New Hermite–Hadamard and Ostrowski-Type Inequalities for Newly Introduced Co-Ordinated Convexity with Respect to a Pair of Functions

Abstract

In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls under the umbrella of convexity. However, it is a significant theory that affects practically all areas of mathematics. In this paper, we introduce the notions of $\left(g,h\right)$-convexity or convexity with respect to a pair of functions on co-ordinates and discuss its fundamental properties. Moreover, we establish some novel Hermite–Hadamard- and Ostrowski-type inequalities for newly introduced co-ordinated convexity. Additionally, it is presented that the newly introduced notion of the convexity and given inequalities are generalizations of existing studies in the literature. Lastly, we look at various mathematical examples and graphs to confirm the validity of the newly found inequalities.

1. Introduction

In different branches of mathematical analysis, inequalities play a fundamental role in proving many famous theorems. Additionally, in recent years, such inequalities have been used in most papers discussing fractional mathematical models, fractional boundary value problems, etc. The application of some of them can be found in numerous articles, including [1,2,3,4,5,6,7,8,9,10,11]. These applications show the importance of study on different generalizations of inequalities.

Throughout this paper, we let $I=\left[\sigma ,\rho \right]\subset \mathbb{R}$ and $△:=[\sigma ,\rho ]\times [\varsigma ,d]\subseteq {\mathbb{R}}^2$. The Hermite–Hadamard inequality is a fudamental inequality which is presented as: if $ϝ:I\to \mathbb{R}$ is a convex function, then

$$ϝ\left(\frac{\sigma +\rho }{2}\right)\le \frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }ϝ(\varkappa )d\varkappa \le \frac{ϝ(\sigma )+ϝ(\rho )}{2}.$$

(1)

The double inequality (1) was discussed by Hermite [12] in 1883, and ten years later in 1893, it was proved by Hadamard [13].

Another famous inequality is the Ostrowski inequality, which was established by Ostrowski [14] in 1938.

Theorem 1.

Let $ϝ:I\to \mathbb{R}$ be differentiable on $I^{\circ }$ (interior of I) with a bounded derivative, that is, $\parallel {ϝ}^{\prime }{\parallel }_{\infty }:={sup}_{\varkappa \in (\sigma ,\rho )}\left|{ϝ}^{\prime }(\varkappa )\right|<\infty$. Then, we have

$$\left|ϝ(\varkappa )-\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }ϝ(\varkappa )d\varkappa \right|\le \left[\frac{1}{4}+\frac{(\varkappa -\frac{\sigma +\rho }{2})}{{(\rho -\sigma )}^2}\right](\rho -\sigma ){\parallel {ϝ}^{\prime }\parallel }_{\infty },$$

(2)

for all $\varkappa \in I$. The constant $\frac{1}{4}$ is the best possible.

The inequality (2) can be rewritten in the equivalent form

$$\left|ϝ(\varkappa )-\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }ϝ(\varkappa )d\varkappa \right|\le \left[\frac{{(\varkappa -\sigma )}^2+{(\rho -\varkappa )}^2}{2(\rho -\sigma )}\right]{\parallel {ϝ}^{\prime }\parallel }_{\infty }.$$

The Hermite–Hadamard inequality and Ostrowski inequality have been extensively studied by a number of researchers; see [15,16,17,18,19,20] and the references therein.

In 2001, S. S. Dragomir defined a new concept of a co-ordinated convex function.

Definition 1

([21]). $ϝ:△\to \mathbb{R}$ is a co-ordinated convex map, if

$${ϝ}_{\varkappa }:[\varsigma ,d]\ni v\mapsto ϝ(\varkappa ,v)\in \mathbb{R}\&{ϝ}_{\gamma }:[\sigma ,\rho ]\ni u\mapsto ϝ(u,\gamma )\in \mathbb{R},$$

(3)

are convex ($\forall \varkappa \in (\sigma ,\rho ),\forall \gamma \in (\varsigma ,d)$).

A formal equivalent definition of such functions can be stated as the following:

Definition 2

([21]). $ϝ:△\to \mathbb{R}$ is co-ordinated convex if

$$\begin{array}{cc}\hfill ϝ(\xi \varkappa +(1-\xi )z,\lambda \gamma +(1-\lambda )w)& \le \xi \lambda ϝ(\varkappa ,\gamma )+\xi (1-\lambda )ϝ(\varkappa ,w)\hfill \\ & +(1-\xi )\lambda ϝ(z,\gamma )+(1-\xi )(1-\lambda )ϝ(z,w),\hfill \end{array}$$

(4)

$\forall \xi ,\lambda \in [0,1]$ and $\forall (\varkappa ,\gamma ),(z,w)\in △$.

He also presented the following version of Hermite–Hadamard-type inequalities for the aforesaid functions:

Theorem 2

([21]). If $ϝ:△\to \mathbb{R}$ is a co-ordinated convex function, then

$$\begin{array}{cc}ϝ\left({\displaystyle \frac{\sigma +\rho }{2}},{\displaystyle \frac{\varsigma +d}{2}}\right)\hfill & \le {\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{1}{\rho -\sigma }}{\int }_{\sigma }^{\rho }ϝ\left(\varkappa ,{\displaystyle \frac{\varsigma +d}{2}}\right)d\varkappa +{\displaystyle \frac{1}{d-\varsigma }}{\int }_{\varsigma }^dϝ\left({\displaystyle \frac{\sigma +\rho }{2}},\gamma \right)d\gamma \right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{(\rho -\sigma )(d-\varsigma )}}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(\varkappa ,\gamma )d\gamma d\varkappa \hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}\left[{\displaystyle \frac{1}{\rho -\sigma }}\left({\int }_{\sigma }^{\rho }ϝ\left(\varkappa ,\varsigma \right)d\varkappa +{\int }_{\sigma }^{\rho }ϝ\left(\varkappa ,d\right)d\varkappa \right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{d-\varsigma }}\left({\int }_{\varsigma }^dϝ\left(\sigma ,\gamma \right)d\gamma +{\int }_{\varsigma }^dϝ\left(\rho ,\gamma \right)d\gamma \right)\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}\left[ϝ(\sigma ,\varsigma )+ϝ(\rho ,\varsigma )+ϝ(\sigma ,d)+ϝ(\rho ,d)\right].\hfill \end{array}$$

