# Hermite–Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications

## Abstract

In the article, we present several Hermite–Hadamard type inequalities for the co-ordinated convex and quasi-convex functions and give an application to the product of the moment of two continuous and independent random variables. Our results are generalizations of some earlier results. Additionally, an illustrative example on the probability distribution is given to support our results.

## Introduction

Let $$\mathcal{I}\subseteq \mathbb{R}$$ be a nonempty interval. Then a real-valued function $$\varPhi :\mathcal{I}\rightarrow \mathbb{R}$$ is said to be convex (concave) if the inequality

$$\varPhi \bigl[\varrho \mu + (1-\varrho )\nu \bigr]\leq ( \geq )\ \varrho \varPhi (\mu )+ (1-\varrho ) \varPhi (\nu )$$

holds for all $$\mu ,\nu \in \mathcal{I}$$ and $$\varrho \in [0, 1 ]$$.

It is a fact that the convex (concave) function is one of the most basic and important functions in the theory of geometric function, it has widely applications in pure and applied mathematics, physics, mechanics, statistics and economics, and meteorology . Recently, the generalizations, extensions, variants and refinements for the convexity (concavity) have attracted the interest of several researchers . In particular, many remarkable inequalities and properties in many branches of mathematics can be found in the literature  using the convexity (concavity) theory. The well-known Hermite–Hadamard inequality states that the double inequality

$$\varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2} \biggr)\leq (\geq )\ \frac{1}{ \lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}}\varPhi (\mu )\,dx \leq (\geq )\ \frac{\varPhi (\lambda _{1} )+\varPhi (\lambda _{2} )}{2}$$
(1.1)

holds for all $$\lambda _{1}, \lambda _{2}\in \mathcal{I}$$ with $$\lambda _{1}\neq \lambda _{2}$$ if $$\varPhi :\mathcal{I}\rightarrow \mathbb{R}$$ is a convex (concave) function.

In the past hundred years, inequality (1.1) has inspired many researchers to estimate the bounds for

$$\biggl\vert \frac{\varPhi (\lambda _{1} )+\varPhi (\lambda _{2} )}{2}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}} ^{\lambda _{2}}\varPhi (\mu )\,d\mu \biggr\vert$$

and

$$\biggl\vert \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2} \biggr)-\frac{1}{ \lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}}\varPhi (\mu ) \,d\mu \biggr\vert ,$$

and all the obtained results are called Hermite–Hadamard type inequalities.

It is well known that the multivariable functions also have the concept of convexity (concavity). An an example, we recall the definition of convexity (concavity) for the bivariate functions.

Let $$\lambda _{1}, \lambda _{2}, \xi _{1}, \xi _{2}\in \mathbb{R}$$ such that $$\lambda _{1}<\lambda _{2}$$ and $$\xi _{1}<\xi _{2}$$. Then a bivariate real-valued function $$\varPhi :[\lambda _{1},\lambda _{2}]\times [\xi _{1}, \xi _{2}]\rightarrow \mathbb{R}$$ is said to be convex (concave) if the inequality

$$\varPhi \bigl(\varrho \mu +(1-\varrho )\rho ,\varrho \nu +(1-\varrho )\omega \bigr) \leq (\geq )\ \varrho \varPhi (\mu ,\nu )+(1-\varrho )\varPhi (\rho ,\omega )$$

holds for all $$(\mu ,\nu ),(\rho ,\omega )\in [\lambda _{1},\lambda _{2}]\times [\xi _{1},\xi _{2}]$$ and $$\varrho \in [0,1]$$.

In order to establish the relation involving the convexity between the bivariate and univariate functions, Dragomir  introduced the definition of the bivariate co-ordinated convex function as follows.

### Definition 1.1

(See )

Let $$\lambda _{1}, \lambda _{2}, \xi _{1}, \xi _{2}\in \mathbb{R}$$ such that $$\lambda _{1}<\lambda _{2}$$ and $$\xi _{1}<\xi _{2}$$. Then a bivariate real-valued function $$\varPhi :[\lambda _{1},\lambda _{2}] \times [\xi _{1},\xi _{2}]\rightarrow \mathbb{R}$$ is said to be convex on the co-ordinates if both of the partial mappings $$\varPhi _{\nu }:[\lambda _{1},\lambda _{2}]\rightarrow \mathbb{R}$$ and $$\varPhi _{\mu }:[\xi _{1},\xi _{2}]\rightarrow \mathbb{R}$$ defined by

$$\varPhi _{\nu }(\delta )=\varPhi (\delta ,\nu )$$

and

$$\varPhi _{\mu }(\theta )=\varPhi (\mu ,\theta )$$

are convex for all $$\mu \in [\lambda _{1},\lambda _{2}]$$ and $$\nu \in [\xi _{1},\xi _{2}]$$.

Latif and Alomari  proved that the bivariate real-valued function $$\varPhi :[\lambda _{1},\lambda _{2}]\times [\xi _{1},\xi _{2}]\rightarrow \mathbb{R}$$ is convex on the co-ordinates if and only if

\begin{aligned}& \varPhi \bigl(\varrho \mu +(1-\varrho )\nu ,\varsigma \rho +(1-\varsigma ) \omega \bigr) \\& \quad \leq \varrho \varsigma \varPhi (\mu ,\rho )+\varrho (1-\varsigma )\varPhi ( \mu , \omega )+\varsigma (1-\varrho )\varPhi (\nu ,\rho )+(1-\varrho ) (1- \varsigma )\varPhi (\nu ,\omega ) \end{aligned}

for all $$\varrho ,\varsigma \in [0,1]\times {}[ 0,1]$$ and $$(\mu ,\rho ),(\nu ,\omega )\in [\lambda _{1},\lambda _{2}]\times {}[ \xi _{1},\xi _{2}]$$.

Dragomir  proved that every convex mapping $$\varPhi :[\lambda _{1}, \lambda _{2}]\times {}[ \xi _{1},\xi _{2}]\rightarrow \mathbb{R}$$ is convex on the co-ordinates and the reverse is not true, and established the Hermite–Hadamard type inequality for the co-ordinated convex function on the rectangle of the plane $${\mathbb{R}}^{2}$$.

### Theorem 1.2

(See )

Let $$\lambda _{1}, \lambda _{2}, \xi _{1}, \xi _{2}\in \mathbb{R}$$such that $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$, and $$\varPhi :[\lambda _{1},\lambda _{2}]\times [\xi _{1},\xi _{2}]\rightarrow \mathbb{R}$$be a co-ordinated convex function. Then one has

\begin{aligned}& \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \frac{\xi _{1}+\xi _{2}}{2} \biggr) \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}}\varPhi \biggl( \mu , \frac{\xi _{1}+\xi _{2}}{2} \biggr)\,d\mu +\frac{1}{\xi _{2}-\xi _{1}} \int _{\xi _{1}}^{\xi _{2}}\varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \nu \biggr) \,d\nu \biggr] \\& \quad \leq \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\mu ,\nu )\,d\nu \,d\mu \\& \quad \leq \frac{1}{4} \biggl[\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}}\varPhi ( \mu ,\xi _{1} ) \,d\mu +\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \varPhi (\mu ,\xi _{2} ) \,d\mu \\& \qquad {}+\frac{1}{\xi _{2}-\xi _{1}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\lambda _{1},\nu )\,d\nu +\frac{1}{\xi _{2}-\xi _{1}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\lambda _{2},\nu )\,d\nu \biggr] \\& \quad \leq \frac{\varPhi (\lambda _{1},\xi _{1} )+\varPhi (\lambda _{1},\xi _{2} )+\varPhi (\lambda _{2},\xi _{1} ) +\varPhi (\lambda _{2},\xi _{2} )}{4} \end{aligned}
(1.2)

with the best possible constant $$1/4$$.

In , Latif et al. derived the variants of the Hermite–Hadamard type inequality (1.2), which are the weighted generalizations of (1.2).

### Theorem 1.3

(See )

Let $$\triangle \subseteq {\mathbb{R}}^{2}$$, $$\triangle ^{ \circ }$$be the interior of , $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1}, \lambda _{2}] \times [\xi _{1}, \xi _{2}]\in \triangle ^{\circ }$$, $$\varPhi :\Delta \rightarrow \mathbb{R}$$be a twice differentiable mapping on $$\triangle ^{\circ }$$, $$p: [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous function symmetric with respect to $$(\lambda _{1}+\lambda _{2})/2$$and $$(\xi _{1}+\xi _{2})/2$$, $$\varPhi _{\varrho \varsigma }\in L ( [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{\varrho \varsigma } \vert$$be a co-ordinated convex function on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$. Then

\begin{aligned}& \biggl\vert \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}} ^{\lambda _{2}}p (\mu ,\nu )\,d\mu \,d\nu + \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}}^{\lambda _{2}}\varPhi (\mu ,\nu )p (\mu ,\nu )\,d\mu \,d\nu \\& \qquad {}- \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}}^{\lambda _{2}} \varPhi \biggl(\mu , \frac{\xi _{1}+\xi _{2}}{2} \biggr)p (\mu ,\nu )\,d\mu \,d\nu - \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}}^{\lambda _{2}} \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \nu \biggr)p (\mu , \nu )\,d\mu \,d\nu \biggr\vert \\& \quad \leq \frac{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )}{4} \bigl[ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{1} ) \bigr\vert + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert + \bigl\vert \varPhi _{ \varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert \bigr] \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl( \int _{\xi _{1}}^{L_{2} (\varsigma )} \int _{\lambda _{1}}^{L_{1} (\varrho )}p (\mu ,\nu ) \,d\mu \,d\nu \biggr)\,d\varrho \,d\varsigma , \end{aligned}
(1.3)

where $$L_{1} (\varrho )=\frac{1-\varrho }{2}\lambda _{1}+\frac{1+ \varrho }{2}\lambda _{2}$$and $$L_{2} (\varsigma )=\frac{1- \varsigma }{2}\xi _{1}+\frac{1+\varsigma }{2}\xi _{2}$$.

### Theorem 1.4

(See )

Let $$q>1$$, $$\triangle \subseteq {\mathbb{R}}^{2}$$, $$\triangle ^{\circ }$$be the interior of , $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1}, \lambda _{2}]\times [\xi _{1}, \xi _{2}]\in \triangle ^{\circ }$$, $$\varPhi :\Delta \rightarrow \mathbb{R}$$be a twice differentiable mapping on $$\triangle ^{\circ }$$, $$p: [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous function symmetric with respect to $$(\lambda _{1}+ \lambda _{2})/2$$and $$(\xi _{1}+\xi _{2})/2$$, $$\varPhi _{\varrho \varsigma } \in L ( [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{\varrho \varsigma } \vert ^{q}$$be a co-ordinated convex function on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$. Then we have

\begin{aligned}& \biggl\vert \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}} ^{\lambda _{2}}p (\mu ,\nu )\,d\mu \,d\nu + \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}}^{\lambda _{2}}\varPhi (\mu ,\nu )p (\mu ,\nu )\,d\mu \,d\nu \\& \qquad {}- \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}}^{\lambda _{2}} \varPhi \biggl(\mu , \frac{\xi _{1}+\xi _{2}}{2} \biggr)p (\mu ,\nu )\,d\mu \,d\nu - \int _{\xi _{1}}^{\xi _{2}} \int _{\lambda _{1}}^{\lambda _{2}} \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \nu \biggr)p (\mu , \nu )\,d\mu \,d\nu \biggr\vert \\& \quad \leq (\lambda _{2}-\lambda _{1} ) (\xi _{2}- \xi _{1} ) \biggl[\frac{ \vert \varPhi _{\varrho \varsigma } (\lambda _{1}, \xi _{1} ) \vert + \vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \vert + \vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \vert + \vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{2} ) \vert }{4} \biggr] ^{\frac{1}{q}} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl( \int _{\xi _{1}}^{L_{2} (\varsigma )} \int _{\lambda _{1}}^{L_{1} (\varrho )} p (\mu ,\nu ) \,d\mu \,d\nu \biggr)\,d\varrho \,d\varsigma , \end{aligned}
(1.4)

where $$L_{1} (\varrho )$$and $$L_{2} (\varsigma )$$are defined as in Theorem 1.3.

Özdemir et al.  generalized the co-ordinated convex function to the co-ordinated quasi-convex function.

### Definition 1.5

(See )

A real-valued function $$\varPhi : [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]\subset \mathbb{R} ^{2}\rightarrow \mathbb{R}$$ is said to be quasi-convex if the inequality

$$\varPhi \bigl(\varrho \mu +(1-\varrho )\rho ,\varrho \nu +(1-\varrho )\omega \bigr) \leq \max \bigl\{ \varPhi (\mu ,\nu ),\varPhi (\rho ,\omega ) \bigr\}$$

holds for all $$(\mu ,\nu ),(\rho ,\omega )\in [\lambda _{1}, \lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$ and $$\varrho \in [0,1]$$.

### Definition 1.6

(See )

A real-valued function $$\varPhi : [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]\rightarrow \mathbb{R}$$ is said to be quasi-convex on the co-ordinates if both of the partial mappings $$\varPhi _{\nu }:[\lambda _{1},\lambda _{2}]\rightarrow \mathbb{R}$$ and $$\varPhi _{\mu }:[\xi _{1},\xi _{2}]\rightarrow \mathbb{R}$$ defined by

$$\varPhi _{\nu }(\delta )=\varPhi (\delta ,\nu ), \qquad \varPhi _{\mu }( \theta )= \varPhi (\mu ,\theta )$$

are quasi-convex for all $$\mu \in [\lambda _{1},\lambda _{2}]$$ and $$\nu \in [\xi _{1},\xi _{2}]$$.

In , Latif et al. provided an equivalent definition for the co-ordinated quasi-convex function as follows.

