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 [1–3] 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. [4–9]. 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. [10–29].

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. [34–38].

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. [41–44]. 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.