4}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1},a ( \chi _{1}^{l_{1}} ) ^{+},\omega _{1}}^{ \sigma _{1}}\chi _{2}^{l_{1}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{2}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr), \end{aligned} \end{aligned}
(3.21)
\begin{aligned}& \begin{aligned}[b] \hbar _{3}={}&{-} \frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{2},\chi _{3} ) }{ \chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{2}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{3}^{l_{2}} ) ^{+},\omega _{2}}^{\sigma _{2}}\chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{3}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{ \sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{4}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr), \end{aligned} \end{aligned}
(3.22)

and finally

\begin{aligned} \hbar _{4} ={}&\frac{l_{1}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) } \frac{l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) } \frac{f ( \chi _{2},\chi _{4} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}}\chi _{2}^{l_{1}-1}\Im _{\rho _{2},\eta _{2}, ( \chi _{4}^{l_{2}} ) ^{-},\omega _{2}}^{\sigma _{2}}\chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} \bigr) \\ &{}- \frac{l_{1}l_{2}\mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] }{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }\chi _{4}^{l_{2}-1}\Im _{ \rho _{1},\eta _{1}, ( \chi _{2}^{l_{1}} ) ^{-},\omega _{1}}^{ \sigma _{1}}\chi _{1}^{l_{1}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} \bigr) \\ &{}+ \frac{l_{1}l_{2}}{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}+1} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}+1}} \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma }\chi _{1}^{l_{1}-1} \chi _{3}^{l_{2}-1}f_{g} \bigl( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} \bigr). \end{aligned}
(3.23)

Making use of (3.20)–(3.23) in (3.18) and then multiplying by

$$\frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] },$$

we arrive at the desired result. Thus we get the proof of Lemma 3.1. □

### Theorem 3.3

Let $$f:\Delta \to \mathbb{R}$$ be a partial differentiable function on Δ. If $$\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert$$ is an $$(l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates on Δ, then we have

\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \bigl( h_{1} ( 1-\xi _{1} ) +h_{1} (\xi _{1} ) \bigr) \bigl( h_{2} ( 1- \xi _{2} ) +h_{2} (\xi _{2} ) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr) \\& \qquad {} \times \biggl( \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{3} ) \biggr\vert + \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{4} ) \biggr\vert \biggr) , \end{aligned}

where $$\mathcal{B} (\xi _{1},\xi _{2} )$$ and A are as in (3.16) and (3.17), respectively, and $$f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) )$$ with $$g_{1} ( x ) =x^{\frac{1}{l_{1}}}$$ and $$g_{2} ( y ) =y^{\frac{1}{l_{2}}}$$.

### Proof

By Lemma 3.1 and the properties of modulus, we have

\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \biggr) . \end{aligned}

Using the $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convexity of $$\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert$$ on coordinates, we obtain

\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}}-A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \biggl( h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} ( \xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert + h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl( h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert + h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl( h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert + h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl( h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert \\& \qquad {} +h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert + h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert \\& \qquad {}+h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert \biggr) \,d\xi _{1}\,d\xi _{2} \biggr) \\& \quad = \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \bigl( h_{1} ( 1-\xi _{1} ) +h_{1} (\xi _{1} ) \bigr) \bigl( \bigl( h_{2} ( 1-\xi _{2} ) +h_{2} (\xi _{2} ) \bigr) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr) \\& \qquad {} \times \biggl( \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{3} ) \biggr\vert + \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{4} ) \biggr\vert \biggr). \end{aligned}

This completely ends the proof of Theorem 3.3. □

### Corollary 3.1

Theorem 3.3with $$\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert \leq K$$ gives the new inequality:

\begin{aligned} & \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{3} ) }{4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}}-A \\ &\qquad {}+ \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} (\chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} )^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} (\chi _{1}^{l_{1}},\chi _{4}^{l_{2}} )}{\chi _{1}^{1-l_{1}} \chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{ \sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\ &\qquad {}+ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, (\chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} (\chi _{2}^{l_{1}}, \chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\ &\quad \leq \frac{K ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) (\chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} )^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}} -\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \bigl( h_{1} ( 1-\xi _{1} )+h_{1} (\xi _{1} ) \bigr) \bigl( h_{2} ( 1- \xi _{2} )+h_{2} ( \xi _{2} ) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr). \end{aligned}

### Remark 3.6

Theorem 3.3 with $$l_{1}=l_{2}=1$$ and $$h_{1} (\xi _{1} ) =h_{2} (\xi _{1} ) =\xi _{1}$$ becomes Theorem 3.2 in [39].