(5)

In relation to the convex functions on co-ordinates, Latif et al. [22] presented the following Ostrowski-type inequality in 2010:

Theorem 3

([22]). Let $ϝ:\Delta \to \mathbb{R}$ be a function with the twice-partial differentiability property on ${\Delta }^{\circ }$ and $\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }$ be continuous and integrable on Δ. When $\left|\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }\right|$ is co-ordinated convex on Δ and $\left|\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }(\varkappa ,\gamma )\right|\le M,(\varkappa ,\gamma )\in \Delta$, then

$$\begin{array}{cc}\hfill \left|ϝ(\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)d\eta d\xi -A\right|& \hfill \\ \hfill \le M\left[\frac{{(\varkappa -\sigma )}^2+{(\rho -\varkappa )}^2}{2(\rho -\sigma )}\right]& \left[\frac{{(\gamma -\varsigma )}^2+{(d-\gamma )}^2}{2(d-\varsigma )}\right],\hfill \end{array}$$

(6)

where

$$A(\varkappa ,\gamma )=\frac{1}{d-\varsigma }{\int }_{\varsigma }^dϝ(\varkappa ,v)dv+\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\varsigma }ϝ(u,\gamma )du.$$

(7)

Theorem 4

([22]). Let $ϝ:\Delta \to \mathbb{R}$ be a function with the twice-partial differentiability property on ${\Delta }^{\circ }$, and let $\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }$ be continuous and integrable on Δ. When ${\left|\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }\right|}^q$ is co-ordinated convex on $\Delta ,\mathit{where}q>1\mathit{with}\frac{1}{p}+\frac{1}{q}=1$, and $\left|\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }(\varkappa ,\gamma )\right|\le M,(\varkappa ,\gamma )\in \Delta$, then

$$\begin{array}{ccc}\hfill \left|ϝ(\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A\right|& & \hfill \\ \hfill \le \frac{M}{{(1+p)}^{2/p}}\left[\frac{{(\varkappa -\sigma )}^2+{(\rho -\varkappa )}^2}{\rho -\sigma }\right]& & \left[\frac{{(\gamma -\varsigma )}^2+{(d-\gamma )}^2}{d-\varsigma }\right],\hfill \end{array}$$

(8)

where A is defined as in (7).

Theorem 5

([22]). Let $ϝ:\Delta \to \mathbb{R}$ be a function with the twice-partial differentiability property on ${\Delta }^{\circ }$, and let $\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }$ be continuous and integrable on Δ. When ${\left|\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }\right|}^q$ is co-ordinated convex on $\Delta ,\mathit{where}q\ge 1$, and $\left|\frac{{\partial }^2ϝ}{\partial \eta \partial \xi }(\varkappa ,\gamma )\right|\le M,(\varkappa ,\gamma )\in \Delta$, then

$$\begin{array}{cc}\hfill \left|ϝ(\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A\right|& \hfill \\ \hfill \le \frac{M}{4}\left[\frac{{(\varkappa -\sigma )}^2+{(\rho -\varkappa )}^2}{\rho -\sigma }\right]& \left[\frac{{(\gamma -\varsigma )}^2+{(d-\gamma )}^2}{d-\varsigma }\right],\hfill \end{array}$$

(9)

where A is defined as in (7).

In [23], B. Samet introduced a new class of convex functions with respect to a pair of functions, which is defined as:

Definition 3.

Let $g,h:I\to \mathbb{R}$ be two mappings. A mapping $ϝ:I\to \mathbb{R}$ is called $\left(g,h\right)$ convex if the following inequality holds for $M\left(\varkappa ,\gamma \right)=g\left(\varkappa \right)h\left(\gamma \right)+g\left(\gamma \right)h\left(\varkappa \right)$

$$ϝ\left(\xi \varkappa +\left(1-\xi \right)\gamma \right)\le {\xi }^2ϝ\left(\varkappa \right)+{\left(1-\xi \right)}^2ϝ\left(\gamma \right)+\xi \left(1-\xi \right)M\left(\varkappa ,\gamma \right),$$

(10)

for all $\xi \in \left[0,1\right]$ and $\varkappa ,\gamma \in I.$

Remark 1.

If $g,h:I\subset \mathbb{R}\to \left[0,\infty \right)$ are two convex mappings, then $ϝ=gh$ is $\left(g,h\right)$-convex mapping.

B. Samet also established a Hermite–Hadamard inequality (double inequality) for newly class of convex functions with respect to a pair of functions, presented as:

Theorem 6

([23]). Let $ϝ:I\to \mathbb{R}$ be a $\left(g,h\right)$-convex function over I with $g,h:I\to \mathbb{R}.$ If $ϝ\in L^1\left(I\right)$ and $g,h\in L^2\left(I\right),$ then

$$\begin{array}{ccc}\hfill & \hfill & 2ϝ\left(\frac{\sigma +\rho }{2}\right)-\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }g\left(\sigma +\rho -\varkappa \right)h\left(\varkappa \right)d\varkappa \hfill \\ \hfill & =\hfill & 2ϝ\left(\frac{\sigma +\rho }{2}\right)-\frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }g\left(\varkappa \right)h\left(\sigma +\rho -\varkappa \right)d\varkappa \hfill \\ \hfill & \le \hfill & \frac{1}{\rho -\sigma }{\int }_{\sigma }^{\rho }ϝ\left(\varkappa \right)d\varkappa \hfill \\ \hfill & \le \hfill & \frac{ϝ\left(\sigma \right)+ϝ\left(\rho \right)}{3}+\frac{M\left(\sigma ,\rho \right)}{6}.\hfill \end{array}$$

(11)

Remark 2.