### Definition 1.7

(See )

A real-valued function $$\varPhi : [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]\subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$$ is said to be quasi-convex on the co-ordinates if the inequality

$$\varPhi \bigl(\varrho \mu +(1-\varrho )\rho ,\varsigma \nu +(1-\varsigma ) \omega \bigr) \leq \max \bigl\{ \varPhi (\mu ,\nu ),\varPhi (\mu , \omega ),\varPhi (\rho ,\nu ) , \varPhi (\rho , \omega ) \bigr\}$$

holds for all $$(\mu ,\nu ),(\rho ,\omega )\in [\lambda _{1}, \lambda _{2} ]\times [ \xi _{1},\xi _{2} ]$$ and $$(\varsigma ,\varrho ) \in [0,1]\times [0,1]$$.

The class of co-ordinated quasi-convex functions on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$ is denoted $$QC( [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ])$$. Özdemir  proved that every quasi-convex function is also a co-ordinated quasi-convex function, but the converse does not hold true. The Hermite–Hadamard type inequality (1.2) was generalized to the co-ordinated quasi-convex function by Latif et al. .

### Theorem 1.8

(See )

Let $$\triangle \subseteq {\mathbb{R}}^{2}$$, $$\triangle ^{ \circ }$$be the interior of , $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1}, \lambda _{2}] \times [\xi _{1}, \xi _{2}]\in \triangle ^{\circ }$$, $$\varPhi :\Delta \rightarrow \mathbb{R}$$be a differentiable mapping on $$\triangle ^{ \circ }$$, $$\varPhi _{\varrho \varsigma }\in L ( [\lambda _{1}, \lambda _{2} ]\times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{\varrho \varsigma } \vert$$be a co-ordinated quasi-convex function on $$[\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]$$. Then

\begin{aligned}& \biggl\vert \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}}\varPhi (\mu ,\nu )\,d\mu \,d\nu \\& \qquad {}+\frac{\varPhi (\lambda _{1},\xi _{1} )+\varPhi (\lambda _{1},\xi _{2} )+\varPhi (\lambda _{2},\xi _{1} ) +\varPhi (\lambda _{2},\xi _{2} )}{4} \\& \qquad {}-\frac{1}{2} \biggl[\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \bigl[\varPhi (\mu ,\xi _{1} ) +\varPhi (\mu ,\xi _{2} ) \bigr]\,d\mu \\& \qquad {} + \frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\mathfrak{q_{\lambda _{2}}}} \bigl[\varPhi (\lambda _{1},\nu )+\varPhi (\lambda _{2},\nu ) \bigr]\,d\nu \biggr] \biggr\vert \\& \quad \leq K \biggl[\sup \biggl\{ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{1} ) \bigr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl(\lambda _{1},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert , \\& \qquad \biggl\vert \varPhi _{\varrho \varsigma } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \xi _{1} \biggr) \biggr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert \biggr\} \\& \qquad {}+\sup \biggl\{ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl( \lambda _{1},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert , \\& \qquad \biggl\vert \varPhi _{\varrho \varsigma } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \xi _{2} \biggr) \biggr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl( \frac{\lambda _{1}+\lambda _{2}}{2}, \frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert \biggr\} \\& \qquad {}+\sup \biggl\{ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl( \lambda _{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert , \\& \qquad \biggl\vert \varPhi _{\varrho \varsigma } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2},\xi _{1} \biggr) \biggr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl( \frac{\lambda _{1}+\lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert \biggr\} \\& \qquad {}+\sup \biggl\{ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{2} ) \bigr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl( \lambda _{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert , \\& \qquad \biggl\vert \varPhi _{\varrho \varsigma } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2},\xi _{2} \biggr) \biggr\vert , \biggl\vert \varPhi _{\varrho \varsigma } \biggl( \frac{\lambda _{1}+\lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert \biggr\} \biggr] , \end{aligned}
(1.5)

where $$K= (\lambda _{2}-\lambda _{1} ) ( \xi _{2}-\xi _{1} )/64$$.

More recent results on Hermite–Hadamard type inequalities and their applications can be found in the literature .

Motivated by Theorems 1.21.4 and 1.8, it is natural to ask the question: what are the weighted versions of the Hermite–Hadamard type inequality for the co-ordinated convex and quasi-convex functions?

The main purpose of the article is to present several weighted versions of the Hermite–Hadamard type inequality for the co-ordinated convex and quasi-convex functions, and give an application to the moment of continuous random variables of bivariate distribution functions in the probability theory. Finally, we provide an example on the probability distribution to support our results.

## Some auxiliary results

First of all, we introduce several symbols as follows.

Let $$\omega (\mu ,\nu ): [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$ be a continuous real-valued function such that $$\int _{\lambda _{1}}^{\lambda _{2}}\int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu =1$$. Then we denote the integral $$\int _{\lambda _{1}}^{\lambda _{2}}\int _{\xi _{1}}^{\xi _{2}} \mu \omega (\mu ,\nu )\,d\nu \,d\mu$$ by $$\varUpsilon _{1}$$, the integral $$\int _{\lambda _{1}}^{\lambda _{2}}\int _{\xi _{1}} ^{\xi _{2}}\nu \omega (\mu ,\nu )\,d\nu \,d\mu$$ by $$\varUpsilon _{2}$$ and the integral $$\int _{\lambda _{1}}^{\lambda _{2}}\int _{\xi _{1}}^{\xi _{2}}\mu \nu \omega (\mu ,\nu )\,d\nu \,d\mu$$ by $$\varUpsilon _{3}$$, that is,

\begin{aligned}& \varUpsilon _{1}= \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\mu \omega (\mu ,\nu )\,d\nu \,d\mu , \qquad \varUpsilon _{2}= \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\mu \nu \omega (\mu ,\nu ) \,d\nu \,d\mu , \\& \varUpsilon _{3}= \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\nu \omega (\mu ,\nu )\,d\mu . \end{aligned}

We show an outcome in which the function $$\omega ( \mu ,\nu )$$ is symmetric on the co-ordinates with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$ and $$\frac{\xi _{1}+\xi _{2}}{2}$$.

### Lemma 2.1

If $$\omega (\mu ,\nu ): [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$is symmetric on the co-ordinates with respect to the midpoints $$\frac{\lambda _{1}+\lambda _{2}}{2}$$and $$\frac{\xi _{1}+\xi _{2}}{2}$$. Then

$$\varUpsilon _{1}=\frac{\lambda _{1}+\lambda _{2}}{2},\qquad \varUpsilon _{3}= \frac{ \xi _{1}+\xi _{2}}{2},\qquad \varUpsilon _{2}= \biggl( \frac{\lambda _{1}+ \lambda _{2}}{2} \biggr) \biggl(\frac{\xi _{1}+\xi _{2}}{2} \biggr).$$

### Proof

It follows from the hypothesis that

\begin{aligned} \varUpsilon _{1}&= \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\mu \omega (\mu ,\nu )\,d\nu \,d\mu = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \mu w (\lambda _{1}+ \lambda _{2}-\mu ,\nu )\,d\nu \,d\mu \\ &= \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{1}+\lambda _{2}-\mu )\omega (\mu , \nu )\,d\nu \,d\mu , \end{aligned}

which gives the desired result due to

$$\int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \omega (\mu ,\nu ) \,d\nu \,d\mu =1.$$

Similarly, one can prove that

$$\varUpsilon _{3}=\frac{\xi _{1}+\xi _{2}}{2}$$

and

$$\varUpsilon _{3}= \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\mu \nu \omega (\mu ,\nu )\,d\nu \,d\mu = \biggl(\frac{\lambda _{1}+\lambda _{2}}{2} \biggr) \biggl(\frac{\xi _{1}+ \xi _{2}}{2} \biggr).$$

□

### Lemma 2.2

Let $$\varOmega \subseteq \mathbb{R}^{2}$$, $$\varOmega ^{\circ }$$be the interior ofΩ, $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] \subseteq \varOmega ^{\circ }$$, $$\omega : [\lambda _{1},\lambda _{2} ] \times [ \xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous mapping, and $$\varPhi :\varOmega \rightarrow \mathbb{R}$$be a twice partially differentiable mapping on $$\varOmega ^{\circ }$$such that $$\varPhi _{\varrho \varsigma }\in L ( [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ] )$$. Then

\begin{aligned}& \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \biggl[\varPhi (\lambda _{1},\xi _{1} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}- \mu ) ( \xi _{2}-\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{1},\xi _{2} ) \int _{\lambda _{1}}^{ \lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{2},\xi _{1} ) \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{2},\xi _{2} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \biggr] \\& \qquad {}+ \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) \varPhi (\lambda _{1},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {} - \frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) \varPhi (\lambda _{2},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\xi _{2}-\nu ) \varPhi ( \mu ,\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\nu -\xi _{1} ) \varPhi (\mu ,\xi _{2} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad =\frac{ (\varUpsilon _{1}-\lambda _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} H (\omega ,\lambda _{1},\xi _{1},\varUpsilon _{1},\varUpsilon _{3};r,\rho ) \\& \qquad {}\times\varPhi _{r\rho } \bigl( (1-r ) \lambda _{1}+\varUpsilon _{1}r, (1-\rho )\xi _{1}+ \varUpsilon _{3} \rho \bigr)\,d\rho \,dr \\& \qquad {}+\frac{ (\lambda _{2}-\varUpsilon _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}- \xi _{1} )} \int _{0}^{1} \int _{0}^{1}H (\omega ,\varUpsilon _{1}, \xi _{1},\lambda _{2},\varUpsilon _{3};r,\rho ) \\& \qquad {}\times\varPhi _{r\rho } \bigl( (1-r ) \varUpsilon _{1}+\lambda _{2}r, (1-\rho )\xi _{1}+ \varUpsilon _{3} \rho \bigr)\,d\rho \,dr \\& \qquad {}+\frac{ (\varUpsilon _{1}-\lambda _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} H (\omega ,\lambda _{1}, \varUpsilon _{3},\varUpsilon _{1},\xi _{2};r,\rho ) \\& \qquad {}\times\varPhi _{r\rho } \bigl( (1-r ) \lambda _{1}+\varUpsilon _{1}r, (1-\rho )\varUpsilon _{3}+ \xi _{2} \rho \bigr)\,d\rho \,dr \\& \qquad {}+\frac{ (\lambda _{2}-\varUpsilon _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}- \xi _{1} )} \int _{0}^{1} \int _{0}^{1}H (\omega ,\varUpsilon _{1}, \varUpsilon _{3},\lambda _{2},\xi _{2};r,\rho ) \\& \qquad {}\times\varPhi _{r\rho } \bigl( (1-r ) \varUpsilon _{1}+\lambda _{2}r, (1-\rho )\varUpsilon _{3}+ \xi _{2} \rho \bigr)\,d\rho \,dr, \end{aligned}
(2.1)

where

\begin{aligned}& H (w,\varUpsilon _{1},\gamma ,\beta ,\epsilon ;r,\rho ) \\& \quad = \int _{ ( 1-r )\alpha +\beta r}^{\lambda _{2}} \int _{ (1-\rho )\gamma +\epsilon \rho }^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{ (1-r )\alpha +\beta r}^{\lambda _{2}} \int _{\xi _{1}} ^{ (1-z ) \gamma +\epsilon \rho } (\lambda _{2}-\mu ) (\nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{ (1-r )\alpha +\beta r} \int _{ (1-\rho )\gamma +\epsilon \rho }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{ (1-r )\alpha +\beta r} \int _{\xi _{1}} ^{ (1-\rho )\gamma +\epsilon \rho } (\mu -\lambda _{1} ) (\nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \end{aligned}

and $$(\alpha ,\gamma ), (\beta ,\epsilon ) \in [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$.

### Proof

Let

$$\sigma (\delta )= \textstyle\begin{cases} 0, & \delta < 0, \\ 1, & \delta >0. \end{cases}$$

Then we clearly see that

\begin{aligned}& \varPhi (\mu ,\nu )-\varPhi (\lambda _{1},\nu )- \varPhi (\mu ,\xi _{1} )+\varPhi (\lambda _{1},\xi _{1} ) \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\sigma (\mu -\varrho )\sigma (\nu - \varsigma )\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho , \end{aligned}
(2.2)
\begin{aligned}& \varPhi (\mu ,\nu )-\varPhi (\lambda _{1},\nu )- \varPhi (\mu ,\xi _{2} )+\varPhi (\lambda _{1},\xi _{2} ) \\& \quad =- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\sigma (\mu -\varrho ) \sigma ( \varsigma - \nu )\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d \varrho , \end{aligned}
(2.3)
\begin{aligned}& \varPhi (\mu ,\nu )-\varPhi (\lambda _{2},\nu )- \varPhi (\mu ,\xi _{1} )+\varPhi (\lambda _{2},\xi _{1} ) \\& \quad =- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\sigma (\varrho -\mu ) \sigma (\nu -\varsigma )\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \end{aligned}
(2.4)

and

\begin{aligned}& \varPhi (\mu ,\nu )-\varPhi (\lambda _{2},\nu )- \varPhi (\mu ,\xi _{2} )+\varPhi (\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\sigma (\varrho -\mu )\sigma ( \varsigma - \nu )\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varrho \,d \varsigma . \end{aligned}
(2.5)

It follows from (2.2) that

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu )\varPhi (\mu , \nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \varPhi (\lambda _{1},\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \varPhi (\mu ,\xi _{1} )\omega (\mu ,\nu ) \,d \nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{1},\xi _{1} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \biggl( \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \sigma (\mu -\varrho )\sigma (\nu -\varsigma ) \varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \biggr) \\& \qquad {}\times\omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl( \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \biggr)\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho . \end{aligned}
(2.6)

Similarly, from (2.3)–(2.5) we have

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \nu -\xi _{1} ) \varPhi (\mu , \nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} ( \lambda _{2}-\mu ) ( \nu -\xi _{1} ) \varPhi (\lambda _{1},\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \nu -\xi _{1} ) \varPhi (\mu ,\xi _{1} )\omega (\mu ,\nu )\,d \nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{1},\xi _{2} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad =- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl( \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\varsigma } (\lambda _{2}-\mu ) ( \nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \biggr)\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho , \end{aligned}
(2.7)
\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu )\varPhi (\mu , \nu )\omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu ) \varPhi (\lambda _{2},\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu ) \varPhi (\mu ,\xi _{1} )\omega (\mu ,\nu )\,d \nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{2},\xi _{1} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad =- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl( \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \biggr)\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \end{aligned}
(2.8)

and

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \nu -\xi _{1} )\varPhi (\mu , \nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \nu -\xi _{1} ) \varPhi (\lambda _{2},\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \nu -\xi _{1} ) \varPhi (\mu ,\xi _{2} )\omega (\mu ,\nu )\,d \nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{2},\xi _{2} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl( \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \biggr)\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho . \end{aligned}
(2.9)

