1 Introduction

In the history of calculus development, integral inequalities have been thought of as a key factor in the theory of differential and integral equations. The study of various types of integral inequalities has been in the focus of great attention of a number of scientists interested in both pure and applied mathematics for more than a century. One of the many fundamental mathematical discoveries of A. M. Ostrowski [15] is the following classical integral inequality associated with the differentiable mappings:

Let \(\mathcal{F}:[\rho _{1},\rho _{2}]\mathbb{\rightarrow R}\) be a differentiable mapping on \((\rho _{1},\rho _{2})\) whose derivative \(\mathcal{F}^{{\prime }}:(\rho _{1},\rho _{2})\mathbb{\rightarrow R}\) is bounded on \((\rho _{1},\rho _{2})\), i.e., \(\Vert \mathcal{F}^{\prime } \Vert _{\infty }= \underset{\psi \in (\rho _{1},\rho _{2})}{\overset{}{\sup }} \vert \mathcal{F}^{\prime }(\psi ) \vert <\infty \). Then, the inequality holds:

$$ \biggl\vert \mathcal{F}(\kappa )-\frac{1}{\rho _{2}-\rho _{1}} \int _{\rho _{1}}^{\rho _{2}}\mathcal{F}(\psi )\,d\psi \biggr\vert \leq \biggl[ \frac{1}{4}+ \frac{ ( \kappa -\frac{\rho _{1}+\rho _{2}}{2} ) ^{2}}{(\rho _{2}-\rho _{1})^{2}} \biggr] (\rho _{2}- \rho _{1}) \bigl\Vert \mathcal{F}^{\prime } \bigr\Vert _{\infty }, $$

for all \(\kappa \in {}[ \rho _{1},\rho _{2}]\). The constant \(\frac{1}{4} \) is the best possible.

The other important fundamental result, the Hermite–Hadamard inequality discovered by C. Hermite and J. Hadamard (see, e.g., [7], [18, p. 137]), is one of the most well-established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that if \(\mathcal{F}:I\rightarrow \mathbb{R}\) is a convex function on the interval I of real numbers and \(\rho _{1},\rho _{2}\in I\) with \(\rho _{1}<\rho _{2}\), then

$$ \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2} \biggr) \leq \frac{1}{\rho _{2}-\rho _{1}} \int _{\rho _{1}}^{\rho _{2}} \mathcal{F}(\kappa )\,d\kappa \leq \frac{\mathcal{F} ( \rho _{1} ) +\mathcal{F} ( \rho _{2} ) }{2}. $$

Both inequalities hold in the reversed direction if \(\mathcal{F}\) is concave. We note that the Hermite–Hadamard inequality may be regarded as a refinement of the concept of convexity, and it follows easily from Jensen’s inequality. The Hermite–Hadamard inequality for convex functions has received renewed attention in recent years, and a remarkable variety of refinements and generalizations have been studied.

A formal definition for co-ordinated convex function may be stated as follows:

Definition 1

A function \(\mathcal{F}:\Delta := [ \rho _{1},\rho _{2} ] \times [ \rho _{3},\rho _{4} ] \rightarrow \mathbb{R} \) is called co-ordinated convex on Δ, for all \((\kappa ,u),(\gamma ,v)\in \Delta \) and \(\psi ,\varphi \in {}[ 0,1]\), if it satisfies the following inequality:

$$\begin{aligned}& \mathcal{F}\bigl(\psi \kappa +(1-\psi )\gamma ,\varphi u+(1-\varphi )v\bigr) \\& \quad \leq \psi \varphi \mathcal{F}(\kappa ,u)+\psi (1-\varphi ) \mathcal{F}(\kappa ,v)+\varphi (1-\psi )\mathcal{F}(\gamma ,u)+(1-\psi ) (1- \varphi )\mathcal{F}(\gamma ,v). \end{aligned}$$
(1.1)

The mapping \(\mathcal{F}\) is a co-ordinated concave on Δ if inequality (1.1) holds in reversed direction for all \(\psi ,\varphi \in {}[ 0,1]\) and \((\kappa ,u),(\gamma ,v)\in \Delta \).

Barnet and Dragomir gave the following Ostrowski type inequalities for double integrals in [5].