In Theorem 6, if we set $g=ϝ$ and $h=1$ or $h=ϝ$ and $g=1$, then inequality (11) becomes the inequality (1).

After that, Ali et al. [24] and Xie et al. [25] used the $\left(g,h\right)$-convexity and established the following fractional Hermite–Hadamard-type inequality.

Theorem 7

([24,25]). Consider $ϝ:I\to \mathbb{R}$ as a $\left(g,h\right)$-convex function on I with $g,h:I\to \mathbb{R}.$ If $ϝ\in L^1\left(I\right)$ and $g,h\in L^2\left(I\right),$ then

$$\begin{array}{ccc}& & 2ϝ\left(\frac{\sigma +\rho }{2}\right)-\frac{\Gamma \left(\alpha +1\right)}{2{\left(\rho -\sigma \right)}^{\alpha }}\left[J_{\sigma +}^{\alpha }\Omega \left(\rho \right)+J_{\rho -}^{\alpha }\Omega \left(\sigma \right)\right]\hfill \\ & \le & \frac{\Gamma \left(\alpha +1\right)}{2{\left(\rho -\sigma \right)}^{\alpha }}\left[J_{\sigma +}^{\alpha }ϝ\left(\rho \right)+J_{\rho -}^{\alpha }ϝ\left(\sigma \right)\right]\hfill \\ & \le & \frac{ϝ\left(\sigma \right)+ϝ\left(\rho \right)}{2}\left[\frac{\alpha }{\alpha +2}+\frac{2}{{\alpha }^2+3\alpha +2}\right]+\frac{\alpha }{{\alpha }^2+3\alpha +2}M\left(\sigma ,\rho \right),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & 2ϝ\left(\frac{\sigma +\rho }{2}\right)-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }}\left[J_{\sigma +}^{\alpha }\Omega \left(\frac{\sigma +\rho }{2}\right)+J_{\rho -}^{\alpha }\Omega \left(\frac{\sigma +\rho }{2}\right)\right]\hfill \\ & \le & \frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }}\left[J_{\sigma +}^{\alpha }ϝ\left(\frac{\sigma +\rho }{2}\right)+J_{\rho -}^{\alpha }ϝ\left(\frac{\sigma +\rho }{2}\right)\right]\hfill \\ & \le & \frac{\left[ϝ\left(\sigma \right)+ϝ\left(\rho \right)\right]\left(\alpha +1\right)}{2\left(\alpha +2\right)}+\frac{M\left(\sigma ,\rho \right)}{2\left(\alpha +2\right)},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill & \hfill & 2ϝ\left(\frac{\sigma +\rho }{2}\right)-\frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }}\left[J_{\frac{\sigma +\rho }{2}+}^{\alpha }\Omega \left(\rho \right)+J_{\frac{\sigma +\rho }{2}-}^{\alpha }\Omega \left(\sigma \right)\right]\hfill \\ \hfill & \le \hfill & \frac{2^{\alpha -1}\Gamma \left(\alpha +1\right)}{{\left(\rho -\sigma \right)}^{\alpha }}\left[J_{\frac{\sigma +\rho }{2}+}^{\alpha }ϝ\left(\rho \right)+J_{\frac{\sigma +\rho }{2}-}^{\alpha }ϝ\left(\sigma \right)\right]\hfill \\ \hfill & \le \hfill & \frac{\left[ϝ\left(\sigma \right)+ϝ\left(\rho \right)\right]}{4}\left[2-\frac{\alpha \left(\alpha +3\right)}{\left(\alpha +1\right)\left(\alpha +2\right)}\right]+\frac{M\left(\sigma ,\rho \right)\alpha \left(\alpha +3\right)}{4\left(\alpha +1\right)\left(\alpha +2\right)},\hfill \end{array}$$

(13)

where $\Omega (\varkappa )=g\left(\varkappa \right)h\left(\sigma +\rho -\varkappa \right).$

For more results on Simpson- and Ostrowski-type inequalities via $\left(g,h\right)$-convexity, one can consult [23,24,25].

Motivated by the above literature, we conduct an anlaysis on some new generalizations of Hermite–Hadamard and Ostrowski type inequalities via the co-ordinated convex functions by defining a new notion of co-ordinated $(g,h)$-convexity or convexity with respect to a pair of functions. It is also shown that the newly established inequalities and class of convex functions are generalizations of the existing results in the literature.

A description of the paper is as follows: in Section 2, the notion of co-ordinated $(g,h)$-convexity is introduced. We also prove some of its important properties and give an example of a co-ordinated $(g,h)$-convex function. In Section 3, we establish some Hermite–Hadamard-type inequalities for co-ordinated $(g,h)$-convex functions. In Section 4, we derive some new Ostrowski-type inequalities under the differentiable $\left(g,h\right)$-convexity. The examples showing the consistency of newly discussed inequalities are in Section 5. Section 6 concludes our work briefly.

2. Co-Ordinated $(\mathit{g},\mathit{h})$-Convex Functions

In this section, we introduce a new concept of the co-ordinated $(g,h)$-convex function, and then some of its properties are proved. We also give an example of a co-ordinated $(g,h)$-convex function at the end of the section.

