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

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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 [130]. Recently, the generalizations, extensions, variants and refinements for the convexity (concavity) have attracted the interest of several researchers [3140]. In particular, many remarkable inequalities and properties in many branches of mathematics can be found in the literature [4170] 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 [71] introduced the definition of the bivariate co-ordinated convex function as follows.

Definition 1.1

(See [71])

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 [72] 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 [71] 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 [71])

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 [73], 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 [73])

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 [73])

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. [74] generalized the co-ordinated convex function to the co-ordinated quasi-convex function.

Definition 1.5

(See [74])

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 [74])

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

Definition 1.7

(See [75])

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 [74] 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. [75].

Theorem 1.8

(See [75])

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 [7696].

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 [75].

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 [75].

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.

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

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

Correspondence to Yu-Ming Chu.

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Latif, M.A., Rashid, S., Dragomir, S.S. et al. Hermite–Hadamard type inequalities for co-ordinated convex and qausi-convex functions and their applications. J Inequal Appl 2019, 317 (2019). https://doi.org/10.1186/s13660-019-2272-7

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MSC

  • 26D15
  • 26D20
  • 26D07

Keywords

  • Hermite–Hadamard inequality
  • Co-ordinated convex function
  • Co-ordinated quasi-convex function
  • Hölder integral inequality
  • Moment of random variable