From (2.6)–(2.9), we get

\begin{aligned}& \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \biggl[\varPhi (\lambda _{1},\xi _{1} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}- \mu ) ( \xi _{2}-\nu )w (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{1},\xi _{2} ) \int _{\lambda _{1}}^{ \lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{2},\xi _{1} ) \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+\varPhi (\lambda _{2},\xi _{2} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} ( \mu -\lambda _{1} ) (\nu -\xi _{1} )\omega ( \mu ,\nu )\,d\nu \,d\mu \biggr] \\& \qquad {}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\xi _{2}-\nu ) \varPhi ( \mu , \xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\nu -\xi _{1} )\varPhi ( \mu , \xi _{2} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) \varPhi (\lambda _{1},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) \varPhi (\lambda _{1},\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu \\& \quad =\frac{1}{ ( \lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}} \biggl[ \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\varsigma } (\lambda _{2}- \mu ) (\nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega (\mu , \nu ) \,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr] \varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \end{aligned}
(2.10)

and

\begin{aligned}& \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl[ \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\varsigma } (\lambda _{2}- \mu ) (\nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu ) \,d\nu \,d\mu \biggr]\varPhi _{\varrho \varsigma } (\varrho , \varsigma )\,d\varsigma \,d\varrho \\& \quad =\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{ \varUpsilon _{3}} \biggl[ \int _{\varrho }^{\lambda _{2}} \int _{\varsigma } ^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\lambda _{2}-\mu ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr]\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \\& \qquad {}+\frac{1}{ (\lambda _{2}-\lambda _{1} ) ( \xi _{2}-\xi _{1} )} \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{ \varUpsilon _{3}} \biggl[ \int _{\varrho }^{\lambda _{2}} \int _{\varsigma } ^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\lambda _{2}-\mu ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr]\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \\& \qquad {}+\frac{1}{ ( \lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}} ^{\xi _{2}} \biggl[ \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\lambda _{2}-\mu ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \biggr]\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho \\& \qquad {}+\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}} ^{\xi _{2}} \biggl[ \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\xi _{2}-\nu )\omega ( \mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\lambda _{2}-\mu ) (\nu -\xi _{1} ) \omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}+ \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr]\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho . \end{aligned}
(2.11)

Therefore, inequality (2.1) follows from (2.10) and (2.11). □

### Remark 2.3

Let $$\omega (\mu ,\nu )=\frac{1}{ (\lambda _{2}- \lambda _{1} ) (\xi _{2}-\xi _{1} )}$$. Then (2.1) reduces to

\begin{aligned}& \frac{\varPhi (\lambda _{1},\xi _{1} )+\varPhi (\lambda _{1}, \xi _{2} ) +\varPhi (\lambda _{2},\xi _{1} )+\varPhi (\lambda _{2},\xi _{2} )}{4} \\& \qquad {}+\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{2 ( \lambda _{2}-\lambda _{1} ) } \int _{\lambda _{1}}^{\lambda _{2}} \bigl[\varPhi (\mu ,\xi _{1} )+ \varPhi (\mu ,\xi _{2} ) \bigr]\,d\mu \\& \qquad {}- \frac{1}{2 (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \bigl[\varPhi (\lambda _{1},\nu ) +\varPhi (\lambda _{2},\nu ) \bigr]\,d\mu \\& \quad = \frac{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )}{16} \biggl[ \int _{0}^{1} \int _{0}^{1}\varrho \varsigma \varPhi _{\varrho \varsigma } \biggl(\frac{1-\varrho }{2}\lambda _{1}+ \frac{1+\varrho }{2}\lambda _{2}, \frac{1-\varsigma }{2}\xi _{1}+\frac{1+\varsigma }{2}\xi _{2} \biggr)\,d\varsigma \,d\varrho \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} ( -\varrho )\varsigma \varPhi _{\varsigma \varrho } \biggl(\frac{1+\varrho }{2}\lambda _{1} + \frac{1- \varrho }{2}\lambda _{2},\frac{1-\varsigma }{2}\xi _{1}+\frac{1+\varsigma }{2}\xi _{2} \biggr)\,d\varsigma \,d\varrho \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1}\varrho (-\varsigma ) \varPhi _{\varrho \varsigma } \biggl(\frac{1-\varrho }{2}\lambda _{1}+ \frac{1+ \varrho }{2}\lambda _{2},\frac{1+\varsigma }{2}\xi _{1} +\frac{1-\varsigma }{2}\xi _{2} \biggr)\,d\varsigma \,d\varrho \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} (-\varsigma ) (- \varrho ) \varPhi _{\varsigma \varrho } \biggl(\frac{1+\varrho }{2} \lambda _{1}+ \frac{1-\varrho }{2} \lambda _{2},\frac{1+\varsigma }{2}\xi _{1}+\frac{1-\varsigma }{2}\xi _{2} \biggr)\,d\varsigma \,d\varrho \biggr]. \end{aligned}
(2.12)

The identity (2.12) was established in .

### Corollary 2.4

If the function $$\omega (\mu ,\nu )$$is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$and $$\frac{\xi _{1}+\xi _{2}}{2}$$on the co-ordinates. Then Lemma 2.2leads to

\begin{aligned}& \frac{\varPhi (\lambda _{1},\xi _{1} )+\varPhi (\lambda _{1}, \xi _{2} )+\varPhi (\lambda _{2},\xi _{1} ) +\varPhi (\lambda _{2},\xi _{2} )}{4}+ \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu ) \omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}-\frac{1}{ \xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\xi _{2}-\nu )\varPhi ( \mu , \xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\nu -\xi _{1} )\varPhi ( \mu ,\xi _{2} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) \varPhi (\lambda _{2},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) \varPhi (\lambda _{2},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \quad = \int _{0}^{1} \int _{0}^{1}H (\omega ,\lambda _{1},\xi _{1},\lambda _{2},\xi _{2};\varrho ,\varsigma )\varPhi _{\varrho \varsigma } \bigl(\lambda _{1}\varrho + (1-\varrho )\lambda _{2},\xi _{1}\varsigma + (1-\varsigma )\xi _{2} \bigr)\,d\varsigma \,d\varrho . \end{aligned}
(2.13)

### Lemma 2.5

Let $$\mu \in [\lambda _{1},\lambda _{2} ]$$, $$\nu \in [\xi _{1},\xi _{2} ]$$, $$A: C ( [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ] )\rightarrow \mathbb{R}$$be a positive linear functional, $$e_{i} (\mu )=\mu ^{i}$$and $$k_{j} (\nu )=\nu ^{j}$$be the monomials $$(i, j\in \mathbb{N})$$, andgbe a co-ordinated convex function on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$. Then

\begin{aligned}& A \bigl(g (e_{1}, k_{1} ) \bigr) \\& \quad \leq \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \bigl[A \bigl( (\lambda _{2}-e_{1} ) (\xi _{2}-k_{1} ) \bigr)g (\lambda _{1},\xi _{1} )+A \bigl( (\lambda _{2}-e_{1} ) (k_{1}-\xi _{1} ) \bigr) g (\lambda _{1},\xi _{2} ) \\& \qquad {}+A \bigl( (e_{1}-\lambda _{1} ) (\xi _{2}-k_{1} ) \bigr)g (\lambda _{2},\xi _{1} ) +A \bigl( (e_{1}-\lambda _{1} ) (k_{1}-\xi _{1} ) \bigr)g (\lambda _{2},\xi _{2} ) \bigr]. \end{aligned}
(2.14)

### Proof

It follows from the convexity of g on the co-ordinates that

\begin{aligned} g (e_{1},k_{1} ) \leq& \frac{ (\lambda _{2}-e_{1} ) g (\lambda _{1}, k_{1} )+ (e_{1}-\lambda _{1} ) g (\lambda _{2},k_{1} )}{\lambda _{2}-\lambda _{1}} \\ \leq& \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \bigl[ (\lambda _{2}-e_{1} ) (\xi _{2}-k _{1} )g (\lambda _{1},\xi _{1} ) + (\lambda _{2}-e _{1} ) (k_{1}-\xi _{1} )g (\lambda _{1},\xi _{2} ) \\ &{}+ (e_{1}-\lambda _{1} ) (\xi _{2}-k_{1} ) g (\lambda _{2},\xi _{1} )+ (e_{1}-\lambda _{1} ) (k_{1}-\xi _{1} )g (\lambda _{2},\xi _{2} ) \bigr]. \end{aligned}
(2.15)

Therefore, inequality (2.14) follows from (2.15) and the assumption of the functional A. □

## Main results

In order to provide compact demonstration, we use the notations for our coming results as follows:

\begin{aligned}& B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\lambda _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu ) ^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\varUpsilon _{1}}^{ \lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) ^{2} (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu , \end{aligned}
(3.1)
\begin{aligned}& B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu ) ( \varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu ) ( \varUpsilon _{3}-\nu )^{2}\omega ( \mu ,\nu )\,d\nu \,d\mu , \end{aligned}
(3.2)
\begin{aligned}& B_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d \mu \\& \qquad {}- \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu ) \omega (\mu , \nu )\,d\nu \,d \mu \\& \qquad {}- \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu ,\nu )\,d\nu \,d \mu \\& \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d \mu \end{aligned}
(3.3)

and

\begin{aligned}& B_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu ) ( \varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu ) ( \varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d\mu . \end{aligned}
(3.4)

### Theorem 3.1

Let $$\varOmega \subseteq \mathbb{R}^{2}$$, $$\varOmega ^{\circ }$$be the interior ofΩ, $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] \subseteq \varOmega ^{\circ }$$, $$\varPhi :\varOmega \rightarrow \mathbb{R}$$be a twice partially differentiable mapping on $$\varOmega ^{\circ }$$such that $$\varPhi _{\varrho \varsigma }\in L ( [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{ \varrho \varsigma } \vert$$is co-ordinated convex on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$, and $$\omega : [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous mapping. Then one has

\begin{aligned}& \biggl\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\mu ,\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\varUpsilon _{1},\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}}\varPhi (\mu ,\varUpsilon _{3} ) \omega (\mu , \nu )\,d\nu \,d\mu +\varPhi (\varUpsilon _{1},\varUpsilon _{3} ) \biggr\vert \\& \quad \leq \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \bigl[ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1}, \xi _{1} ) \bigr\vert A_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert A_{2} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ) \\& \qquad {}+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert A_{3} (\lambda _{1},\xi _{1},\lambda _{2}, \xi _{2} )+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{2} ) \bigr\vert A_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ) \bigr], \end{aligned}
(3.5)

where

\begin{aligned}& A_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) +\frac{ (\lambda _{2}-\varUpsilon _{1} )}{2}B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+\frac{ (\xi _{2}-\varUpsilon _{3} )}{2}B_{3} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) + (\lambda _{2}-\varUpsilon _{1} ) (\xi _{2}-\varUpsilon _{3} )B_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ), \\& A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )+\frac{ (\lambda _{2}-\varUpsilon _{1} )}{2}B_{2} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+\frac{ (\varUpsilon _{3}-\xi _{1} )}{2}B_{3} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) + (\lambda _{2}-\varUpsilon _{1} ) (\varUpsilon _{3}-\xi _{1} )B_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ), \\& A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )+\frac{ (\varUpsilon _{1}-\lambda _{1} )}{2}B_{2} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+\frac{ (\xi _{2}-\varUpsilon _{3} )}{2}B_{3} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) + (\varUpsilon _{1}-\lambda _{1} ) (\xi _{2}-\varUpsilon _{3} )B_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ) \end{aligned}

and

\begin{aligned}& A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) +\frac{ (\varUpsilon _{1}-\lambda _{1} )}{2}B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+\frac{ (\varUpsilon _{3}-\xi _{1} )}{2}B_{3} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} )+ (\varUpsilon _{1}-\lambda _{1} ) (\varUpsilon _{3}-\xi _{1} )B_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ). \end{aligned}

### Proof

We clearly see that

\begin{aligned}& \varPhi (\mu ,\nu )-\varPhi ( \varUpsilon _{1},\nu )- \varPhi (\mu ,\varUpsilon _{3} )+\varPhi (\varUpsilon _{1}, \varUpsilon _{3} ) \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \bigl[\sigma (\mu -\varrho ) \sigma (\nu - \varsigma )-\sigma (\varUpsilon _{1}-\varrho )\sigma (\nu - \varsigma ) \\& \qquad {}-\sigma (\mu -\varrho )\sigma (\varUpsilon _{3}- \varsigma )+\sigma ( \varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}-\varsigma ) \bigr]\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d\varrho . \end{aligned}
(3.6)

It follows from (3.6) that

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\mu ,\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\varUpsilon _{1},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\varUpsilon _{3} ) \omega (\mu ,\nu )\,d\nu \,d\mu +\varPhi (\varUpsilon _{1},\varUpsilon _{3} ) \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl( \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma ( \varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr) \\& \qquad {}\times\varPhi _{\varrho \varsigma } (\varrho ,\varsigma )\,d\varsigma \,d \varrho . \end{aligned}
(3.7)