Definition 4.

Let $g,h:△\to \mathbb{R}$ be two given functions. A function $ϝ:△\to \mathbb{R}$ is called co-ordinated $(g,h)$-convex, if

$$\begin{array}{cc}\hfill ϝ(\xi \varkappa & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)+{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)+\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )+\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)+\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill M_1(\varkappa ,\gamma ,w)& =h(\varkappa ,w)g(\varkappa ,\gamma )+h(\varkappa ,\gamma )g(\varkappa ,w),\hfill \\ \hfill M_2(\varkappa ,z,\gamma ,w)& =h(\varkappa ,w)g(z,\gamma )+h(z,\gamma )g(\varkappa ,w),\hfill \\ \hfill M_3(\varkappa ,z,\gamma )& =h(\varkappa ,\gamma )g(z,\gamma )+h(z,\gamma )g(\varkappa ,\gamma ),\hfill \\ \hfill M_4(\varkappa ,z,\gamma ,w)& =h(\varkappa ,\gamma )g(z,w)+h(z,w)g(\varkappa ,\gamma ),\hfill \\ \hfill M_5(\varkappa ,z,w)& =h(\varkappa ,w)g(z,w)+h(z,w)g(\varkappa ,w),\hfill \end{array}$$

and

$$M_6(z,\gamma ,w)=h(z,\gamma )g(z,w)+h(z,w)g(z,\gamma ),$$

for all $\xi ,\eta \in [0,1]$, $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$.

Some properties of co-ordinated $(g,h)$-convexity are proved as follows:

Proposition 1.

Let $g,h:△\to \mathbb{R}$ be two given functions. Then, we have that

(i) 

ϝ is co-ordinated $(g,h)$-convex if ϝ is co-ordinated $(h,g)$-convex.

(ii) 

If ϝ is co-ordinated $(g,h)$-convex, then ϝ is co-ordinated $({\sigma }^{-1}g,\sigma h)$-convex for all $\sigma \in \mathbb{R},$$\sigma \ne 0$.

(iii) 

If $ϝ,\overline{ϝ}$ are co-ordinated $(g,h)$-convex, in this case, $ϝ+\overline{ϝ}$ is co-ordinated $(2g,h)$-convex and co-ordinated $(g,2h)$-convex.

(iv) 

If ϝ is co-ordinated $(g,h)$-convex, then $\sigma ϝ$ is co-ordinated $(\sigma g,h)$-convex and co-ordinated $(g,\sigma h)$-convex for all $\sigma >0$.

(v) 

Let ϝ be co-ordinated $(g,h)$-convex, $ϝ\ge 0$ and $(g\ge 0,h\le 0)$ or $(g\le 0,h\ge 0)$. Then ϝ is co-ordinated convex.

(vi) 

If ϝ is co-ordinated $(g,h)$-convex and $\overline{ϝ}$ are co-ordinated $(g,\overline{h})$-convex, then $ϝ+\overline{ϝ}$ is co-ordinated $(g,h+\overline{h})$-convex.

(vii) 

If ϝ is co-ordinated $(g,h)$-convex and $\overline{ϝ}$ are co-ordinated $(\overline{g},h)$-convex, then $ϝ+\overline{ϝ}$ is co-ordinated $(g+\overline{g},h)$-convex.

(viii) 

If ϝ is co-ordinated $(g,h)$-convex, then

$$\begin{array}{cc}\hfill & ϝ\left(\frac{\varkappa +z}{2},\frac{\gamma +w}{2}\right)\le \frac{ϝ(\varkappa ,\gamma )+ϝ(\varkappa ,w)+ϝ(z,\gamma )+ϝ(z,w)}{16}\hfill \\ \hfill & +\frac{M_1(\varkappa ,\gamma ,w)+M_2(\varkappa ,z,\gamma ,w)+M_3(\varkappa ,z,\gamma )+M_4(\varkappa ,z,\gamma ,w)+M_5(\varkappa ,z,w)+M_6(z,\gamma ,w)}{16},\hfill \end{array}$$

for all $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$.

Proof .

(i)

It is immediately true by Definition 4.

(ii)

We simply consider that

$$\begin{array}{cc}\hfill h(\varkappa ,w)g(\varkappa ,\gamma )+h(\varkappa ,\gamma )g(\varkappa ,w)& =\left(\sigma h(\varkappa ,w)\right)\left({\sigma }^{-1}g(\varkappa ,\gamma )\right)+\left(\sigma h(\varkappa ,\gamma )\right)\left({\sigma }^{-1}g(\varkappa ,w)\right),\hfill \\ \hfill h(\varkappa ,w)g(z,\gamma )+h(z,\gamma )g(\varkappa ,w)& =\left(\sigma h(\varkappa ,w)\right)\left({\sigma }^{-1}g(z,\gamma )\right)+\left(\sigma h(z,\gamma )\right)\left({\sigma }^{-1}g(\varkappa ,w)\right),\hfill \\ \hfill h(\varkappa ,\gamma )g(z,\gamma )+h(z,\gamma )g(\varkappa ,\gamma )& =\left(\sigma h(\varkappa ,\gamma )\right)\left({\sigma }^{-1}g(z,\gamma )\right)+\left(\sigma h(z,\gamma )\right)\left({\sigma }^{-1}g(\varkappa ,\gamma )\right),\hfill \\ \hfill h(\varkappa ,\gamma )g(z,w)+h(z,w)g(\varkappa ,\gamma )& =\left(\sigma h(\varkappa ,\gamma )\right)\left({\sigma }^{-1}g(z,w)\right)+\left(\sigma h(z,w)\right)\left({\sigma }^{-1}g(\varkappa ,\gamma )\right),\hfill \\ \hfill h(\varkappa ,w)g(z,w)+h(z,w)g(\varkappa ,w)& =\left(\sigma h(\varkappa ,w)\right)\left({\sigma }^{-1}g(z,w)\right)+\left(\sigma h(z,w)\right)\left({\sigma }^{-1}g(\varkappa ,w)\right),\hfill \end{array}$$

and

$$h(z,\gamma )g(z,w)+h(z,w)g(z,\gamma )=\left(\sigma h(z,\gamma )\right)\left({\sigma }^{-1}g(z,w)\right)+\left(\sigma h(z,w)\right)\left({\sigma }^{-1}g(z,\gamma )\right).$$