Theorem 1

Let \(\mathcal{F}:\Delta := [ \rho _{1},\rho _{2} ] \times [ \rho _{3},\rho _{4} ] \rightarrow \mathbb{R} \) be continuous on Δ, \(\mathcal{F}_{\kappa ,\gamma }^{\prime \prime }= \frac{\partial ^{2}\mathcal{F}}{\partial \kappa \partial \gamma }\) exists on \(( \rho _{1},\rho _{2} ) \times ( \rho _{3},\rho _{4} ) \) and is bounded, i. e.,

$$ \bigl\Vert \mathcal{F}_{\kappa ,\gamma }^{\prime \prime } \bigr\Vert _{\infty }=\sup_{ ( \kappa ,\gamma ) \in ( \rho _{1},\rho _{2} ) \times ( \rho _{3},\rho _{4} ) } \biggl\vert \frac{\partial ^{2}\mathcal{F}(\kappa ,\gamma )}{\partial \kappa \partial \gamma } \biggr\vert < \infty . $$

Then, we have the inequality:

$$\begin{aligned}& \biggl\vert \int _{\rho _{1}}^{\rho _{2}} \int _{ \rho _{3}}^{\rho _{4}}\mathcal{F}(\psi ,\varphi )\,d\varphi \,d \psi - ( \rho _{2}-\rho _{1} ) ( \rho _{4}-\rho _{3} ) \mathcal{F}(\kappa ,\gamma ) \\& \qquad {}- \biggl[ ( \rho _{2}-\rho _{1} ) \int _{ \rho _{3}}^{\rho _{4}}\mathcal{F}(\kappa ,\varphi )\,d\varphi + ( \rho _{4}-\rho _{3} ) \int _{\rho _{1}}^{\rho _{2}} \mathcal{F}(\psi ,\gamma )\,d\psi \biggr] \biggr\vert \\& \quad \leq \biggl[ \frac{1}{4} ( \rho _{2}-\rho _{1} ) ^{2}+ \biggl( \kappa -\frac{\rho _{1}+\rho _{2}}{2} \biggr) ^{2} \biggr] \biggl[ \frac{1}{4} ( \rho _{4}-\rho _{3} ) ^{2}+ \biggl( \gamma - \frac{\rho _{3}+\rho _{4}}{2} \biggr) ^{2} \biggr] \bigl\Vert \mathcal{F}_{\kappa ,\gamma }^{\prime \prime } \bigr\Vert _{\infty }, \end{aligned}$$
(1.2)

for all \(( \kappa ,\gamma ) \in\Delta \).

In [6], Dragomir proved the following inequalities which are the Hermite–Hadamard type inequalities for co-ordinated convex functions on the rectangle from the plane \(\mathbb{R} ^{2}\).

Theorem 2

Suppose that \(\mathcal{F}:\Delta \rightarrow \mathbb{R} \) is co-ordinated convex, then we have the following inequalities:

$$\begin{aligned}& \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \\& \quad \leq \frac{1}{2} \biggl[ \frac{1}{\rho _{2}-\rho _{1}} \int _{\rho _{1}}^{\rho _{2}}\mathcal{F} \biggl( \kappa , \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,d\kappa + \frac{1}{\rho _{4}-\rho _{3}} \int _{\rho _{3}}^{\rho _{4}} \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},\gamma \biggr)\,d\gamma \biggr] \\& \quad \leq \frac{1}{(\rho _{2}-\rho _{1})(\rho _{4}-\rho _{3})} \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}} \mathcal{F}(\kappa ,\gamma )\,d\gamma \,d\kappa \\& \quad \leq \frac{1}{4} \biggl[ \frac{1}{\rho _{2}-\rho _{1}} \int _{ \rho _{1}}^{\rho _{2}}\mathcal{F}(\kappa ,\rho _{3})\,d\kappa + \frac{1}{\rho _{2}-\rho _{1}} \int _{\rho _{1}}^{\rho _{2}} \mathcal{F}(\kappa ,\rho _{4})\,d\kappa \\& \qquad {} + \frac{1}{\rho _{4}-\rho _{3}} \int _{\rho _{3}}^{ \rho _{4}}\mathcal{F}(\rho _{1},\gamma )\,d\gamma +\frac{1}{\rho _{4}-\rho _{3}} \int _{\rho _{3}}^{\rho _{4}}\mathcal{F}(\rho _{2},\gamma )\,d\gamma \biggr] \\& \quad \leq \frac{\mathcal{F}(\rho _{1},\rho _{3})+\mathcal{F}(\rho _{1},\rho _{4})+\mathcal{F}(\rho _{2},\rho _{3})+\mathcal{F}(\rho _{2},\rho _{4})}{4}. \end{aligned}$$
(1.3)

The above inequalities are sharp. The inequalities in (1.3) hold in the reverse direction if the mapping \(\mathcal{F}\) is a co-ordinated concave mapping.