From Lemma 2.5 and (3.7) we get

\begin{aligned}& \biggl\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\mu ,\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\varUpsilon _{1},\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}} \varPhi (\mu ,\varUpsilon _{3} )\omega (\mu , \nu )\,d\nu \,d\mu +\varPhi (\varUpsilon _{1},\varUpsilon _{3} ) \biggr\vert \\& \quad \leq \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma ( \varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \bigl\vert \varPhi _{\varrho \varsigma } (\varrho ,\varsigma ) \bigr\vert \,ds\,dt \\& \quad \leq \frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \biggl[ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1}, \lambda _{2} ) \bigr\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{ \lambda _{2}} \int _{\varsigma }^{d}\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu-\sigma ( \varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega ( \mu , \nu )\,d\nu \,d\mu \\& \qquad {} + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert (\lambda _{2}- \varrho ) (\xi _{2}-\varsigma )\,d\varsigma \,d\varrho \\& \qquad {}+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu , \nu ) \,d\nu \,d\mu \\& \qquad {}- \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega ( \mu , \nu )\,d\nu \,d\mu \\& \qquad {} + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert (\lambda _{2}- \varrho ) (\varsigma -\xi _{1} )\,d\varsigma \,d\varrho \\& \qquad {}+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu-\sigma ( \varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {} + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert (\varrho - \lambda _{1} ) (\xi _{2}-\varsigma )\,d\varsigma \,d\varrho \\& \qquad {}+ \bigl\vert \varPhi _{\varrho \varsigma } ( \lambda _{2},\xi _{2} ) \bigr\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}- \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu -\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}+ \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert (\varrho - \lambda _{1} ) (\varsigma -\xi _{1} )\,d\varsigma \,d\varrho \biggr] \end{aligned}
(3.8)

and

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert (\lambda _{2}- \varrho ) (\xi _{2}-\varsigma )\,d\varsigma \,d\varrho \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} \biggl( \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } \omega (\mu , \nu )\,d\nu \,d\mu \biggr) (\lambda _{2}-\varrho ) (\xi _{2}-\varsigma )\,d \varsigma \,d\varrho \\& \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} \biggl( \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\varsigma }\omega (\mu , \nu )\,d\nu \,d\mu \biggr) (\lambda _{2}-\varrho ) (\xi _{2}-\varsigma )\,d \varsigma \,d\varrho \\ & \qquad {}+ \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} \biggl( \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \biggr) (\lambda _{2}-\varrho ) (\xi _{2}-\varsigma )\,d \varsigma \,d\varrho \\ & \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} \biggl( \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \biggr) (\lambda _{2}-\varrho ) (\xi _{2}-\varsigma )\,d \varsigma \,d\varrho \\ & \quad =\frac{ (\xi _{2}-\varUpsilon _{3} )^{2} (\lambda _{2}- \varUpsilon _{1} )^{2}}{4} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}}\omega (\mu ,\nu )\,d\nu \,d\mu \\ & \qquad {}- \frac{ (\xi _{2}-\varUpsilon _{3} )^{2} (\lambda _{2}- \varUpsilon _{1} )^{2}}{4} \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}}\omega (\mu ,\nu )\,d\nu \,d\mu \\ & \qquad {}-\frac{ (\xi _{2}-\varUpsilon _{3} )^{2} (\lambda _{2}- \varUpsilon _{1} )^{2}}{4} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu \\ & \qquad {}+ \frac{ (\xi _{2}-\varUpsilon _{3} )^{2} (\lambda _{2}- \varUpsilon _{1} )^{2}}{4} \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu \\ & \qquad {}-\frac{ (\xi _{2}-\varUpsilon _{3} )^{2}}{2} \int _{\lambda _{1}} ^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\lambda _{2}- \mu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}+\frac{ (\xi _{2}-\varUpsilon _{3} )^{2}}{2} \int _{\varUpsilon _{1}}^{ \lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\lambda _{2}- \mu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}+\frac{ (\xi _{2}-\varUpsilon _{3} )^{2}}{2} \int _{\lambda _{1}} ^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}}\frac{ (\lambda _{2}- \mu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}-\frac{ (\xi _{2}-\varUpsilon _{3} )^{2}}{2} \int _{\varUpsilon _{1}}^{ \lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}}\frac{ (\lambda _{2}- \mu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}-\frac{ (\lambda _{2}-\varUpsilon _{1} )^{2}}{2} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\xi _{2}-\nu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}+\frac{ (\lambda _{2}-\varUpsilon _{1} )^{2}}{2} \int _{\varUpsilon _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\xi _{2}- \nu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}+\frac{ (\lambda _{2}-\varUpsilon _{1} )^{2}}{2} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}}\frac{ (\xi _{2}-\nu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}-\frac{ (\lambda _{2}-\varUpsilon _{1} )^{2}}{2} \int _{\varUpsilon _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\xi _{2}- \nu )^{2}}{2}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}+ \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\xi _{2}-\nu )^{2} (\lambda _{2}-\mu )^{2}}{4} \omega (\mu ,\nu )\,d\nu \,d\mu \\ & \qquad {}- \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}}\frac{ (\xi _{2}-\nu ) ^{2} (\lambda _{2}-\nu )^{2}}{4}\omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}- \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}}\frac{ (\xi _{2}-\nu )^{2} (\lambda _{2}-\mu )^{2}}{4} \omega (\mu , \nu )\,d\nu \,d\mu \\ & \qquad {}+ \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}}\frac{ (\xi _{2}-\nu ) ^{2} (\lambda _{2}-\mu )^{2}}{4} \omega (\mu , \nu )\,d\nu \,d\mu . \end{aligned}
(3.9)

Equation (3.9) can be rewritten as

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}-s ) \biggr\vert \\& \qquad {}\times(\lambda _{2}- \varrho ) (\xi _{2}- \varsigma )\,d\varsigma \,d\varrho \\& \quad =\frac{1}{4}B_{1} ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\lambda _{2}-\varUpsilon _{1} )}{2}B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\xi _{2}- \varUpsilon _{3} )}{2}B_{3} (\lambda _{1},\xi _{1},\lambda _{2}, \xi _{2} ) \\& \qquad {}+ (\lambda _{2}-\varUpsilon _{1} ) (\xi _{2}- \varUpsilon _{3} )B _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) =A_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ). \end{aligned}
(3.10)

Similarly, we have

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \\& \qquad {}\times(\lambda _{2}- \varrho ) (\varsigma -\xi _{1} )\,d\varsigma \,d\varrho \\& \quad =\frac{1}{4}B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )+ \frac{ (\lambda _{2}-\varUpsilon _{1} )}{2}B_{2} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} )+ \frac{ (\varUpsilon _{3}-\xi _{1} )}{2}B_{3} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ (\lambda _{2}-\varUpsilon _{1} ) (\varUpsilon _{3}-\xi _{1} )B _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) =A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ), \end{aligned}
(3.11)
\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \\& \qquad {}\times (\lambda _{1}- \varrho ) (\xi _{2}-\varsigma )\,d\varsigma \,d\varrho \\& \quad =\frac{1}{4}B_{1} ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\varUpsilon _{1}-\lambda _{1} )}{2}B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\xi _{2}- \varUpsilon _{3} )}{2}B_{3} (\lambda _{1},\xi _{1},\lambda _{2}, \xi _{2} ) \\& \qquad {}+ (\varUpsilon _{1}-\lambda _{1} ) (\xi _{2}- \varUpsilon _{3} )B _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) =A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \end{aligned}
(3.12)

and

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega ( \mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \\& \qquad {}\times(\lambda _{1}- \varrho ) (\varsigma -\xi _{1} )\,d\varsigma \,d\varrho \\& \quad =\frac{1}{4}B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\varUpsilon _{1}-\lambda _{1} )}{2}B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\varUpsilon _{3}- \xi _{1} )}{2}B_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ (\varUpsilon _{1}-\lambda _{1} ) ( \varUpsilon _{3}-\xi _{1} )B _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) =A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ). \end{aligned}
(3.13)

Therefore, inequality (3.5) follows from (3.8) and (3.10)–(3.13). □

### Corollary 3.2

If all the conditions of Theorem 3.1are satisfied and $$\omega (\mu , \nu )$$is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$and $$\frac{\xi _{1}+\xi _{2}}{2}$$on the co-ordinates. Then one has

\begin{aligned}& \biggl\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \nu \biggr) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}}\varPhi \biggl( \mu , \frac{\xi _{1}+\xi _{2}}{2} \biggr) \omega (\mu ,\nu )\,d\nu \,d\mu +\varPhi \biggl( \frac{\lambda _{1}+ \lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert \\& \quad \leq \bigl[ \bigl\vert \varPhi _{\varrho \varsigma } ( \lambda _{1}, \xi _{1} ) \bigr\vert + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert + \bigl\vert \varPhi _{\varrho \varsigma } ( \lambda _{2},\xi _{2} ) \bigr\vert \bigr] \\& \qquad {}\times \int _{\frac{\lambda _{1}+\lambda _{2}}{2}}^{\lambda _{2}} \int _{\frac{\xi _{1}+\xi _{2}}{2}}^{\xi _{2}} \biggl(\mu -\frac{\lambda _{1}+\lambda _{2}}{2} \biggr) \biggl(\nu -\frac{\xi _{1}+\xi _{2}}{2} \biggr) \omega (\mu ,\nu )\,d\nu \,d\mu. \end{aligned}
(3.14)

### Proof

Making use of the hypothesis of Theorem 3.1 and the symmetry of the function $$\omega (\mu ,\nu )$$ with respect to $$\frac{ \lambda _{1}+\lambda _{2}}{2}$$ and $$\frac{\xi _{1}+\xi _{2}}{2}$$ on the co-ordinates that the function $$(\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2}\omega (\mu ,\nu )$$ is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$ and $$\frac{\xi _{1}+\xi _{2}}{2}$$ on the co-ordinates, the function $$(\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )^{2} \omega (\mu ,\nu )$$ is symmetric with respect to $$\frac{\xi _{1}+\xi _{2}}{2}$$ on the co-ordinates, and the function $$(\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu ) \omega (\mu ,\nu )$$ is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$ on the co-ordinates. Therefore,

\begin{aligned}& \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2} \omega (\mu , \nu )\,d\nu \,d\mu \\& \quad = \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2} \omega (\mu , \nu )\,d\nu \,d\mu \\& \quad = \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu , \\& \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )^{2}\omega (\mu , \nu )\,d\nu \,d\mu \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu ) ( \varUpsilon _{3}-\nu )^{2}\omega (\mu ,\nu )\,d\nu \,d\mu , \\& \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu ) (\varUpsilon _{3}-\nu )^{2} \omega (\mu , \nu )\,d\nu \,d \mu \\& \quad = \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu ) ( \varUpsilon _{3}-\nu )^{2}\omega (\mu ,\nu )\,d\nu \,d\mu , \\& \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d \mu \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu ,\nu )\,d\nu \,d \mu \end{aligned}

and

$$\int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu , \nu )\,d\nu \,d \mu = \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\varUpsilon _{1}-\mu )^{2} (\varUpsilon _{3}-\nu )\omega (\mu ,\nu ) \,d\nu \,d \mu .$$

From the above equations, we obtain

$$B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) =B_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=0$$

and

$$B_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=4 \int _{\frac{\lambda _{1}+\lambda _{2}}{2}}^{b} \int _{\frac{\xi _{1}+\xi _{2}}{2}}^{d} \biggl(\mu -\frac{\lambda _{1}+ \lambda _{2}}{2} \biggr) \biggl(\nu -\frac{\xi _{1}+\xi _{2}}{2} \biggr) \omega (\mu ,\nu )\,d\nu \,d\mu .$$

Therefore, inequality (3.14) follows from (3.10) and the above quantities. □

### Remark 3.3

Inequality (1.3) can be derived from (3.5) if

$$\omega (\mu ,\nu )=\frac{g (\mu ,\nu )}{ \int _{\lambda _{1}}^{\lambda _{2}}\int _{\xi _{1}}^{\xi _{2}}g (\mu ,\nu )\,d\nu \,d\mu }$$

and $$g (\mu ,\nu )$$ is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$ and $$\frac{\xi _{1}+\xi _{2}}{2}$$ on the co-ordinates.

### Theorem 3.4

Let $$q>1$$, $$\varOmega \subseteq \mathbb{R}^{2}$$, $$\varOmega ^{\circ }$$be the interior ofΩ, $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]\subseteq \varOmega ^{\circ }$$, $$\varPhi :\varOmega \rightarrow \mathbb{R}$$be a twice partially differentiable mapping on $$\varOmega ^{\circ }$$such that $$\varPhi _{\varrho \varsigma } \in L ( [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{\varrho \varsigma } \vert ^{q}$$is co-ordinated convex on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$, and $$\omega : [\lambda _{1}, \lambda _{2} ]\times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous mapping. Then

\begin{aligned}& \biggl\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu ) \omega (\mu , \nu )\,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\varUpsilon _{1},\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}}\varPhi (\mu ,\varUpsilon _{3} )\omega (\mu , \nu )\,d\nu \,d\mu +\varPhi (\varUpsilon _{1},\varUpsilon _{3} ) \biggr\vert \\& \quad \leq 4^{1-\frac{1}{q}} \biggl( \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\lambda _{2}} (\mu -\varUpsilon _{1} ) (\nu -\varUpsilon _{3} ) \omega (\mu ,\nu )\,d\nu \,d\mu \biggr)^{1-\frac{1}{q}} \biggl(\frac{1}{ ( \lambda _{2}- \lambda _{1} ) (\xi _{2}-\xi _{1} )} \biggr)^{ \frac{1}{q}} \\& \qquad {}\times \bigl[ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1}, \xi _{1} ) \bigr\vert ^{q}A_{1} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} )+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert ^{q}A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert ^{q}A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )+ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{2} ) \bigr\vert ^{q}A_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ) \bigr]^{\frac{1}{q}}, \end{aligned}
(3.15)

where $$A_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )$$, $$A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )$$, $$A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )$$and $$A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )$$are defined in Theorem 3.1.