### Remark 3.7

Theorem 3.3 with $$l_{1}=l_{2}=1$$, $$\eta _{1}=\eta _{2}=\alpha$$, $$\sigma _{1} ( 0 ) =\sigma _{2} ( 0 ) =1$$, $$\omega _{1}=\omega _{2}=0$$, and $$h_{1} (\xi _{1} ) =h_{2} (\xi _{1} )=\xi _{1}$$ becomes Theorem 3 in [33].

### Theorem 3.4

Let $$f:\Delta \to \mathbb{R}$$ be a partial differentiable function on Δ. If $$\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert ^{q}$$ is an $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex function on coordinates on Δ, then for $$q>1$$ and $$\frac{1}{p}+\frac{1}{q}=1$$, we have

\begin{aligned} & \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\ &\qquad {}+ \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\ &\qquad {}+ \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\ &\quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\ &\qquad {}\times \biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}} \biggl[ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert ^{q}+ h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {} +h_{1} ( 1- \xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\ &\qquad {}+ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q} \\ &\qquad {}+ h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} ( 1- \xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\ &\qquad {}+ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q}+ h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {} +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\ &\qquad {}+ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\ &\qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q}+ h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\ &\qquad {} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \biggr], \end{aligned}

where $$\mathcal{B} (\xi _{1},\xi _{2} )$$ and A are defined as in (3.16) and (3.17), respectively, and $$f_{g} ( x,y ) =f ( g_{1} ( x ) ,g_{2} ( y ) )$$ with $$g_{1} ( x ) =x^{\frac{1}{l_{1}}}$$ and $$g_{2} ( y ) =y^{\frac{1}{l_{2}}}$$.

### Proof

Making use of Lemma 3.1 and the properties of modulus, we get

\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} -A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0}\mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B } (\xi _{1},\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+ \xi _{1}\chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \\& \qquad \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \\& \qquad {} + \int^{1}_{0} \int^{1}_{0} \mathcal{B} (\xi _{1}, \xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1} \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \\& \qquad \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \biggr\vert \,d\xi _{1}\,d\xi _{2} \biggr) . \end{aligned}

Making use of the $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convexity of $$\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert ^{q}$$ on coordinates and Hölder’s inequality, we obtain

\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} + \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}}-A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times\biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}} \\& \qquad {} \times \biggl( \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1-\xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}}} \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl(\xi _{1}\chi _{1}^{l_{1}}+ ( 1-\xi _{1} ) \chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1}\chi _{2}^{l_{1}} \bigr) ^{\frac{1}{l_{1}} }, \bigl(\xi _{2}\chi _{3}^{l_{2}}+ ( 1- \xi _{2} ) \chi _{4}^{l_{2}} \bigr) ^{\frac{1}{l_{2}} } \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \bigl( \bigl( ( 1-\xi _{1} ) \chi _{1}^{l_{1}}+\xi _{1}\chi _{2}^{l_{1}} \bigr) ^{ \frac{1}{l_{1}}}, \bigl( ( 1-\xi _{2} ) \chi _{3}^{l_{2}}+ \xi _{2}\chi _{4}^{l_{2}} \bigr) ^{ \frac{1}{l_{2}}} \bigr) \biggr\vert ^{q} \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \biggr) \\& \quad \leq \frac{ ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{4l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times\biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}}\biggl[ \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\& \qquad {} +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{4} ) \biggr\vert ^{q}+ h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\& \qquad {} +h_{1} ( 1- \xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q} \\& \qquad {} + h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\& \qquad {}+h_{1} ( 1- \xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\& \qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q} \\& \qquad {} + h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} +h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f }{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{ \frac{1}{q}} \\& \qquad {} + \biggl\{ \int^{1}_{0} \int^{1}_{0} \biggl( h_{1} ( 1-\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1},\chi _{3} ) \biggr\vert ^{q} \\& \qquad {}+h_{1} ( 1-\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{ \partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{1}, \chi _{4} ) \biggr\vert ^{q}+ h_{1} (\xi _{1} ) h_{2} ( 1-\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} ( \chi _{2}, \chi _{3} ) \biggr\vert ^{q} \\& \qquad {} +h_{1} (\xi _{1} ) h_{2} (\xi _{2} ) \biggl\vert \frac{\partial ^{2}f}{ \partial \xi _{1}\partial \xi _{2}} ( \chi _{2},\chi _{4} ) \biggr\vert ^{q} \biggr) \,d\xi _{1}\,d\xi _{2} \biggr\} ^{\frac{1}{q}} \biggr]. \end{aligned}