Over the years, many papers have been dedicated to the generalizations and new versions of the inequalities (1.2) and (1.3) using the different types of convex functions. For the other Ostrowski and Hermite–Hadamard type inequalities for co-ordinated convex functions, please refer to ([1–4, 8–14, 16, 17, 19–29])

In [30], Yıldız and Sarıkaya proved the following Lemma.

Lemma 1

Let \(w:\Delta := [ \rho _{1},\rho _{2} ] \times [ \rho _{3}, \rho _{4} ] \rightarrow {}[ 0,\infty )\) be an integrable function on Δ and let \(\mathcal{F}:\Delta \rightarrow \mathbb{R} \) be an absolutely continuous function such that the partial derivatives of order \(\frac{\partial ^{2}\mathcal{F}(\psi ,\varphi )}{\partial \psi \partial \varphi }\) exist for all \(( \psi ,\varphi ) \in \Delta \). Then we have the identity

$$\begin{aligned}& \biggl( \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{ \rho _{4}}w(u,v)\,dv\,du \biggr) \mathcal{F}(\kappa , \gamma )- \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,\gamma )\,dv\,du \\& \qquad {}- \int _{\rho _{1}}^{\rho _{2}} \int _{ \rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F}(\kappa ,v)\,dv\,du \\& \qquad {}+ \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{ \rho _{4}}w(u,v)\mathcal{F}(u,v)\,dv\,du \\& \quad = \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{ \rho _{4}}P(\kappa ,\tau ;\gamma ,\eta ) \frac{\partial ^{2}\mathcal{F}(\tau ,\eta )}{\partial \psi \partial \varphi }\,d\eta \,d\tau , \end{aligned}$$

where

$$ P(\kappa ,\tau ;\gamma ,\eta )=\textstyle\begin{cases} \int _{\rho _{1}}^{\tau }\int _{\rho _{3}}^{\eta }w(u,v)\,dv\,du, & \rho _{1}\leq \tau < \kappa , \rho _{3}\leq \eta < \gamma, \\ \int _{\rho _{1}}^{\tau }\int _{\rho _{4}}^{\eta }w(u,v)\,dv\,du, & \rho _{1}\leq \tau < \kappa , \gamma \leq \eta \leq \rho _{4}, \\ \int _{\rho _{2}}^{\tau }\int _{\rho _{3}}^{\eta }w(u,v)\,dv\,du, & \kappa \leq \tau \leq \rho _{2}, \rho _{3}\leq \eta < \gamma, \\ \int _{\rho _{2}}^{\tau }\int _{\rho _{4}}^{\eta }w(u,v)\,dv\,du, & \kappa \leq \tau \leq \rho _{2}, \gamma \leq \eta \leq \rho _{4}.\end{cases} $$

The aim of this paper is to establish some weighted generalizations of the Ostrowski type integral inequalities. The results presented in this paper provide extensions of those given in [14].

2 Weighted Ostrowski type inequalities

In this section, using Lemma 1, we established some weighted Ostrowski type inequalities for co-ordinated convex mapping.

First, we define the following mapping

$$\begin{aligned}& \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4};\mathcal{F},w) \\& \quad = \mathcal{F}(\kappa ,\gamma )- \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,\gamma )\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(\kappa ,v)\,dv\,du \\& \qquad {}+\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \end{aligned}$$
(2.1)

where

$$ m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})= \int _{\rho _{1}}^{ \rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\,dv\,du. $$

Using the change of variables in Lemma 1, we have the following identity:

$$\begin{aligned}& \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4};\mathcal{F},p) \\& \quad = \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}( \varphi ) \bigr)\,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}( \varphi ) \bigr)\,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}( \varphi ) \bigr)\,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{0}^{1} \int _{0}^{1} \biggl[ \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr] \\& \qquad {}\times \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}( \varphi ) \bigr)\,d\varphi \,d\psi , \end{aligned}$$
(2.2)

where \(U_{1}(\psi )=\psi \kappa +(1-\psi )\rho _{1}\), \(U_{2}(\psi )=\psi \kappa +(1-\psi )\rho _{2}\), \(V_{1}(\varphi )=\varphi \gamma +(1-\varphi )\rho _{3}\) and \(V_{2}(\varphi )=\varphi \gamma +(1-\varphi )\rho _{4}\).