### Proof

It follows from (3.8) and the Hölder inequality that

\begin{aligned}& \biggl\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\mu ,\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\varUpsilon _{1},\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}}\varPhi (\mu ,\varUpsilon _{3} ) \omega ( \mu , \nu )\,d\nu \,d\mu +\varPhi (\varUpsilon _{1},\varUpsilon _{3} ) \biggr\vert \\& \quad \leq \biggl( \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}-\sigma ( \varUpsilon _{1}-\varsigma ) \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\lambda _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho )\sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \,d\varsigma \,d\varrho \biggr)^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}-\varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma ( \varUpsilon _{3}-\varsigma ) \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \\& \qquad {}\times\bigl\vert \varPhi _{\varrho \varsigma } (\varrho ,\varsigma ) \bigr\vert ^{q}\,d\varsigma \,d\varrho \biggr) ^{\frac{1}{q}}. \end{aligned}
(3.16)

Making use of Lemma 2.5, we have

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \bigl\vert \varPhi _{\varrho \varsigma } (\varrho ,\varsigma ) \bigr\vert ^{q}\,d\varsigma \,d\varrho \\& \quad \leq \frac{1}{ ( \lambda _{2}-\lambda _{1} ) (\xi _{2}- \xi _{1} )} \bigl[ \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{1} ) \bigr\vert ^{q}A_{1} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ) + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \bigr\vert ^{q}A_{2} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ \bigl\vert \phi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \bigr\vert ^{q}A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \bigl\vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{2} ) \bigr\vert ^{q}A_{4} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ) \bigr]. \end{aligned}
(3.17)

On the other hand, we also have

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \biggl\vert \int _{\varrho }^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu - \sigma (\varUpsilon _{1}- \varrho ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\varsigma }^{\xi _{2}} \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\sigma (\varUpsilon _{3}-\varsigma ) \int _{\varrho } ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu + \sigma (\varUpsilon _{1}-\varrho ) \sigma (\varUpsilon _{3}- \varsigma ) \biggr\vert \,d\varsigma \,d\varrho \\& \quad = \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{\varUpsilon _{3}} \biggl( \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma }\omega (\mu , \nu )\,d\nu \,d\mu \biggr)\,d\varsigma \,d\varrho \\& \qquad {}+ \int _{\varUpsilon _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\varUpsilon _{3}} \biggl( \int _{\varrho }^{ \lambda _{2}} \int _{\xi _{1}}^{\varsigma }\omega (\mu ,\nu )\,d\nu \,d\mu \biggr) \,d\varsigma \,d\varrho \\& \qquad {}+ \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\varUpsilon _{3}}^{\xi _{2}} \biggl( \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}}\omega (\mu , \nu )\,d\nu \,d\mu \biggr)\,d\varsigma \,d\varrho \\& \qquad {}+ \int _{\varUpsilon _{1}} ^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} \biggl( \int _{\varrho }^{ \lambda _{2}} \int _{\varsigma }^{\xi _{2}}\omega (\mu ,\nu )\,d\nu \,d\mu \biggr)\,d\varsigma \,d\varrho \\& \quad =4 \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}}^{\xi _{2}} (\mu - \varUpsilon _{1} ) (\nu -\varUpsilon _{2} )\omega (\mu , \nu )\,d\nu \,d\mu . \end{aligned}
(3.18)

Therefore, inequality (3.15) follows from (3.16)–(3.18). □

### Remark 3.5

If $$\omega (\mu ,\nu )$$ is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$ and $$\frac{\xi _{1}+\xi _{2}}{2}$$ on the co-ordinates, then inequality (3.15) leads to

\begin{aligned}& \biggl\vert \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi ( \mu ,\nu )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \nu \biggr)\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}} \varPhi \biggl(\mu , \frac{\xi _{1}+\xi _{2}}{2} \biggr)\omega (\mu ,\nu )\,d\nu \,d\mu +\varPhi \biggl( \frac{\lambda _{1}+ \lambda _{2}}{2},\frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\vert \\& \quad \leq 4 \biggl[\frac{ \vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{1} ) \vert ^{q} + \vert \varPhi _{\varrho \varsigma } (\lambda _{1},\xi _{2} ) \vert ^{q} + \vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{1} ) \vert ^{q} + \vert \varPhi _{\varrho \varsigma } (\lambda _{2},\xi _{2} ) \vert ^{q}}{4} \biggr]^{\frac{1}{q}} \\& \qquad {}\times \int _{\frac{\lambda _{1}+\lambda _{2}}{2}}^{\lambda _{2}} \int _{\frac{\xi _{1}+\xi _{2}}{2}}^{\xi _{2}} \biggl(\mu -\frac{\lambda _{1}+\lambda _{2}}{2} \biggr) \biggl(\nu -\frac{\xi _{1}+\xi _{2}}{2} \biggr) \omega (\mu ,\nu )\,d\nu \,d\mu. \end{aligned}
(3.19)

### Remark 3.6

If $$g (\mu ,\nu )$$ is symmetric with respect to $$\frac{\lambda _{1}+\lambda _{2}}{2}$$ and $$\frac{\xi _{1}+\xi _{2}}{2}$$ on the co-ordinates and

$$\omega (\mu ,\nu )=\frac{g (\mu ,\nu )}{ \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}g (\mu ,\nu )\,d\nu \,d\mu },$$

then inequality (3.15) gives the result proved in .

Next, we use the symbols for our upcoming results as follows:

\begin{aligned} \varPsi (\omega ,\varPhi ) =&\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \biggl[\varPhi (\lambda _{1},\xi _{1} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}+\varPhi (\lambda _{1},\xi _{2} ) \int _{\lambda _{1}}^{ \lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) ( \nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}+\varPhi (\lambda _{2},\xi _{1} ) \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) ( \xi _{2}-\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}+\varPhi (\lambda _{2},\xi _{2} ) \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} ( \mu - \lambda _{1} ) (\nu -\xi _{1} ) \omega ( \mu , \nu ) \,d\nu \,d\mu \biggr] \\ &{}+ \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu ) \omega (\mu ,\nu ) \,d\nu \,d\mu \\ &{}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}-\mu ) \varPhi (\lambda _{1},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}- \frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) \varPhi (\lambda _{2},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\xi _{2}-\nu )\varPhi ( \mu , \xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}- \frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\nu -\xi _{1} )\varPhi ( \mu , \xi _{2} )\omega (\mu ,\nu )\,d\nu \,d\mu . \end{aligned}
(3.20)

From (3.20) we clearly see that

\begin{aligned}& \varPsi \biggl(\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )},\varPhi \biggr) \\& \quad =\frac{\varPhi (\lambda _{1},\xi _{1} )+\varPhi (\lambda _{1},\xi _{2} ) +\varPhi (\lambda _{2},\xi _{1} )+\varPhi (\lambda _{2},\xi _{2} )}{4} \\& \qquad {}-\frac{1}{2 (\lambda _{2}-\lambda _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \bigl[\varPhi (\mu ,\xi _{2} )+ \varPhi (\mu ,\xi _{1} ) \bigr]\,d\mu \\& \qquad {}- \frac{1}{2 (\xi _{2}-\xi _{1} )} \int _{\xi _{1}}^{\xi _{2}} \bigl[\varPhi (\lambda _{1},\nu ) +\varPhi (\lambda _{2},\nu ) \bigr]\,d\nu \\& \qquad {}+\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu )\,d\nu \,d\mu . \end{aligned}
(3.21)

We also use the following notations:

\begin{aligned}& \sup \bigl\{ \varPhi _{r\rho } (\lambda _{1},\xi _{1} ), \varPhi _{r\rho } (\varUpsilon _{1},\xi _{1} ),\varPhi _{r\rho } (\lambda _{1},\varUpsilon _{3} ),\varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ) \bigr\} =\eta (\lambda _{1},\xi _{1}, \lambda _{2}, \xi _{2} ), \\& \sup \bigl\{ \varPhi _{r\rho } (\varUpsilon _{1},\xi _{1} ), \varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ),\varPhi _{r\rho } (\lambda _{2},\xi _{1} ),\varPhi _{r\rho } (\lambda _{2}, \varUpsilon _{3} ) \bigr\} =\eta _{2} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ), \\& \sup \bigl\{ \varPhi _{r\rho } (\lambda _{1},\varUpsilon _{3} ), \varPhi _{r\rho } (\lambda _{1},\xi _{2} ),\varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ),\varPhi _{r\rho } (\varUpsilon _{1},\xi _{2} ) \bigr\} =\eta _{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ), \\& \sup \bigl\{ \varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ), \varPhi _{r\rho } (\varUpsilon _{1},\xi _{2} ),\varPhi _{r\rho } (\lambda _{2},\varUpsilon _{3} ),\varPhi _{r\rho } (\lambda _{2},\xi _{2} ) \bigr\} =\eta _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ). \end{aligned}

The following results present uppermost estimates for $$\vert \varPsi (\omega ,\varPhi ) \vert$$ if the function $$\varPhi (\mu ,\nu )$$ is quasi-convex on the co-ordinates.

### Theorem 3.7

Let $$\varOmega \subseteq \mathbb{R}^{2}$$, $$\varOmega ^{\circ }$$be the interior ofΩ, $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] \subseteq \varOmega ^{\circ }$$, $$\varPhi :\varOmega \rightarrow \mathbb{R}$$be a twice partially differentiable mapping on $$\varOmega ^{\circ }$$such that $$\varPhi _{\varrho \varsigma }\in L ( [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{ \varrho \varsigma } \vert$$is quasi-convex on the co-ordinates on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$, and $$\omega : [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous mapping. Then

\begin{aligned} \bigl\vert \varPsi (\omega ,\varPhi ) \bigr\vert \leq& \frac{ (\varUpsilon _{1}-\lambda _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \eta ( \lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\ &{}\times \int _{0} ^{1} \int _{0}^{1} \bigl\vert H (\omega ,\lambda _{1},\xi _{1}, \varUpsilon _{1},\varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\ & {}+\frac{ (\lambda _{2}-\varUpsilon _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )}\eta _{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\ &{}\times \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega ,\varUpsilon _{1},\xi _{1},\lambda _{2},\varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\ & {}+\frac{ (\varUpsilon _{1}-\lambda _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\lambda _{2}-\xi _{1} )}\eta _{3} (\lambda _{1},\xi _{1},\lambda _{2}, \xi _{2} ) \\ &{}\times \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \lambda _{1},\varUpsilon _{3},\varUpsilon _{1},\xi _{2};r,\rho ) \bigr\vert \,d\rho \,dr \\ & {}+\frac{ (\lambda _{2}-\varUpsilon _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )}\eta _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\ &{}\times \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega ,\varUpsilon _{1}, \varUpsilon _{3},\lambda _{2},\xi _{2};r,\rho ) \bigr\vert \,d\rho \,dr. \end{aligned}
(3.22)

### Proof

It follows from the quasi-convexity of the function $$\vert \varPhi _{ \varrho \varsigma } \vert$$ on the co-ordinates that

\begin{aligned}& \varPhi _{r\rho } \bigl( (1-r )\lambda _{1}+\varUpsilon _{1}r, (1- \rho )\xi _{1}+\varUpsilon _{3}\rho \bigr) \\& \quad \leq \sup \bigl\{ \varPhi _{r\rho } (\lambda _{1},\xi _{1} ), \varPhi _{r\rho } (\varUpsilon _{1},\xi _{1} ),\varPhi _{r\rho } (\lambda _{1},\varUpsilon _{3} ),\varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ) \bigr\} =\eta (\lambda _{1},\xi _{1}, \lambda _{2}, \xi _{2} ), \\& \varPhi _{r\rho } \bigl( (1-r )\varUpsilon _{1}+br, (1-\rho ) \xi _{1}+\varUpsilon _{3}\rho \bigr) \\& \quad \leq \sup \bigl\{ \varPhi _{r\rho } (\varUpsilon _{1},\xi _{1} ), \varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ),\varPhi _{r\rho } (\lambda _{2},\xi _{1} ),\varPhi _{r\rho } (\lambda _{2}, \varUpsilon _{3} ) \bigr\} =\eta _{2} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} ), \\& \varPhi _{r\rho } \bigl( (1-r )\lambda _{1}+\varUpsilon _{1}r, (1- \rho )\varUpsilon _{3}+\xi _{2}\rho \bigr) \\& \quad \leq \sup \bigl\{ \varPhi _{r\rho } (\lambda _{1},\varUpsilon _{3} ), \varPhi _{r\rho } (\lambda _{1},\xi _{2} ),\varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ),\varPhi _{r\rho } (\varUpsilon _{1},\xi _{2} ) \bigr\} =\eta _{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \end{aligned}

and

\begin{aligned}& \varPhi _{r\rho } \bigl( (1-r )\varUpsilon _{1}+\lambda _{2}r, (1- \rho )\varUpsilon _{3}+\xi _{2}\rho \bigr) \\& \quad \leq \sup \bigl\{ \varPhi _{r\rho } (\varUpsilon _{1},\varUpsilon _{3} ), \varPhi _{r\rho } (\varUpsilon _{1},\xi _{2} ),\varPhi _{r\rho } (\lambda _{2},\varUpsilon _{3} ),\varPhi _{r\rho } (\lambda _{2},\xi _{2} ) \bigr\} =\eta _{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \end{aligned}

for all $$(r,\rho )\in [0,1 ]\times [0,1 ]$$.