This rearranges to the proof of Theorem 3.4 □

### Corollary 3.2

Theorem 3.4with $$\vert \frac{\partial ^{2}f}{\partial \xi _{1}\partial \xi _{2}} \vert ^{q}\leq K$$, give the new inequality:

\begin{aligned}& \biggl\vert \frac{f ( \chi _{1},\chi _{3} ) }{4\chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{ f ( \chi _{1},\chi _{4} ) }{4\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{3} ) }{ 4\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}}+ \frac{f ( \chi _{2},\chi _{4} ) }{4\chi _{2}^{1-l_{1}} \chi _{4}^{1-l_{2}}}-A \\& \qquad {} + \frac{1}{4 ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\eta _{1}} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\eta _{2}}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {} \times \biggl( \Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{4}^{l_{2}} ) }{\chi _{1}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{2}^{l_{1}} ) ^{-}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{1}^{l_{1}},\chi _{3}^{l_{2}} ) }{ \chi _{1}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) \\& \qquad {} + \Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{4}^{l_{2}} ) ^{-},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{3}^{l_{2}} ) }{\chi _{2}^{1-l_{1}}\chi _{3}^{1-l_{2}}} \biggr) +\Im _{\rho ,\eta , ( \chi _{1}^{l_{1}} ) ^{+}, ( \chi _{3}^{l_{2}} ) ^{+},\omega }^{\sigma } \biggl( \frac{f_{g} ( \chi _{2}^{l_{1}},\chi _{4}^{l_{2}} ) }{ \chi _{2}^{1-l_{1}}\chi _{4}^{1-l_{2}}} \biggr) \biggr) \biggr\vert \\& \quad \leq \frac{K ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) }{l_{1}l_{2}\mathfrak{F} _{\rho _{1},\eta _{1}+1}^{\sigma _{1}} [ \omega _{1} ( \chi _{2}^{l_{1}}-\chi _{1}^{l_{1}} ) ^{\rho _{1}} ] \mathfrak{F} _{\rho _{2},\eta _{2}+1}^{\sigma _{2}} [ \omega _{2} ( \chi _{4}^{l_{2}}-\chi _{3}^{l_{2}} ) ^{\rho _{2}} ] } \\& \qquad {}\times\biggl( \int^{1}_{0} \int^{1}_{0} \bigl[\mathcal{B} (\xi _{1},\xi _{2} ) \bigr]^{p} \,d\xi _{1}\,d\xi _{2} \biggr)^{\frac{1}{p}} \\& \qquad {} \times \biggl( \int^{1}_{0} \int^{1}_{0} \bigl( h_{1} ( 1-\xi _{1} ) +h_{1} (\xi _{1} ) \bigr) \bigl( h_{2} ( 1-\xi _{2} ) +h_{2} (\xi _{2} ) \bigr) \,d\xi _{1}\,d\xi _{2} \biggr)^{ \frac{1}{q}}. \end{aligned}

## 4 Conclusion

Since convexity has wide applications in many mathematical areas, the general class of $$( l_{1},h_{1} )$$-$$( l_{2},h_{2} )$$-convex functions on coordinates can be applied to obtain several results in convex analysis, special functions, related optimization theory, mathematical inequalities and may stimulate further research in different areas of pure and applied sciences.

Not applicable.