Theorem 3

Suppose that the mapping w is as in Lemma 1. Moreover, let w is bounded on Δ, i.e., \(\Vert w \Vert _{\infty }:=\sup_{(\kappa , \gamma )\in \Delta } \vert w(\kappa ,\gamma ) \vert \). If \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert \) is a co-ordinated convex function on Δ, then for all \((\kappa ,\gamma )\in \Delta \) we have the following inequality

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ \Vert w \Vert _{\infty }}{36\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl\{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert \\& \qquad {} +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \biggr] \\& \qquad {}+ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}- \gamma ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert \\& \qquad {}+2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert \biggr] \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert \\& \qquad {} +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert \biggr] \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}- \gamma ) ^{2} \biggl[ 4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert \\& \qquad {} +2 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr] \biggr\} , \end{aligned}$$
(2.3)

where the mapping Θ is defined as in (2.1).

Proof

By taking the modulus of the equality (2.2), we have

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad = \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi . \end{aligned}$$
(2.4)

Since \(w(\kappa ,\gamma )\) is bounded on Δ, and \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert \) is co-ordinated convex on Δ, we obtain

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq \Vert w \Vert _{\infty } \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}( \psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) \Vert w \Vert _{\infty } \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl[ \psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert +\psi ( 1- \varphi ) \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert \\& \qquad {}+ ( 1-\psi ) \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1}, \gamma ) \biggr\vert + ( 1-\psi ) ( 1- \varphi ) \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \biggr]\,d\varphi \,d \psi \\& \quad = ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert \\& \qquad {} +\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \biggr] . \end{aligned}$$
(2.5)

Similarly, we have

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \end{aligned}$$
(2.6)
$$\begin{aligned}& \quad \leq ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert \biggr] , \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert \biggr], \end{aligned}$$
(2.7)

and

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert \,d\varphi \,d\psi \\& \quad \leq ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) \Vert w \Vert _{\infty } \\& \qquad {}\times \biggl[ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert +\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert + \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr] . \end{aligned}$$
(2.8)

If we substitute the inequalities (2.5)–(2.8) in (2.4), then we obtain the required result (2.3). This completes the proof. □

Corollary 1

Under the same assumption of Theorem 3with \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \vert \leq M\), \(( \kappa ,\gamma ) \in \Delta \), we have the following weighted Ostrowski type inequality

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{M ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl[ \frac{1}{4}+ \frac{ ( \kappa -\frac{\rho _{1}+\rho _{2}}{2} ) ^{2}}{ ( \rho _{2}-\rho _{1} ) ^{2}} \biggr] \biggl[ \frac{1}{4}+\frac{ ( \gamma -\frac{\rho _{3}+\rho _{4}}{2} ) ^{2}}{ ( \rho _{4}-\rho _{3} ) ^{2}} \biggr] \Vert w \Vert _{\infty }. \end{aligned}$$

Remark 1

If we choose \(w(\kappa ,\gamma )=1\) in Corollary 1, then Corollary 1 reduces to [14, Theorem 3].

Corollary 2

Under the same assumption of Theorem 3with \(\kappa =\frac{\rho _{1}+\rho _{2}}{2}\) and \(\gamma =\frac{\rho _{3}+\rho _{4}}{2}\), we have the following weighted Hermite–Hadamard type inequality

$$\begin{aligned}& \biggl\vert \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) + \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \\& \qquad {}-\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F} \biggl( u, \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},v \biggr)\,dv\,du \biggr\vert \\& \quad \leq \frac{ ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{576} \frac{ \Vert w \Vert _{\infty }}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl\{ 16 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert +4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert \\& \qquad {}+4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert +4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{4} \biggr) \biggr\vert \\& \qquad {}+4 \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr\} \\& \quad \leq \frac{ ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{64} \frac{ \Vert w \Vert _{\infty }}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert \biggr] . \end{aligned}$$

Theorem 4

Let w be as in Theorem 3. If \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, then for all \((\kappa ,\gamma )\in \Delta \), we have the following inequality

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ \Vert w \Vert _{\infty }}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})(p+1)^{\frac{2}{p}}} \\& \qquad {}\times \biggl\{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \\& \qquad {}\times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr] \biggr) ^{\frac{1}{q}} \\& \qquad {}+ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}- \gamma ) ^{2} \\& \qquad {}\times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr] \biggr) ^{\frac{1}{q}} \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \\& \qquad {}\times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr] \biggr) ^{\frac{1}{q}} \\& \qquad {}+ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}-\gamma ) ^{2} \\& \qquad {} \times \biggl( \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr] \biggr) ^{ \frac{1}{q}} \biggr\} , \end{aligned}$$
(2.9)

where the mapping Θ is defined as in (2.1), and \(\frac{1}{p}+\frac{1}{q}=1\).