Therefore, inequality (3.22) follows from (2.1) and the above inequalities. □

### Theorem 3.8

Let $$\varOmega \subseteq \mathbb{R}^{2}$$, $$\varOmega ^{\circ }$$be the interior ofΩ, $$\lambda _{1}<\lambda _{2}$$and $$\xi _{1}<\xi _{2}$$such that $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ] \subseteq \varOmega ^{\circ }$$, $$\varPhi :\varOmega \rightarrow \mathbb{R}$$be a twice partially differentiable mapping on $$\varOmega ^{\circ }$$such that $$\varPhi _{\varrho \varsigma }\in L ( [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ] )$$and $$\vert \varPhi _{ \varrho \varsigma } \vert$$is quasi-convex on the co-ordinates on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$, and $$\omega : [\lambda _{1},\lambda _{2} ] \times [\xi _{1},\xi _{2} ]\rightarrow [0,\infty )$$be a continuous function symmetric with respect to $$\frac{\lambda _{1}+ \lambda _{2}}{2}$$and $$\frac{\xi _{1}+\xi _{2}}{2}$$on the co-ordinates. Then

\begin{aligned}& \biggl\vert \frac{\varPhi (\lambda _{1},\xi _{1} )+\varPhi (\lambda _{1},\xi _{2} ) +\varPhi (\lambda _{2},\xi _{1} )+\varPhi (\lambda _{2},\xi _{2} )}{4} + \int _{\lambda _{1}}^{ \lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varPhi (\mu ,\nu ) \omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\lambda _{2}- \mu ) \varPhi (\lambda _{1},\nu ) \omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu - \lambda _{1} ) \varPhi (\lambda _{2},\nu )\omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\xi _{2}-\nu ) \varPhi ( \mu ,\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}-\frac{1}{\xi _{2}} (\nu -\xi _{1} ) \varPhi (\mu , \xi _{2} )\omega (\mu ,\nu ) \,d\nu \,d\mu \biggr\vert \\& \quad \leq \biggl( \int _{\lambda _{1}}^{\frac{\lambda _{1}+\lambda _{2}}{2}} \int _{\xi _{1}}^{\frac{\xi _{1}+\xi _{2}}{2}} (\mu -\lambda _{1} ) (\nu -\xi _{1} )\omega (\mu ,\nu ) \,d\nu \,d\mu \biggr) \\& \qquad {}\times \biggl[\sup \biggl\{ \varPhi _{r\rho } (\lambda _{1},\xi _{1} ), \varPhi _{r\rho } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2},\xi _{1} \biggr), \\& \qquad\varPhi _{r\rho } \biggl(\lambda _{1},\frac{\xi _{1}+\xi _{2}}{2} \biggr), \varPhi _{r\rho } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \frac{\xi _{1}+ \xi _{2}}{2} \biggr) \biggr\} \\& \qquad {}+\sup \biggl\{ \varPhi _{r\rho } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \xi _{1} \biggr),\varPhi _{r\rho } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \frac{ \xi _{1}+\xi _{2}}{2} \biggr),\varPhi _{r\rho } (\lambda _{2},\xi _{1} ), \varPhi _{r\rho } \biggl(\lambda _{2}, \frac{\xi _{1}+\xi _{2}}{2} \biggr) \biggr\} \\& \qquad {}+\sup \biggl\{ \varPhi _{r\rho } \biggl(\lambda _{1}, \frac{\xi _{1}+\xi _{2}}{2} \biggr),\varPhi _{r\rho } (\lambda _{1},\xi _{2} ),\varPhi _{r\rho } \biggl(\frac{\lambda _{1} +\lambda _{2}}{2}, \frac{ \xi _{1}+\xi _{2}}{2} \biggr),\varPhi _{r\rho } \biggl(\frac{\lambda _{1}+ \lambda _{2}}{2}, \xi _{2} \biggr) \biggr\} \\& \qquad {}+\sup \biggl\{ \varPhi _{r\rho } \biggl(\frac{\lambda _{1}+\lambda _{2}}{2}, \frac{\xi _{1}+\xi _{2}}{2} \biggr),\varPhi _{r\rho } \biggl(\frac{ \lambda _{1}+\lambda _{2}}{2}, \xi _{2} \biggr), \\& \qquad \varPhi _{r\rho } \biggl(\lambda _{2},\frac{ \xi _{1}+\xi _{2}}{2} \biggr),\varPhi _{r\rho } (\lambda _{2},\xi _{2} ) \biggr\} \biggr]. \end{aligned}
(3.23)

### Proof

From the hypothesis of Theorem 3.8 we have

\begin{aligned} \varPsi (\omega ,\varPhi ) =&\frac{\varPhi (\lambda _{1},\xi _{1} ) +\varPhi (\lambda _{1},\xi _{2} )+\varPhi (\lambda _{2},\xi _{1} ) +\varPhi (\lambda _{2},\xi _{2} )}{4} \\ &{}+ \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\mu ,\nu )\omega (\mu ,\nu ) \,d\nu \,d\mu + \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}} ^{\xi _{2}} \varPhi (\mu ,\nu )\omega ( \mu ,\nu )\,d\nu \,d\mu \\ &{}-\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} ( \lambda _{2}-\mu ) \varPhi (\lambda _{1},\nu )\omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}- \frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}} ^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\mu -\lambda _{1} ) \varPhi (\lambda _{2},\nu ) \omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}-\frac{1}{\xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\xi _{2}-\nu ) \varPhi ( \mu , \xi _{1} ) \omega (\mu ,\nu ) \,d\nu \,d\mu \\ &{}- \frac{1}{ \xi _{2}-\xi _{1}} \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} (\nu -\xi _{1} )\varPhi ( \mu , \xi _{2} )\omega (\mu ,\nu )\,d\nu \,d\mu . \end{aligned}
(3.24)

We also observe that

\begin{aligned}& \frac{ (\varUpsilon _{1}-\lambda _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}- \xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \lambda _{1},\xi _{1},\varUpsilon _{1}, \varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{\xi _{1}}^{ \varUpsilon _{3}} \biggl\vert \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu ) \,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varrho }^{\xi _{2}} \int _{\varsigma }^{\xi _{1}} (\mu - \lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu , \nu )\,d\nu \,d\mu \biggr\vert \,d \varsigma \,d\varrho , \\& \frac{ (\lambda _{2}-\varUpsilon _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}- \xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \varUpsilon _{1},\xi _{1},\lambda _{2}, \varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{ \varUpsilon _{3}} \biggl\vert \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varrho }^{\xi _{2}} \int _{\varsigma }^{\xi _{1}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr\vert \,d \varsigma \,d\varrho , \\& \frac{ (\varUpsilon _{1}-\lambda _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \varUpsilon _{1},\xi _{1},\lambda _{2}, \varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\lambda _{1}}^{\varUpsilon _{1}} \int _{ \varUpsilon _{3}} ^{\xi _{2}} \biggl\vert \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varrho }^{\xi _{2}} \int _{\varsigma }^{\xi _{1}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr\vert \,d \varsigma \,d\varrho \end{aligned}

and

\begin{aligned}& \frac{ (\lambda _{2}-\varUpsilon _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \varUpsilon _{1}, \varUpsilon _{3}, \lambda _{2},\xi _{2};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{1}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{\varUpsilon _{1}}^{\lambda _{2}} \int _{\varUpsilon _{3}} ^{\xi _{2}} \biggl\vert \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{ \varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu - \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\varrho }^{\xi _{2}} \int _{\varsigma }^{\xi _{1}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \biggr\vert \,d \varsigma \,d\varrho . \end{aligned}

Let $$p: [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]\rightarrow \mathbb{R}$$ defined by

\begin{aligned} p (\varrho ,\varsigma ) =& \int _{\lambda _{1}}^{\varrho } \int _{\xi _{1}}^{\varsigma } (\mu -\lambda _{1} ) (\nu - \xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}- \int _{\lambda _{1}}^{\varrho } \int _{\varsigma }^{\xi _{2}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu \\ &{}- \int _{\varrho }^{\lambda _{2}} \int _{\xi _{1}} ^{\varsigma } (\mu -\lambda _{1} ) (\nu -\xi _{1} ) \omega (\mu ,\nu )\,d\nu \,d\mu \\ &{}+ \int _{\varrho }^{\xi _{2}} \int _{\varsigma }^{\xi _{1}} (\mu - \lambda _{1} ) (\nu -\xi _{1} )\omega (\mu , \nu )\,d\nu \,d\mu . \end{aligned}

Then we clearly see that

$$p_{\varrho \varsigma } (\varrho ,\varsigma )= (\lambda _{2}- \lambda _{1} ) (\xi _{2}-\xi _{1} )\omega (\varrho , \varsigma )>0$$

for $$(\varrho ,\varsigma )\in [\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$, which implies that $$p (\varrho ,\varsigma )$$ is an increasing function on $$[\lambda _{1},\lambda _{2} ]\times [\xi _{1},\xi _{2} ]$$ and

$$p (\varUpsilon _{1}, \varUpsilon _{3} )=0.$$

Now it is easy to see that

\begin{aligned}& \frac{ (\varUpsilon _{1}-\lambda _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \lambda _{1},\xi _{1},\varUpsilon _{1}, \varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{ (\lambda _{2}-\varUpsilon _{1} ) (\varUpsilon _{3}-\xi _{1} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \varUpsilon _{1},\xi _{1},\lambda _{2}, \varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{ (\varUpsilon _{1}-\lambda _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \varUpsilon _{1},\xi _{1},\lambda _{2}, \varUpsilon _{3};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad =\frac{ (\lambda _{2}-\varUpsilon _{1} ) (\xi _{2}-\varUpsilon _{3} )}{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \int _{0}^{1} \int _{0}^{1} \bigl\vert H (\omega , \varUpsilon _{1}, \varUpsilon _{3},\lambda _{2},\xi _{2};r,\rho ) \bigr\vert \,d\rho \,dr \\& \quad = \int _{\lambda _{1}}^{\frac{\lambda _{1}+\lambda _{2}}{2}} \int _{\xi _{1}} ^{\frac{\xi _{1}+\xi _{2}}{2}} (\mu -\lambda _{1} ) (\nu - \xi _{1} )\omega (\mu ,\nu )\,d\nu \,d\mu . \end{aligned}

Thus, inequality (3.23) can be derived from (3.24). □

## Applications to random variables

Let $$\alpha ,\beta \in \mathbb{R}$$, $$0<\lambda _{1}<\lambda _{2}$$, $$0<\xi _{1}<\xi _{2}$$, $$\mathcal{X}$$ and $$\mathcal{Y}$$ be two independent continuous random variables with the common continuous probability density function $$\omega :[\lambda _{1},\lambda _{2}]\times [\xi _{1},\xi _{2} ]\rightarrow {}[ 0,\infty )$$. Then the α-moment of $$\mathcal{X}$$ and β-moment of $$\mathcal{Y}$$ about the origin are, respectively, defined by

$$\mathcal{E}_{\alpha } (\mathcal{X} )= \int _{\lambda _{1}}^{\lambda _{2}}\varrho ^{\alpha }\omega _{1} (\varrho )\,d\varrho , \mathcal{E}_{\beta } (\mathcal{Y} )= \int _{\xi _{1}}^{\xi _{2}}\varsigma ^{\beta }\omega _{2} (\varsigma )\,d\varsigma$$

if both $$\mathcal{E}_{\alpha } (\mathcal{X} )$$ and $$\mathcal{E}_{\beta } (\mathcal{Y} )$$ are finite, where $$\omega _{1}:[\lambda _{1},\lambda _{2}]\rightarrow {}[ 0,\infty )$$ and $$\omega _{2}: [ \xi _{1},\xi _{2} ]\rightarrow {}[ 0, \infty )$$ are, respectively, the marginal probability density functions of $$\mathcal{X}$$ and $$\mathcal{Y}$$. Since $$\mathcal{X}$$ and $$\mathcal{Y}$$ are two independent random variables, one has

$$\omega (\varrho ,\varsigma )=\omega _{1} (\varrho ) \omega _{2} (\varsigma )$$

for all $$(\varrho ,\varsigma )\in [\lambda _{1},\lambda _{2}]\times [\xi _{1},\xi _{2} ]$$, and

\begin{aligned} \mathcal{E}_{\alpha ,\beta } ( \mathcal{XY} ) =& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varrho ^{\alpha }\varsigma ^{\beta } \omega (\varrho ,\varsigma )\,d\varsigma \,d\varrho \\ =& \int _{\lambda _{1}}^{\lambda _{2}}\varrho ^{\alpha }\omega _{1} (\varrho )\,d\varrho \int _{\xi _{1}}^{\xi _{2}} \varsigma ^{\beta }\omega _{2} (\varsigma )\,d\varsigma = \mathcal{E}_{\alpha } ( \mathcal{X} )\mathcal{E}_{\beta } (\mathcal{Y} ). \end{aligned}

Making use of the above notations, we clearly see that

\begin{aligned}& B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\mathcal{E} ( \mathcal{X} )} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} )-\nu \bigr) ^{2}\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} ) -\nu \bigr)^{2}\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} )-\nu \bigr) ^{2}\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} )- \nu \bigr)^{2} \omega (\mu ,\nu )\,d\nu \,d\mu , \\& B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} ) -\nu \bigr) ^{2}\omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} )- \nu \bigr)^{2} \omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} )-\nu \bigr) ^{2}\omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} ) -\nu \bigr)^{2}\omega (\mu ,\nu )\,d\nu \,d\mu , \\& B_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} ) - \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} )-\nu \bigr) \omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} ) -\nu \bigr)\omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} )-\nu \bigr) \omega ( \mu ,\nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr)^{2} \bigl(\mathcal{E} (\mathcal{Y} ) -\nu \bigr)\omega (\mu ,\nu )\,d\nu \,d\mu \end{aligned}

and

\begin{aligned}& B_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} ) -\nu \bigr) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\lambda _{1}}^{\mathcal{E} (\mathcal{X} )} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} ) - \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} )- \nu \bigr) \omega (\mu ,\nu )\,d\nu \,d\mu \\& \qquad {}- \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\xi _{1}}^{\mathcal{E} (\mathcal{Y} )} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) (\varUpsilon _{3}-\nu ) \omega (\mu , \nu )\,d\nu \,d\mu \\& \qquad {}+ \int _{\mathcal{E} (\mathcal{X} )}^{\lambda _{2}} \int _{\mathcal{E} (\mathcal{Y} )}^{\xi _{2}} \bigl(\mathcal{E} (\mathcal{X} )- \mu \bigr) \bigl(\mathcal{E} (\mathcal{Y} )- \nu \bigr) \omega (\mu ,\nu )\,d\nu \,d\mu . \end{aligned}

Moreover, we have

\begin{aligned}& A_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\lambda _{2}-\mathcal{E} (\mathcal{X} ) )}{2}B _{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ \frac{ (\xi _{2}-\mathcal{E} (\mathcal{Y} ) )}{2}B _{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \bigl(\lambda _{2}-\mathcal{E} (\mathcal{X} ) \bigr) \bigl(\xi _{2}- \mathcal{E} (\mathcal{Y} ) \bigr)B_{4} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ), \\& A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\lambda _{2}-\mathcal{E} (\mathcal{X} ) )}{2}B _{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ \frac{ (\mathcal{E} (\mathcal{Y} )-\xi _{1} )}{2}B _{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \bigl(\lambda _{2}-\mathcal{E} (\mathcal{X} ) \bigr) \bigl(\mathcal{E} (\mathcal{Y} )-\xi _{1} \bigr)B_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ), \\& A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\mathcal{E} (\mathcal{X} )-\lambda _{1} )}{2}B _{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ \frac{ (\xi _{2}-\mathcal{E} (\mathcal{Y} ) )}{2}B _{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \bigl( \mathcal{E} (\mathcal{X} )-\lambda _{1} \bigr) \bigl(\xi _{2}- \mathcal{E} (\mathcal{Y} ) \bigr)B_{4} (\lambda _{1}, \xi _{1},\lambda _{2},\xi _{2} ) \end{aligned}

and

\begin{aligned}& A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \quad = \frac{1}{4}B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \frac{ (\mathcal{E} (\mathcal{X} )-\lambda _{1} )}{2} B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+ \frac{ (\mathcal{E} (\mathcal{Y} )-\xi _{1} )}{2}B _{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \bigl( \mathcal{E} (\mathcal{X} )-\lambda _{1} \bigr) \bigl(\mathcal{E} ( \mathcal{Y} )-\xi _{1} \bigr)B_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ). \end{aligned}

Now, we give an application for our obtained results to the random variables.