## References

1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

2. Atangana, A.: Fractional Operators with Constant and Variable Order with Application to Geo-Hydrology. Academic Press, New York (2017)

3. Srivastava, H.M., Mohammed, P.O.: A correlation between solutions of uncertain fractional forward difference equations and their paths. Front. Phys. 8, 280 (2020). https://doi.org/10.3389/fphy.2020.00280

4. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)

5. Abdeljawad, T., Baleanu, D.: Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10(3), 1098–1107 (2017)

6. Mohammed, P.O., Abdeljawad, T., Jarad, F., Chu, Y.-M.: Existence and uniqueness of uncertain fractional backward difference equations of Riemann–Liouville type. Math. Probl. Eng. 2020, Article ID 6598682 (2020)

7. Alqudah, M.A., Mohammed, P.O., Abdeljawad, T.: Solution of singular integral equations via Riemann–Liouville fractional integrals. Math. Probl. Eng. 2020, Article ID 1250970 (2020)

8. Agarwal, P., Choi, J., Paris, R.B.: Extended Riemann–Liouville fractional derivative operator and its applications. J. Nonlinear Sci. Appl. 8(5), 451–466 (2015)

9. Agarwal, P., Nieto, J.J., Luo, M.J.: Extended Riemann–Liouville type fractional derivative operator with applications. Open Math. 15(1), 1667–1681 (2017)

10. Yasemin, B., Baleanu, D.: Ostrowski type inequalities involving psi-Hilfer fractional integrals. Mathematics 7, 770 (2019)

11. Yasemin, B., Baleanu, D.: New aspects of Opial-type integral inequalities. Adv. Differ. Equ. 2018, 452 (2018)

12. Liu, X., Zhang, L., Agarwal, P., Wang, G.: On some new integral inequalities of Gronwall–Bellman–Bihari type with delay for discontinuous functions and their applications. Indag. Math. 27(1), 1–10 (2016)

13. Mehrez, K., Agarwal, P.: New Hermite–Hadamard type integral inequalities for convex functions and their applications. J. Comput. Appl. Math. 350, 274–285 (2019)

14. Agarwal, P., Tariboon, J., Ntouyas, S.K.: Some generalized Riemann–Liouville k-fractional integral inequalities. J. Inequal. Appl. 2016, 122 (2016)

15. Denton, Z., Vatsala, A.S.: Fractional integral inequalities and applications. Comput. Math. Appl. 59(3), 1087–1094 (2010)

16. Anastassiou, G.A.: Opial type inequalities involving Riemann–Liouville fractional derivatives of two functions with applications. Math. Comput. Model. 48, 344–374 (2008)

17. Fernandez, A., Mohammed, P.: Hermite–Hadamard inequalities in fractional calculus defined using Mittag-Leffler kernels. Math. Methods Appl. Sci., 1–18 (2020). https://doi.org/10.1002/mma.6188

18. Mohammed, P.O.: New integral inequalities for preinvex functions via generalized beta function. J. Interdiscip. Math. 22(4), 539–549 (2019)

19. Mohammed, P.O., Brevik, I.: A new version of the Hermite–Hadamard inequality for Riemann–Liouville fractional integrals. Symmetry 12, 610 (2020). https://doi.org/10.3390/sym12040610

20. Mohammed, P.O., Abdeljawad, T., Kashuri, A.: Fractional Hermite–Hadamard–Fejér inequalities for a convex function with respect to an increasing function involving a positive weighted symmetric function. Symmetry 12, 1503 (2020)

21. Mohammed, P.O., Abdeljawad, T., Zeng, S., Kashuri, A.: Fractional Hermite–Hadamard integral inequalities for a new class of convex functions. Symmetry 12, 1485 (2020)

22. Mohammed, P.O., Sarikaya, M.Z., Baleanu, D.: On the generalized Hermite–Hadamard inequalities via the tempered fractional integrals. Symmetry 12, 595 (2020). https://doi.org/10.3390/sym12040595

23. Mohammed, P.O.: Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals of a convex function with respect to a monotone function. Math. Methods Appl. Sci., 1–11 (2019). https://doi.org/10.1002/mma.5784

24. Mohammed, P.O., Abdeljawad, T.: Modification of certain fractional integral inequalities for convex functions. Adv. Differ. Equ. 2020, 69 (2020)