(iii)

Since $ϝ,\overline{ϝ}$ are co-ordinated $(g,h)$-convex, we have

$$\begin{array}{cc}ϝ(\xi \varkappa \hfill & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)+\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )+\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)+\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w),\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\overline{ϝ}(\xi \varkappa \hfill & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2\overline{ϝ}(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2\overline{ϝ}(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2\overline{ϝ}(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2\overline{ϝ}(z,w)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)+\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )+\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)+\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w),\hfill \end{array}$$

(15)

for all $\xi ,\eta \in [0,1]$, $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$. Adding (14) and (15), we obtain

$$\begin{array}{cc}\hfill \left(ϝ+\overline{ϝ}\right)& (\xi \varkappa +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2(ϝ+\overline{ϝ})(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2(ϝ+\overline{ϝ})(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2(ϝ+\overline{ϝ})(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2(ϝ+\overline{ϝ})(z,w)\hfill \\ \hfill & +2{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)+2\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +2\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )+2\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +2\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)+2\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w),\hfill \end{array}$$

which means that $ϝ+\overline{ϝ}$ is co-ordinated $(2g,h)$-convex and co-ordinated $(g,2h)$-convex.

(iv)

Let $ϝ$ be co-ordinated $(g,h)$-convex. In this case, multiplying (14) by $\sigma \ge 0$ yields

$$\begin{array}{cc}\hfill (\sigma ϝ)(\xi \varkappa & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2(\sigma ϝ)(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2(\sigma ϝ)(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2(\sigma ϝ)(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2(\sigma ϝ)(z,w)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )\sigma M_1(\varkappa ,\gamma ,w)+\xi \eta (1-\xi )(1-\eta )\sigma M_2(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi {\eta }^2(1-\xi )\sigma M_3(\varkappa ,z,\gamma )+\xi \eta (1-\xi )(1-\eta )\sigma M_4(\varkappa ,z,\gamma ,w)\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2\sigma M_5(\varkappa ,z,w)+\eta {(1-\xi )}^2(1-\eta )\sigma M_6(z,\gamma ,w),\hfill \end{array}$$

which shows that $\sigma ϝ$ is co-ordinated $(\sigma g,h)$-convex and co-ordinated $(g,\sigma h)$-convex.

(v)

Let $ϝ\ge 0$ be co-ordinated $(g,h)$-convex, where $g\ge 0$ and $h\le 0$. Then, for all $\xi ,\eta \in [0,1]$, $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$, we have

$$\begin{array}{cc}\hfill ϝ(\xi \varkappa & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)+{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )\hfill \\ \hfill & +{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)+{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)+\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)+\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)\hfill \\ \hfill & +\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w)\hfill \\ \hfill & \le \xi \eta ϝ(\varkappa ,\gamma )+\xi (1-\eta )ϝ(\varkappa ,w)+(1-\xi )\eta ϝ(z,\gamma )+(1-\xi )(1-\eta )ϝ(z,w),\hfill \end{array}$$

which shows that $ϝ$ is co-ordinated convex. This holds similarly in the case of $g\le 0$ and $h\ge 0$.

(vi)

If $ϝ$ is co-ordinated $(g,h)$-convex and $\overline{ϝ}$ are co-ordinated $(g,\overline{h})$-convex, we may write

$$\begin{array}{cc}ϝ(\xi \varkappa \hfill & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)+{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)+\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)+\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)\hfill \\ \hfill & +\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w),\hfill \end{array}$$

(16)

and

$$\begin{array}{cc}\overline{ϝ}(\xi \varkappa \hfill & +(1-\xi )z,\eta \gamma +(1-\eta )w)\le {\xi }^2{\eta }^2\overline{ϝ}(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2\overline{ϝ}(\varkappa ,w)\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2\overline{ϝ}(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2\overline{ϝ}(z,w)+{\xi }^2\eta (1-\eta )M_7(\varkappa ,\gamma ,w)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_8(\varkappa ,z,\gamma ,w)+\xi {\eta }^2(1-\xi )M_9(\varkappa ,z,\gamma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_{10}(\varkappa ,z,\gamma ,w)+\xi (1-\xi ){(1-\eta )}^2{M}_{11}(\varkappa ,z,w)\hfill \\ \hfill & +\eta {(1-\xi )}^2(1-\eta )M_{12}(z,\gamma ,w),\hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\hfill M_7(\varkappa ,\gamma ,w)& =\overline{h}(\varkappa ,w)g(\varkappa ,\gamma )+\overline{h}(\varkappa ,\gamma )g(\varkappa ,w),\hfill \\ \hfill M_8(\varkappa ,z,\gamma ,w)& =\overline{h}(\varkappa ,w)g(z,\gamma )+\overline{h}(z,\gamma )g(\varkappa ,w),\hfill \\ \hfill M_9(\varkappa ,z,\gamma )& =\overline{h}(\varkappa ,\gamma )g(z,\gamma )+\overline{h}(z,\gamma )g(\varkappa ,\gamma ),\hfill \\ \hfill M_{10}(\varkappa ,z,\gamma ,w)& =\overline{h}(\varkappa ,\gamma )g(z,w)+\overline{h}(z,w)g(\varkappa ,\gamma ),\hfill \\ \hfill M_{11}(\varkappa ,z,w)& =\overline{h}(\varkappa ,w)g(z,w)+\overline{h}(z,w)g(\varkappa ,w),\hfill \end{array}$$

and

$$M_{12}(z,\gamma ,w)=\overline{h}(z,\gamma )g(z,w)+\overline{h}(z,w)g(z,\gamma ),$$

for all $\xi ,\eta \in [0,1]$, $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$. Adding (16) and (17), we obtain that $ϝ+\overline{ϝ}$ is co-ordinated $(g,h+\overline{h})$-convex.