Proof

Using the well-known Hölder inequality in (2.4), we obtain

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}( \psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}( \psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}( \psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}( \psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(2.10)

Since w is bounded on Δ, we have

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \\& \quad \leq \Vert w \Vert _{ \infty }^{p} \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}( \varphi )}\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \\& \quad = \Vert w \Vert _{\infty }^{p} ( \kappa -\rho _{1} ) ^{p} ( \gamma -\rho _{3} ) ^{p} \int _{0}^{1} \int _{0}^{1}\varphi ^{p}\psi ^{p}\,d\varphi \,d\psi \\& \quad = \frac{ ( \kappa -\rho _{1} ) ^{p} ( \gamma -\rho _{3} ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}. \end{aligned}$$
(2.11)

Similarly, we get

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \leq \frac{ ( \kappa -\rho _{1} ) ^{p} ( \rho _{4}-\gamma ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}, \end{aligned}$$
(2.12)
$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \leq \frac{ ( \rho _{2}-\kappa ) ^{p} ( \gamma -\rho _{3} ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}, \end{aligned}$$
(2.13)

and

$$ \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{ \rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert ^{p}\,d\varphi \,d\psi \leq \frac{ ( \rho _{2}-\kappa ) ^{p} ( \rho _{4}-\gamma ) ^{p}}{(p+1)^{2}} \Vert w \Vert _{\infty }^{p}. $$
(2.14)

On the other hand, as \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, we obtain

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \end{aligned}$$
(2.15)
$$\begin{aligned}& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr] , \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \end{aligned}$$
(2.16)
$$\begin{aligned}& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr] , \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \\& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr] , \end{aligned}$$
(2.17)

and

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}( \varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \\& \quad \leq \frac{1}{4} \biggl[ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr] . \end{aligned}$$
(2.18)

If we substitute the inequalities (2.11)–(2.18) in (2.10), then we obtain the required inequality (2.9). □

Corollary 3

Under the same assumption of Theorem 4with \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \vert \leq M\), \(( \kappa ,\gamma ) \in \Delta \), then we have the following weighted Ostrowski type inequality

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{4M ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})(p+1)^{\frac{2}{p}}} \biggl[ \frac{1}{4}+ \frac{ ( \kappa -\frac{\rho _{1}+\rho _{2}}{2} ) ^{2}}{ ( \rho _{2}-\rho _{1} ) ^{2}} \biggr] \biggl[ \frac{1}{4}+ \frac{ ( \gamma -\frac{\rho _{3}+\rho _{4}}{2} ) ^{2}}{ ( \rho _{4}-\rho _{3} ) ^{2}} \biggr] \Vert w \Vert _{\infty }. \end{aligned}$$

Remark 2

If we choose \(w(\kappa ,\gamma )=1\) in Corollary 3, then Corollary 3 reduces to [14, Theorem 4].

Corollary 4

Under the same assumption of Theorem 4with \(\kappa =\frac{\rho _{1}+\rho _{2}}{2}\) and \(\gamma =\frac{\rho _{3}+\rho _{4}}{2}\), then we have the following weighted Hermite–Hadamard type inequality

$$\begin{aligned}& \biggl\vert \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) + \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F} \biggl( u, \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},v \biggr)\,dv\,du \biggr\vert \\& \quad \leq \frac{ \Vert w \Vert _{\infty } ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2}}{2^{4+\frac{2}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})(p+1)^{\frac{2}{p}}} \\& \qquad {}\times \biggl\{ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \biggl( \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {} + \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \biggr\} . \end{aligned}$$

Theorem 5

Let w be as in Theorem 3. If the function \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, then for all \((\kappa ,\gamma )\in \Delta \) and \(q\geq 1 \), we have the following inequality

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \end{aligned}$$
(2.19)

where the mapping Θ is defined as in (2.1).