25. Han, J., Mohammed, P.O., Zeng, H.: Generalized fractional integral inequalities of Hermite–Hadamard-type for a convex function. Open Math. 18(1), 794–806 (2020)

26. Mohammed, P.O., Sarikaya, M.Z.: Hermite–Hadamard type inequalities for F-convex function involving fractional integrals. J. Inequal. Appl. 2018, 359 (2018)

27. Mohammed, P.O., Sarikaya, M.Z.: On generalized fractional integral inequalities for twice differentiable convex functions. J. Comput. Appl. Math. 372, 112740 (2020)

28. Abdeljawad, T., Ali, M.A., Mohammed, P.O., Kashuri, A.: On inequalities of Hermite–Hadamard–Mercer type involving Riemann-Liouville fractional integrals. AIMS Math. 5, 7316–7331 (2020)

29. Abdeljawad, T., Mohammed, P.O., Kashuri, A.: New modified conformable fractional integral inequalities of Hermite–Hadamard type with applications. J. Funct. Spaces 2020, Article ID 4352357 (2020)

30. Raina, R.K.: On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Math. J. 21(2), 191–203 (2005)

31. Tomar, M., Agarwal, P., Jleli, M., Samet, B.: Certain Ostrowski type inequalities for generalized s-convex functions. J. Nonlinear Sci. Appl. 10, 5947–5957 (2017)

32. Dragomir, S.S.: On the Hadamard’s inequality for convex functions on the coordinates in a rectangle from the plane. Taiwan. J. Math. 5, 775–788 (2001)

33. Sarikaya, M.Z.: On the Hermite–Hadamard-type inequalities for co-ordinated convex function via fractional integrals. Integral Transforms Spec. Funct. 25(2), 134–147 (2014)

34. Noor, M.A., Noor, K.I., Iftikhar, S., Rashid, S., Awan, M.U.: Coordinated convex functions and inequalities. Appl. Math. E-Notes 19, 189–198 (2019)

35. Latif, M.A., Alomari, M.: On Hadamard-type inequalities for h-convex functions on the co-ordinates. Int. J. Math. Anal. 3(33), 1645–1656 (2009)

36. Cao, H.: A new Hermite–Hadamard type inequality for coordinate convex function. J. Inequal. Appl. 2020, 162 (2020)

37. Raees, M., Anwar, M.: On Hermite–Hadamard type inequalities of coordinated $$(p_{1},h_{1})$$-$$(p_{2},h_{2})$$-convex functions via Katugampola fractional integrals. Filomat 33(15), 4785–4802 (2019)

38. Yaldız, H., Sarikaya, M.Z., Dahmani, Z.: On the Hermite–Hadamard–Féjer-type inequalities for co-ordinated convex functions via fractional integrals. Int. J. Optim. Control Theor. Appl. 7(2), 205–215 (2017)

39. Tunç, T., Sarikaya, M.Z.: On Hermite–Hadamard type inequalities via fractional integral operators. Filomat 33(3), 837–854 (2019)

40. Yang, W.: Hermite–Hadamard type inequalities for $$(p_{1},h_{1})$$-$$(p_{2},h_{2})$$-convex functions on the coordinates. Tamkang J. Math. 47(3), 289–322 (2016)

41. Akkurt, A., Sarıkaya, M.Z., Budak, H., Yıldırım, H.: On the Hadamard’s type inequalities for co-ordinated convex functions via fractional integrals. J. King Saud Univ., Sci. 29, 380–387 (2017)

42. Mohammed, P.O.: Some new Hermite–Hadamard type inequalities for MT-convex functions on differentiable coordinates. J. King Saud Univ., Sci. 30, 258–262 (2018)

43. Kara, H., Budak, H., Kiris, M.E.: On Fejer type inequalities for co-ordinated hyperbolic ρ-convex functions. AIMS Math. 5(5), 4681–4701 (2020)

44. Noor, M.A., Awan, M.U., Noor, K.I.: Integral inequalities for two-dimensional pq-convex functions. Filomat 30(2), 343–351 (2016)

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Baleanu, D., Kashuri, A., Mohammed, P.O. et al. General Raina fractional integral inequalities on coordinates of convex functions. Adv Differ Equ 2021, 82 (2021). https://doi.org/10.1186/s13662-021-03241-y