(vii)

The proof is similar to that of (vi).

(viii)

Taking $\xi =\eta =\frac{1}{2}$ in (14), the desired inequality is obtained.

 □

Proposition 2.

Let $ϝ:△\to \mathbb{R}$ be a co-ordinated convex function. Then, ϝ is co-ordinated $(ϝ,1_{\mathbb{R}\times \mathbb{R}})$-convex.

Proof. 

Let $\xi ,\eta \in [0,1]$, $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$. By the co-ordinated convexity of $ϝ$, we write

$$\begin{array}{cc}\hfill & ϝ(\xi \varkappa +(1-\xi )z,\eta \gamma +(1-\eta )w)\hfill \\ \hfill & \le \xi \eta ϝ(\varkappa ,\gamma )+\xi (1-\eta )ϝ(\varkappa ,v)+(1-\xi )\eta ϝ(z,\gamma )+(1-\xi )(1-\eta )ϝ(z,w)\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)+{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )(ϝ(\varkappa ,\gamma )+ϝ(\varkappa ,w))+\xi \eta (1-\xi )(1-\eta )(ϝ(z,\gamma )+ϝ(\varkappa ,w))\hfill \\ \hfill & +\xi {\eta }^2(1-\xi )(ϝ(z,\gamma )+ϝ(\varkappa ,\gamma ))+\xi \eta (1-\xi )(1-\eta )(ϝ(z,w)+ϝ(\varkappa ,\gamma ))\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2(ϝ(z,w)+ϝ(\varkappa ,w))+\eta {(1-\xi )}^2(1-\eta )(ϝ(z,w)+ϝ(z,\gamma ))\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)+{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )(1_{\mathbb{R}\times \mathbb{R}}(\varkappa ,w)ϝ(\varkappa ,\gamma )+1_{\mathbb{R}\times \mathbb{R}}(\varkappa ,\gamma )ϝ(\varkappa ,w))\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )(1_{\mathbb{R}\times \mathbb{R}}(\varkappa ,w)ϝ(z,\gamma )+1_{\mathbb{R}\times \mathbb{R}}(z,\gamma )ϝ(\varkappa ,w))\hfill \\ \hfill & +\xi {\eta }^2(1-\xi )(1_{\mathbb{R}\times \mathbb{R}}(\varkappa ,\gamma )ϝ(z,\gamma )+1_{\mathbb{R}\times \mathbb{R}}(z,\gamma )ϝ(\varkappa ,\gamma ))\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )(1_{\mathbb{R}\times \mathbb{R}}(\varkappa ,\gamma )ϝ(z,w)+1_{\mathbb{R}\times \mathbb{R}}(z,w)ϝ(\varkappa ,\gamma ))\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2(1_{\mathbb{R}\times \mathbb{R}}(\varkappa ,w)ϝ(z,w)+1_{\mathbb{R}\times \mathbb{R}}(z,w)ϝ(\varkappa ,w))\hfill \\ \hfill & +\eta {(1-\xi )}^2(1-\eta )(1_{\mathbb{R}\times \mathbb{R}}(z,\gamma )ϝ(z,w)+1_{\mathbb{R}\times \mathbb{R}}(z,w)ϝ(z,\gamma )),\hfill \end{array}$$

which proves that $ϝ$ is co-ordinated $(ϝ,1_{\mathbb{R}\times \mathbb{R}})$-convex. □

Proposition 3.

Let $g,h:△\to \mathbb{R}$ be two co-ordinated convex functions. Then, $ϝ=gh$ is co-ordinated $(g,h)$-convex.

Proof. 

Let $\xi ,\eta \in [0,1]$, $\varkappa ,z\in [\sigma ,\rho ]$ and $\gamma ,w\in [\varsigma ,d]$. Since $g,h:△\to \mathbb{R}$ are two functions with the co-ordinated convexity property, we have

$$\begin{array}{cc}\hfill ϝ(& \xi \varkappa +(1-\xi )z,\eta \gamma +(1-\eta )w)\hfill \\ \hfill & =g(\xi \varkappa +(1-\xi )z,\eta \gamma +(1-\eta )w)h(\xi \varkappa +(1-\xi )z,\eta \gamma +(1-\eta )w)\hfill \\ \hfill & \le \left[\xi \eta g(\varkappa ,\gamma )+\xi (1-\eta )g(\varkappa ,w)+(1-\xi )\eta g(z,\gamma )+(1-\xi )(1-\eta )g(z,w)\right]\hfill \\ \hfill & \times \left[\xi \eta h(\varkappa ,\gamma )+\xi (1-\eta )h(\varkappa ,w)+(1-\xi )\eta h(z,\gamma )+(1-\xi )(1-\eta )h(z,w)\right]\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\varkappa ,\gamma )+{\xi }^2{(1-\eta )}^2ϝ(\varkappa ,w)+{(1-\xi )}^2{\eta }^2ϝ(z,\gamma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(z,w)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,w)+\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,z,\gamma ,w)+\xi {\eta }^2(1-\xi )M_3(\varkappa ,z,\gamma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,z,\gamma ,w)+\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,z,w)\hfill \\ \hfill & +\eta {(1-\xi )}^2(1-\eta )M_6(z,\gamma ,w).\hfill \end{array}$$

This gurantees that $ϝ$ is co-ordinated $(g,h)$-convex. □

Example 1.