### Theorem 4.1

The inequality

\begin{aligned}& \bigl\vert \mathcal{E}_{\alpha } (\mathcal{X} ) \mathcal{E}_{\beta } (\mathcal{Y} )- \bigl[\mathcal{E} (\mathcal{X} ) \bigr]^{\alpha } \mathcal{E}_{\beta } (\mathcal{Y} )-\mathcal{E}_{\alpha } ( \mathcal{X} ) \bigl[\mathcal{E} (\mathcal{Y} ) \bigr]^{\beta }+ \bigl[ \mathcal{E} (\mathcal{X} ) \bigr]^{\alpha } \bigl[\mathcal{E} (\mathcal{Y} ) \bigr] ^{\beta } \bigr\vert \\& \quad \leq \frac{\alpha \beta }{ (\lambda _{2}-\lambda _{1} ) (\xi _{2}-\xi _{1} )} \bigl[\lambda _{1}^{\alpha -1}\xi _{1}^{\beta -1} \varUpsilon _{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) + \lambda _{1}^{\alpha -1}\xi _{2}^{\beta -1}A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \\& \qquad {}+\lambda _{2}^{\alpha -1}\xi _{1}^{\beta -1}A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) +\lambda _{2}^{\alpha -1}\xi _{2}^{\beta -1}A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} ) \bigr] \end{aligned}
(4.1)

holds for $$0<\lambda _{1}<\lambda _{2}$$, $$0<\xi _{1}<\xi _{2}$$and $$\alpha ,\beta \geq 2$$, where $$\varUpsilon _{1} (\lambda _{1},\xi _{1}, \lambda _{2},\xi _{2} )$$, $$A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )$$, $$A_{3} (\lambda _{1},\xi _{1},\lambda _{2}, \xi _{2} )$$and $$A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )$$are defined by the previously shown equations.

### Proof

Let $$\alpha ,\beta \geq 2$$, $$\varPhi (\varrho ,\varsigma )=\varrho ^{ \alpha }\varsigma ^{\beta }$$ be defined on $$[\lambda _{1},\lambda _{2}] \times [\xi _{1},\xi _{2} ]$$. Then we clearly see that $$\vert \varPhi _{\varrho \varsigma } (\varrho ,\varsigma ) \vert = \alpha \beta \varrho ^{\alpha -1}\varsigma ^{\beta -1}$$ is co-ordinated convex on $$[\lambda _{1},\lambda _{2}]\times [\xi _{1},\xi _{2} ]$$ and

\begin{aligned}& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\varrho ,\varsigma ) \omega (\varrho ,\varsigma )\,d\varsigma \,d\varrho \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}}\varrho ^{\alpha }\varsigma ^{\beta } \omega _{1} (\varrho ) \omega _{2} ( \varsigma )\,d\varsigma \,d\varrho \\& \quad = \int _{\lambda _{1}}^{\lambda _{2}}\varrho ^{\alpha }\omega _{1} (\varrho )\,d\varrho \int _{\xi _{1}}^{\xi _{2}} \varsigma ^{\beta }\omega _{2} (\varsigma )\,d\varsigma = \mathcal{E}_{\alpha } ( \mathcal{X} )\mathcal{E}_{\beta } (\mathcal{Y} ), \\& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\varUpsilon _{1}, \varrho )\omega (\varrho , \varsigma )\,d\varsigma \,d\varrho = \bigl[\mathcal{E} (\mathcal{X} ) \bigr] ^{\alpha }\mathcal{E}_{\beta } (\mathcal{Y} ), \\& \int _{\lambda _{1}}^{\lambda _{2}} \int _{\xi _{1}}^{\xi _{2}} \varPhi (\varrho ,\varUpsilon _{3} )\omega (\varrho , \varsigma )\,d\varsigma \,d\varrho = \mathcal{E}_{\alpha } (\mathcal{X} ) \bigl[\mathcal{E} (\mathcal{Y} ) \bigr]^{\beta } \end{aligned}

and

$$\varPhi (\varUpsilon _{1},\varUpsilon _{3} )= \bigl[\mathcal{E} (\mathcal{X} ) \bigr] ^{\alpha } \bigl[\mathcal{E} (\mathcal{Y} ) \bigr]^{ \beta }.$$

Therefore, inequality (4.1) follows from (3.5) and the above identities. □

Next, we provide an example to support our obtained results.

### Example 4.2

Let

$$\varPhi ( \mu ,\nu ) = \textstyle\begin{cases} \frac{4}{9}\mu \nu , & 1\leq \mu ,\nu \leq 2, \\ 0, & \text{otherwise}, \end{cases}$$

be the joint probability density function of the random variables $$\mathcal{X}$$ and $$\mathcal{Y}$$. Then the marginal probability density functions of the random variables $$\mathcal{X}$$ and $$\mathcal{Y}$$ are given by

$$\mathcal{G} ( \mu ) = \textstyle\begin{cases} \frac{2}{3}\mu , & 1\leq \mu \leq 2, \\ 0, & \text{otherwise}, \end{cases}$$

and

$$\mathcal{H} ( \nu ) = \textstyle\begin{cases} \frac{2}{3}\nu , & 1\leq \nu \leq 2, \\ 0, & \text{otherwise}, \end{cases}$$

respectively, and the random variables $$\mathcal{X}$$ and $$\mathcal{Y}$$ are independent due to $$\varPhi (\mu ,\nu )=\mathcal{G} (\mu )\times \mathcal{H} (\nu )$$. Elaborate computations give

\begin{aligned}& \mathcal{E} (\mathcal{X} )=\mathcal{E} (\mathcal{Y} )= \frac{14}{9}, \\& B_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{597 \text{,}529}{13\text{,} 947\text{,}137\text{,}604}, \\& B_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{618 \text{,}400}{387\text{,}420\text{,}489}, \\& B_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{618 \text{,}400}{387\text{,}420\text{,}489}, \end{aligned}

and

$$B_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{2\text{,}560 \text{,}000}{43\text{,}046\text{,}721}.$$

Hence,

\begin{aligned}& A_{1} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{77 \text{,}281\text{,}681}{6\text{,}198\text{,}727\text{,}824}, \\& A_{2} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{288 \text{,}107\text{,}443}{18\text{,} 596\text{,}183\text{,}472}, \\& A_{3} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{288 \text{,}107\text{,}443}{18\text{,} 596\text{,}183\text{,}472}, \end{aligned}

and

$$A_{4} (\lambda _{1},\xi _{1},\lambda _{2},\xi _{2} )=\frac{1\text{,}074 \text{,}069\text{,}529}{55\text{,} 788\text{,}550\text{,}416}.$$

Therefore, it follows from (4.1) that

\begin{aligned}& \biggl\vert \frac{4}{9} \biggl( \int _{1}^{2}\mu ^{\alpha +1}\,d\mu \biggr) \biggl( \int _{1}^{2}\mu ^{\beta +1}\,d\mu \biggr)- \frac{2}{3} \biggl(\frac{14}{9} \biggr)^{\alpha } \int _{1}^{2} \mu ^{\beta +1}\,d\mu \\& \qquad {}- \frac{2}{3} \biggl(\frac{14}{9} \biggr)^{\beta } \int _{1}^{2}\mu ^{\alpha +1}\,d\mu + \biggl(\frac{14}{9} \biggr) ^{\alpha +\beta } \biggr\vert \\& \quad \leq \alpha \beta \biggl[\frac{77\text{,}281\text{,}681}{6\text{,}198\text{,}727\text{,}824}+ \bigl(2^{ \beta -1}+2^{\alpha -1} \bigr) \times \frac{288\text{,}107\text{,}443}{18\text{,} 596\text{,}183\text{,}472} \\& \qquad {}+2^{\alpha +\beta -2}\times \frac{1\text{,}074 \text{,}069\text{,}529}{55\text{,} 788\text{,}550\text{,}416} \biggr] \end{aligned}

for $$\alpha ,\beta \geq 2$$, that is,

\begin{aligned}& \biggl\vert \frac{ [2 (9^{\alpha }-2^{2+\alpha }\times 9^{ \alpha }+3\times 14^{\alpha } )+3\times 14^{\alpha }\times \alpha ] [2 (9^{\beta }-2^{2+\beta }\times 9^{ \beta }+3\times 14^{\beta } )+3\times 14^{\beta }\times \beta ]}{9^{1+ \alpha +\beta }(2+\alpha )(2+\beta )} \biggr\vert \\& \quad \leq \frac{\alpha \beta (32\text{,}773\times 2^{\alpha }+52\text{,}746 ) (32\text{,}773\times 2^{\beta }+52\text{,}746 )}{223\text{,} 154\text{,}201\text{,}664} \end{aligned}

for $$\alpha ,\beta \geq 2$$.

## Conclusion

In the article, we have established several weighted Hermite–Hadamard type inequalities for the co-ordinated convex and quasi-convex functions, provided an application to the moment of continuous random variables of bivariate distribution functions in the probability theory and presented an example on the probability distribution to support our results. Our results are generalizations of some previous results, and our approach may have further applications in the theory of convexity and Hermite–Hadamard inequality.

## References

1. Shi, H.-P., Zhang, H.-Q.: Existence of gap solitons in periodic discrete nonlinear Schrödinger equations. J. Math. Anal. Appl. 361(2), 411–419 (2010)

2. Zhou, W.-J., Zhang, L.: Convergence of a regularized factorized quasi-Newton method for nonlinear least squares problems. Comput. Appl. Math. 29(2), 195–214 (2010)

3. Yang, X.-S., Zhu, Q.-X., Huang, C.-X.: Generalized lag-synchronization of chaotic mix-delayed systems with uncertain parameters and unknown perturbations. Nonlinear Anal., Real World Appl. 12(1), 93–105 (2011)

4. Zhu, Q.-X., Huang, C.-X., Yang, X.-S.: Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays. Nonlinear Anal. Hybrid Syst. 5(1), 52–77 (2011)

5. Dai, Z.-F., Wen, F.-H.: A modified CG-DESCENT method for unconstrained optimization. J. Comput. Appl. Math. 235(11), 3332–3341 (2011)

6. Gou, K., Sun, B.: Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions. Appl. Math. Comput. 217(21), 8765–8777 (2011)

7. Huang, C.-X., Liu, L.-Z.: Sharp function inequalities and boundness for Toeplitz type operator related to general fractional singular integral operator. Publ. Inst. Math. 92(106), 165–176 (2012)

8. Zhang, L., Jian, S.-Y.: Further studies on the Wei–Yao–Liu nonlinear conjugate gradient method. Appl. Math. Comput. 219(14), 7616–7621 (2013)

9. Li, X.-F., Tang, G.-J., Tang, B.-Q.: Stress field around a strike-slip fault in orthotropic elastic layers via a hypersingular integral equation. Comput. Math. Appl. 66(11), 2317–2326 (2013)

10. Huang, C.-X., Guo, S., Liu, L.-Z.: Boundedness on Morrey space for Toeplitz type operator associated to singular integral operator with variable Calderón–Zygmund kernel. J. Math. Inequal. 8(3), 453–464 (2014)

11. Huang, C.-X., Yang, Z.-C., Yi, T.-S., Zou, X.-F.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014)

12. Xie, D.-X., Li, J.: A new analysis of electrostatic free energy minimization and Poisson–Boltzmann equation for protein in ionic solvent. Nonlinear Anal., Real World Appl. 21, 185–196 (2015)

13. Liu, Y.-C., Wu, J.: Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations. Adv. Differ. Equ. 2015, Article ID 379 (2015)

14. Zhou, W.-J., Wang, F.: A PRP-based residual method for large-scale monotone nonlinear equations. Appl. Math. Comput. 261, 1–7 (2015)

15. Dai, Z.-F.: Comments on a new class of nonlinear conjugate gradient coefficients with global convergence properties. Appl. Math. Comput. 276, 297–300 (2016)

16. Li, J.-L., Sun, G.-Y., Zhang, R.-M.: The numerical solution of scattering by infinite rough interfaces based on the integral equation method. Comput. Math. Appl. 71(7), 1491–1502 (2016)

17. Hu, H.-J., Liu, L.-Z.: Weighted inequalities for a general commutator associated to a singular integral operator satisfying a variant of Hörmander’s condition. Math. Notes 101(5–6), 830–840 (2017)

18. Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Generalized Lyapunov–Razumikhin method for retarded differential inclusions: applications to discontinuous neural networks. Discrete Contin. Dyn. Syst., Ser. B 22(9), 3591–3614 (2017)