Proof

By utilizing the power mean inequality in (2.4), we obtain

$$\begin{aligned}& \bigl\vert \Theta (\rho _{1},\rho _{2},\rho _{3},\rho _{4}; \mathcal{F},p) \bigr\vert \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d \psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{3}}^{V_{1}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d\psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{1}}^{U_{1}(\psi )} \int _{\rho _{4}}^{V_{2}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \gamma -\rho _{3} ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{3}}^{V_{1}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d \psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{3}}^{V_{1}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ( \rho _{4}-\gamma ) }{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}( \psi )} \int _{\rho _{4}}^{V_{2}(\varphi )}w(u,v)\,dv\,du \biggr\vert \,d\varphi \,d\psi \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \int _{\rho _{2}}^{U_{2}(\psi )} \int _{\rho _{4}}^{V_{2}( \varphi )}w(u,v)\,dv\,du \biggr\vert \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \quad \leq \frac{ ( \kappa -\rho _{1} ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \kappa -\rho _{1} ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \gamma -\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \frac{ ( \rho _{2}-\kappa ) ^{2} ( \rho _{4}-\gamma ) ^{2} \Vert w \Vert _{\infty }}{4^{1-\frac{1}{q}}\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}}. \end{aligned}$$
(2.20)

Since \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \vert ^{q}\) is a co-ordinated convex function on Δ, we have the following inequalities

$$\begin{aligned}& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \end{aligned}$$
(2.21)
$$\begin{aligned}& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \\& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{1}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \end{aligned}$$
(2.22)
$$\begin{aligned}& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \\& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{1}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{3} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}, \end{aligned}$$
(2.23)

and

$$\begin{aligned}& \biggl( \int _{0}^{1} \int _{0}^{1}\psi \varphi \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \bigl( U_{2}(\psi ),V_{2}(\varphi ) \bigr) \biggr\vert ^{q}\,d\varphi \,d\psi \biggr) ^{\frac{1}{q}} \\& \quad \leq \biggl( \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa , \gamma ) \biggr\vert ^{q}+\frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\rho _{4} ) \biggr\vert ^{q}+ \frac{1}{18} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\gamma ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{36} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{ \frac{1}{q}}. \end{aligned}$$
(2.24)

By utilizing the equalities (2.21)–(2.24) in (2.20), we obtain the desired inequality (2.19). This completes the proof. □

Remark 3

Under the same assumption of Theorem 5 with \(\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \kappa ,\gamma ) \vert \leq M\), \(( \kappa ,\gamma ) \in \Delta \), then Theorem 5 reduces to Corollary 1.

Corollary 5

Under the same assumption of Theorem 5with \(\kappa =\frac{\rho _{1}+\rho _{2}}{2}\) and \(\gamma =\frac{\rho _{3}+\rho _{4}}{2}\), we have the following weighted Hermite–Hadamard type inequality

$$\begin{aligned}& \biggl\vert \mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) + \frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F}(u,v)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \int _{ \rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v) \mathcal{F} \biggl( u, \frac{\rho _{3}+\rho _{4}}{2} \biggr)\,dv\,du \\& \qquad {} -\frac{1}{m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})}\int _{\rho _{1}}^{\rho _{2}} \int _{\rho _{3}}^{\rho _{4}}w(u,v)\mathcal{F} \biggl( \frac{\rho _{1}+\rho _{2}}{2},v \biggr)\,dv\,du \biggr\vert \\& \quad \leq \frac{ ( \rho _{2}-\rho _{1} ) ^{2} ( \rho _{4}-\rho _{3} ) ^{2} \Vert w \Vert _{\infty }}{64\times m(\rho _{1},\rho _{2};\rho _{3},\rho _{4})} \\& \qquad {}\times \biggl[ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{3} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {}+\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{1},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{1},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \rho _{3} \biggr) \biggr\vert ^{q} \\& \qquad {} + \biggl(\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+\frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{3} ) \biggr\vert ^{q} \biggr) \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \frac{4}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2},\frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \frac{\rho _{1}+\rho _{2}}{2}, \rho _{4} \biggr) \biggr\vert ^{q} \\& \qquad {} +\frac{2}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } \biggl( \rho _{2}, \frac{\rho _{3}+\rho _{4}}{2} \biggr) \biggr\vert ^{q}+ \frac{1}{9} \biggl\vert \frac{\partial ^{2}\mathcal{F}}{\partial \psi \partial \varphi } ( \rho _{2},\rho _{4} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \biggr] . \end{aligned}$$

3 Conclusion

In this paper, we consider the identity given by Yıldız and Sarıkaya in [30] to obtain some weighted Ostrowski type inequalities for co-ordinated convex functions. We also present some weighted Hermite–Hadamard type inequalities by the special cases of our main results. In future works, the authors can try to generalize their results by utilizing different kinds of co-ordinated convex function classes.