Let $ϝ:{[0,1]}^2\to \mathbb{R}$ be a function as

$$ϝ(\varkappa ,\gamma )=\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }.$$

Consider the functions $g,h:[0,1]\times [0,1]\to \mathbb{R}$ defined by

$$g(\varkappa ,\gamma )=\varkappa \gamma ,$$

and

$$h(\varkappa ,\gamma )={\mathrm{e}}^{-\varkappa -\gamma }.$$

Since $g,h$ are non-negative co-ordinated convex functions, by Proposition 3, $ϝ=gh$ is a co-ordinated $(g,h)$-convex function.

3. Hermite–Hadamard-Type Inequalities

Now, we here derive some new Hermite–Hadamard-type inequalities for co-ordinated $(g,h)$-convex functions. We also show that our results can be reduced to previous work.

Theorem 8.

Let $ϝ:△\to \mathbb{R}$ be a function with the co-ordinated $(g,h)$-convexity on △. If $ϝ\in L^1(△)$ and $g,h\in L^2(△)$, then the following Hermite–Hadamard-type inequalities are formulated as

$$\begin{array}{cc}\hfill & 4ϝ\left(\frac{\sigma +\rho }{2},\frac{\varsigma +d}{2}\right)-\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )\left[g(\varkappa ,\varsigma +d-\gamma )+g(\sigma +\rho -\varkappa ,\gamma )\right.\hfill \\ \hfill & \left.+g(\sigma +\rho -\varkappa ,\varsigma +d-\gamma )\right]d\gamma d\varkappa \hfill \\ \hfill & \le \frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(\varkappa ,\gamma )d\gamma d\varkappa \hfill \\ \hfill & \le \frac{1}{9}\left[ϝ(\sigma ,\varsigma )+ϝ(\sigma ,d)+ϝ(\rho ,\varsigma )+ϝ(\rho ,d)+\frac{K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)+M(\sigma ,\rho ,\varsigma )+O(\sigma ,\rho ,d)}{2}\right.\hfill \\ \hfill & +\left.\frac{L(\sigma ,\rho ,\varsigma ,d)+N(\sigma ,\rho ,\varsigma ,d)}{4}\right],\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill K(\sigma ,\varsigma ,d)& =h(\sigma ,d)g(\sigma ,\varsigma )+h(\sigma ,\varsigma )g(\sigma ,d),\hfill \\ \hfill L(\sigma ,\rho ,\varsigma ,d)& =h(\sigma ,d)g(\rho ,\varsigma )+h(\rho ,\varsigma )g(\sigma ,d),\hfill \\ \hfill M(\sigma ,\rho ,\varsigma )& =h(\sigma ,\varsigma )g(\rho ,\varsigma )+h(\rho ,\varsigma )g(\sigma ,\varsigma ),\hfill \\ \hfill N(\sigma ,\rho ,\varsigma ,d)& =h(\sigma ,\varsigma )g(\rho ,d)+h(\rho ,d)g(\sigma ,\varsigma ),\hfill \\ \hfill O(\sigma ,\rho ,d)& =h(\sigma ,d)g(\rho ,d)+h(\rho ,d)g(\sigma ,d),\hfill \end{array}$$

and

$$P(\rho ,\varsigma ,d)=h(\rho ,\varsigma )g(\rho ,d)+h(\rho ,d)g(\rho ,\varsigma ).$$

Proof. 

Let

$$\begin{array}{cc}\hfill u_1(\xi )& =\xi \sigma +(1-\xi )\rho ,u_2(\xi )=(1-\xi )\sigma +\xi \rho ,\hfill \\ \hfill v_1(\eta )& =\eta \varsigma +(1-\eta )d,v_2(\eta )=(1-\eta )\varsigma +\eta d,\hfill \end{array}$$

for $\xi ,\eta \in [0,1]$.

By the co-ordinated $(g,h)$-convexity of $ϝ$ on $[\sigma ,\rho ]\times [\varsigma ,d]$, we have

$$\begin{array}{cc}\hfill & 16ϝ\left(\frac{\sigma +\rho }{2},\frac{\varsigma +d}{2}\right)=16ϝ\left(\frac{u_1(\xi )+u_2(\xi )}{2},\frac{v_1(\eta )+v_2(\eta )}{2}\right)\hfill \\ \hfill & \le ϝ\left(u_1(\xi ),v_1(\eta )\right)+ϝ\left(u_1(\xi ),v_2(\eta )\right)+ϝ\left(u_2(\xi ),v_1(\eta )\right)+ϝ\left(u_2(\xi ),v_2(\eta )\right)\hfill \\ \hfill & +M_1\left(u_1(\xi ),v_1(\eta ),v_2(\eta )\right)+M_2\left(u_1(\xi ),u_2(\xi ),v_1(\eta ),v_2(\eta )\right)+M_3\left(u_1(\xi ),u_2(\xi ),v_1(\eta )\right)\hfill \\ \hfill & +M_4\left(u_1(\xi ),u_2(\xi ),v_1(\eta ),v_2(\eta )\right)+M_5\left(u_1(\xi ),u_2(\xi ),v_2(\eta )\right)+M_6\left(u_2(\xi ),v_1(\eta ),v_2(\eta )\right).\hfill \end{array}$$