19. Wang, W.-S.: On A-stable one-leg methods for solving nonlinear Volterra functional differential equations. Appl. Math. Comput. 314, 380–390 (2017)

20. Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017)

21. Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018)

22. Wu, S.-H., Chu, Y.-M.: Schur m-power convexity of generalized geometric Bonferroni mean involving three parameters. J. Inequal. Appl. 2019, Article ID 57 (2019)

23. Qian, W.-M., He, Z.-Y., Zhang, H.-W., Chu, Y.-M.: Sharp bounds for Neuman means in terms of two-parameter contraharmonic and arithmetic mean. J. Inequal. Appl. 2019, Article ID 168 (2019)

24. Zhao, T.-H., Zhou, B.-C., Wang, M.-K., Chu, Y.-M.: On approximating the quasi-arithmetic mean. J. Inequal. Appl. 2019, Article ID 42 (2019)

25. Wang, J.-L., Qian, W.-M., He, Z.-Y., Chu, Y.-M.: On approximating the Toader mean by other bivariate means. J. Funct. Spaces 2019, Article ID 6082413 (2019)

26. Peng, J., Zhang, Y.: Heron triangles with figurate number sides. Acta Math. Hung. 157(2), 478–488 (2019)

27. Tian, Z.-L., Liu, Y., Zhang, Y., Liu, Z.-Y., Tian, M.-Y.: The general inner-outer iteration method based on regular splittings for the PageRank problem. Appl. Math. Comput. 356, 479–501 (2019)

28. Wang, W.-S., Chen, Y.-Z., Fang, H.: On the variable two-step IMEX BDF method for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J. Numer. Anal. 57(3), 1289–1317 (2019)

29. Wang, M.-K., Zhang, W., Chu, Y.-M.: Monotonicity, convexity and inequalities involving the generalized elliptic integrals. Acta Math. Sci. Ser. B Engl. Ed. 39(5), 1440–1450 (2019)

30. Wang, M.-K., Chu, H.-H., Chu, Y.-M.: Precise bounds for the weighted Hölder mean of the complete p-elliptic integrals. J. Math. Anal. Appl. (2019). https://doi.org/10.1016/j.jmaa.2019.123388

31. Chu, Y.-M., Wang, G.-D., Zhang, X.-H.: The Schur multiplicative and harmonic convexities of the complete symmetric function. Math. Nachr. 284(5–6), 653–663 (2011)

32. Chu, Y.-M., Xia, W.-F., Zhang, X.-H.: The Schur concavity, Schur multiplicative and harmonic convexities of the second dual form of the Hamy symmetric function with applications. J. Multivar. Anal. 105, 412–421 (2012)

33. Adil Khan, M., Hanif, M., Khan, Z.A., Ahmad, K., Chu, Y.-M.: Association of Jensen’s inequality for s-convex function with Csiszár divergence. J. Inequal. Appl. 2019, Article ID 162 (2019)

34. Song, Y.-Q., Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: Integral inequalities involving strongly convex functions. J. Funct. Spaces 2018, Article ID 6595921 (2018)

35. Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: Majorization theorems for strongly convex functions. J. Inequal. Appl. 2019, Article ID 58 (2019)

36. Zaheer Ullah, S., Adil Khan, M., Khan, Z.A., Chu, Y.-M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, Article ID 9487823 (2019)

37. Adil Khan, M., Zaheer Ullah, S., Chu, Y.-M.: The concept of coordinate strongly convex functions and related inequalities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2235–2251 (2019)

38. Adik Khan, M., Wu, S.-H., Ullah, H., Chu, Y.-M.: Discrete majorization type inequalities for convex functions on rectangles. J. Inequal. Appl. 2019, Article ID 16 (2019)

39. Qian, W.-M., Yang, Y.-Y., Zhang, H.-W., Chu, Y.-M.: Optimal two-parameter geometric and arithmetic mean bounds for the Sándor–Yang mean. J. Inequal. Appl. 2019, Article ID 287 (2019)

40. Zaheer Ullah, S., Adil Khan, M., Chu, Y.-M.: A note on generalized convex functions. J. Inequal. Appl. 2019, Article ID 291 (2019)

41. Lin, L., Liu, Z.-Y.: An alternating projected gradient algorithm for nonnegative matrix factorization. Appl. Math. Comput. 217(24), 9997–10002 (2011)

42. Zhang, L., Li, J.-L.: A new globalization technique for nonlinear conjugate gradient methods for nonconvex minimization. Appl. Math. Comput. 217(24), 10295–10304 (2011)

43. Xiao, C.-E., Liu, J.-B., Liu, Y.-L.: An inverse pollution problem in porous media. Appl. Math. Comput. 218(7), 3649–3653 (2011)

44. Zhou, W.-J.: On the convergence of the modified Levenberg–Marquardt method with a nonmonotone second order Armijo type line search. J. Comput. Appl. Math. 239, 152–161 (2013)

45. Qin, G.-X., Huang, C.X., Xie, Y.-Q., Wen, F.-H.: Asymptotic behavior for third-order quasi-linear differential equations. Adv. Differ. Equ. 2013, Article ID 305 (2013)

46. Zhou, W.-J., Chen, X.-L.: On the convergence of a modified regularized Newton method for convex optimization with singular solutions. J. Comput. Appl. Math. 239, 179–188 (2013)

47. Wang, M.-K., Chu, Y.-M.: Asymptotical bounds for complete elliptic integrals of the second kind. J. Math. Anal. Appl. 402(1), 119–126 (2013)

48. Wang, G.-D., Zhang, X.-H., Chu, Y.-M.: A power mean inequality involving the complete elliptic integrals. Rocky Mt. J. Math. 44(5), 1661–1667 (2014)

49. Tang, W.-S., Sun, Y.-J.: Construction of Runge–Kutta type methods for solving ordinary differential equations. Appl. Math. Comput. 234, 179–191 (2014)

50. Dai, Z.-F., Chen, X.-H., Wen, F.-H.: A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations. Appl. Math. Comput. 270, 378–386 (2015)

51. Fang, X.-P., Deng, Y.-J., Li, J.: Plasmon resonance and heat generation in nanostructures. Math. Methods Appl. Sci. 38(18), 4663–4672 (2015)

52. Tan, Y.-X., Jing, K.: Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations. Math. Methods Appl. Sci. 39(11), 2821–2839 (2016)

53. Wang, M.-K., Chu, Y.-M., Jiang, Y.-P.: Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mt. J. Math. 46(2), 679–691 (2016)

54. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: Monotonicity rule for the quotient of two functions and its application. J. Inequal. Appl. 2017, Article ID 106 (2017)

55. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On rational bounds for the gamma function. J. Inequal. Appl. 2017, Article ID 210 (2017)

56. Duan, L., Huang, C.-X.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)

57. Duan, L., Huang, L.-H., Guo, Z.-Y., Fang, X.-W.: Periodic attractor for reaction–diffusion high-order Hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2), 233–245 (2017)

58. Wang, W.-S., Chen, Y.-Z.: Fast numerical valuation of options with jump under Merton’s model. J. Comput. Appl. Math. 318, 79–92 (2017)

59. Huang, C.-X., Liu, L.-Z.: Boundedness of multilinear singular integral operator with a non-smooth kernel and mean oscillation. Quaest. Math. 40(3), 295–312 (2017)

60. Tan, Y.-X., Huang, C.-X., Sun, B., Wang, T.: Dynamics of a class of delayed reaction–diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)

61. Tang, W.-S., Zhang, J.-J.: Symplecticity-preserving continuous-stage Runge–Kutta–Nyström methods. Appl. Math. Comput. 323, 204–219 (2018)

62. Duan, L., Fang, X.-W., Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(5), 1954–1965 (2018)

63. Liu, Z.-Y., Wu, N.-C., Qin, X.-R., Zhang, Y.-L.: Trigonometric transform splitting methods for real symmetric Toeplitz systems. Comput. Math. Appl. 75(8), 2782–2794 (2018)

64. Huang, T.-R., Han, B.-W., Ma, X.-Y., Chu, Y.-M.: Optimal bounds for the generalized Euler–Mascheroni constant. J. Inequal. Appl. 2018, Article ID 118 (2018)

65. Huang, T.-R., Tan, S.-Y., Ma, X.-Y., Chu, Y.-M.: Monotonicity properties and bounds for the complete p-elliptic integrals. J. Inequal. Appl. 2018, Article ID 239 (2018)

66. Yang, Z.-H., Qian, W.-M., Chu, Y.-M.: Monotonicity properties and bounds involving the complete elliptic integrals of the first kind. Math. Inequal. Appl. 21(4), 1185–1199 (2018)

67. Yang, Z.-H., Chu, Y.-M., Zhang, W.: High accuracy asymptotic bounds for the complete elliptic integral of the second kind. Appl. Math. Comput. 348, 552–564 (2019)

68. Qiu, S.-L., Ma, X.-Y., Chu, Y.-M.: Sharp Landen transformation inequalities for hypergeometric functions, with applications. J. Math. Anal. Appl. 474(2), 1306–1337 (2019)

69. Wang, M.-K., Chu, Y.-M., Zhang, W.: Monotonicity and inequalities involving zero-balanced hypergeometric function. Math. Inequal. Appl. 22(2), 601–617 (2019)

70. Wang, M.-K., Chu, Y.-M., Zhang, W.: Precise estimates for the solution of Ramanujan’s generalized modular equation. Ramanujan J. 49(3), 653–668 (2019)

71. Dragomir, S.S.: On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwan. J. Math. 5(4), 775–788 (2001)

72. Latif, M.A., Alomari, M.: Hadamard-type inequalities for product two convex functions on the co-ordinates. Int. Math. Forum 4(45–48), 2327–2338 (2009)

73. Latif, M.A., Dragomir, S.S., Momoniat, E.: Weighted generalization of some integral inequalities for differentiable co-ordinated convex functions. Politeh. Univ. Buchar. Sci. Bull. Ser. A Appl. Math. Phys. 78(4), 197–210 (2016)

74. Özdemir, M.E., Akdemir, A.O., Yıldız, Ç.: On co-ordinated quasi-convex functions. Czechoslov. Math. J. 62(137)(4), 889–900 (2012)

75. Latif, M.A., Hussian, S., Dragomir, S.S.: Refinements of Hermite–Hadamard type inequalities for co-ordinated quasi-convex function. Int. J. Math. Arch. 3(1), 161–171 (2012)

76. Zhang, X.-M., Chu, Y.-M., Zhang, X.-H.: The Hermite–Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl. 2010, Article ID 507560 (2010)

77. Chu, Y.-M., Wang, M.-K., Qiu, S.-L.: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc. Indian Acad. Sci. Math. Sci. 122(1), 41–51 (2012)

78. Hu, H.-J., Zou, X.-F.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763–4771 (2017)

79. Adil Khan, M., Chu, Y.-M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15(1), 1414–1430 (2017)

80. Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R., Ali, G.: Conformable fractional integrals versions of Hermite–Hadamard inequalities and their generalizations. J. Funct. Spaces 2018, Article ID 6928130 (2018)

81. Adil Khan, M., Iqbal, A., Suleman, M., Chu, Y.-M.: Hermite–Hadamard type inequalities for fractional integrals via Green’s function. J. Inequal. Appl. 2018, Article ID 161 (2018)

82. Adil Khan, M., Khurshid, Y., Du, T.-S., Chu, Y.-M.: Generalization of Hermite–Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, Article ID 5357463 (2018)

83. Huang, C.-X., Qiao, Y.-C., Huang, L.-H., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, Article ID 186 (2018)

84. Cai, Z.-W., Huang, J.-H., Huang, L.-H.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146(11), 4667–4682 (2018)

85. Wang, J.-F., Chen, X.-Y., Huang, L.-H.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)

86. Wang, J.-F., Huang, C.-X., Huang, L.-H.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)

87. Jiang, Y.-J., Xu, X.-J.: A monotone finite volume method for time fractional Fokker–Planck equations. Sci. China Math. 62(4), 783–794 (2019)

88. Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Improvements of bounds for the Sándor–Yang means. J. Inequal. Appl. 2019, Article ID 73 (2019)

89. He, X.-H., Qian, W.-M., Xu, H.-Z., Chu, Y.-M.: Sharp power mean bounds for two Sándor–Yang means. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2627–2638 (2019)

90. Khurshid, Y., Adil Khan, M., Chu, Y.-M.: Conformable integral inequalities of the Hermite–Hadamard type in terms of GG- and GA-convexities. J. Funct. Spaces 2019, Article ID 6926107 (2019)

91. Khurshid, Y., Adil Khan, M., Chu, Y.-M., Khan, Z.A.: Hermite–Hadamard–Fejér inequalities for conformable fractional integrals via preinvex functions. J. Funct. Spaces 2019, Article ID 3146210 (2019)

92. Huang, C.-X., Zhang, H., Cao, J.-D., Hu, H.-J.: Stability and Hopf bifurcation of a delayed prey–predator model with disease in the predator. Int. J. Bifurc. Chaos Appl. Sci. Eng. 29(7), Article ID 1950091 (2019)

93. Huang, C.-X., Liu, B.-W., Tian, X.-M., Yang, L.-S., Zhang, X.-X.: Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions. Neural Process. Lett. 49(2), 625–641 (2019)

94. Huang, C.-X., Liu, B.-W.: New studies on dynamic analysis of inertial neural networks involving non-reduced order method. Neurocomputing 325(24), 283–287 (2019). https://doi.org/10.1016/j.neucom.2018.09.065

95. Rashid, S., Noor, M.A., Noor, K.I., Safdar, F., Chu, Y.-M.: Hermite–Hadamard type inequalities for the class of convex functions on time scale. Mathematics 7, Article ID 956 (2019). https://doi.org/10.3390/math7100956

96. Huang, C.-X., Zhang, H.: Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method. Int. J. Biomath. 12(2), Article ID 1950016 (2019)

### Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

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## Funding

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485).

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### Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Yu-Ming Chu.

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### Competing interests

The authors declare that they have no competing interests. 