(19)

Integrating both sides of (19) over $[0,1]\times [0,1]$ with respect to $\xi$ and $\eta$, we obtain

$$\begin{array}{cc}16ϝ\hfill & (\frac{\sigma +\rho }{2},\frac{\varsigma +d}{2})\le {\int }_0^1{\int }_0^1ϝ\left(u_1(\xi ),v_1(\eta )\right)+ϝ\left(u_1(\xi ),v_2(\eta )\right)+ϝ\left(u_2(\xi ),v_1(\eta )\right)\hfill \\ \hfill & +ϝ\left(u_2(\xi ),v_2(\eta )\right)+M_1\left(u_1(\xi ),v_1(\eta ),v_2(\eta )\right)+M_2\left(u_1(\xi ),u_2(\xi ),v_1(\eta ),v_2(\eta )\right)\hfill \\ \hfill & +M_3\left(u_1(\xi ),u_2(\xi ),v_1(\eta )\right)+M_4\left(u_1(\xi ),u_2(\xi ),v_1(\eta ),v_2(\eta )\right)\hfill \\ \hfill & +M_5\left(u_1(\xi ),u_2(\xi ),v_2(\eta )\right)+M_6\left(u_2(\xi ),v_1(\eta ),v_2(\eta )\right)d\eta d\xi .\hfill \end{array}$$

(20)

We consider

$$\begin{array}{cc}\hfill {\int }_0^1{\int }_0^1ϝ\left(u_1(\xi ),v_1(\eta )\right)d\eta d\xi & ={\int }_0^1{\int }_0^1ϝ\left(\xi \sigma +(1-\xi )\rho ,\eta \varsigma +(1-\eta )d\right)d\eta d\xi \hfill \\ \hfill & =\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(\varkappa ,\gamma )d\gamma d\varkappa \hfill \\ \hfill & ={\int }_0^1{\int }_0^1ϝ\left(u_1(\xi ),v_2(\eta )\right)d\eta d\xi \hfill \\ \hfill & ={\int }_0^1{\int }_0^1ϝ\left(u_2(\xi ),v_1(\eta )\right)d\eta d\xi \hfill \\ \hfill & ={\int }_0^1{\int }_0^1ϝ\left(u_2(\xi ),v_2(\eta )\right)d\eta d\xi .\hfill \end{array}$$

Moreover, we have

$$\begin{array}{cc}\hfill {\int }_0^1{\int }_0^1& M_1\left(u_1(\xi ),v_1(\eta ),v_2(\eta )\right)d\eta d\xi ={\int }_0^1{\int }_0^{1}h\left(u_1(\xi ),v_2(\eta )\right)g\left(u_1(\xi ),v_1(\eta )\right)d\eta d\xi \hfill \\ \hfill & +{\int }_0^1{\int }_0^{1}h\left(u_1(\xi ),v_1(\eta )\right)g\left(u_1(\xi ),v_2(\eta )\right)d\eta d\xi \hfill \\ \hfill & ={\int }_0^1{\int }_0^{1}h\left(\xi \sigma +(1-\xi )\rho ,(1-\eta )\varsigma +\eta d\right)g\left(\xi \sigma +(1-\xi )\rho ,\eta \varsigma +(1-\eta )d\right)d\eta d\xi \hfill \\ \hfill & +{\int }_0^1{\int }_0^{1}h\left(\xi \sigma +(1-\xi )\rho ,\eta \varsigma +(1-\eta )d\right)g\left(\xi \sigma +(1-\xi )\rho ,(1-\eta )\varsigma +\eta d\right)d\eta d\xi \hfill \\ \hfill & =\frac{2}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )g(\varkappa ,\varsigma +d-\gamma )d\gamma d\varkappa .\hfill \end{array}$$

Similarly, we get

$$\begin{array}{cc}\hfill {\int }_0^1{\int }_0^1{M}_2\left(u_1(\xi ),u_2(\xi ),v_1(\eta ),v_2(\eta )\right)d\eta d\xi & =\frac{2}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )g(\sigma +\rho -\varkappa ,\varsigma +d-\gamma )d\gamma d\varkappa ,\hfill \\ \hfill {\int }_0^1{\int }_0^1{M}_3\left(u_1(\xi ),u_2(\xi ),v_1(\eta )\right)d\eta d\xi & =\frac{2}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )g(\sigma +\rho -\varkappa ,\varsigma +d-\gamma )d\gamma d\varkappa ,\hfill \\ \hfill {\int }_0^1{\int }_0^1{M}_4\left(u_1(\xi ),u_2(\xi ),v_1(\eta ),v_2(\eta )\right)d\eta d\xi & =\frac{2}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )g(\sigma +\rho -\varkappa ,\varsigma +d-\gamma )d\gamma d\varkappa ,\hfill \\ \hfill {\int }_0^1{\int }_0^1{M}_5\left(u_1(\xi ),u_2(\xi ),v_2(\eta )\right)d\eta d\xi & =\frac{2}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )g(\sigma +\rho -\varkappa ,\gamma )d\gamma d\varkappa ,\hfill \end{array}$$

and

$${\int }_0^1{\int }_0^1{M}_5\left(u_2(\xi ),v_1(\eta ),v_2(\eta )\right)d\eta d\xi =\frac{2}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )g(\varkappa ,\varsigma +d-\gamma )d\gamma d\varkappa .$$

Thus, (20) becomes

$$\begin{array}{cc}\hfill 4ϝ\left(\frac{\sigma +\rho }{2},\frac{\varsigma +d}{2}\right)& \le \frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(\varkappa ,\gamma )d\gamma d\varkappa \hfill \\ \hfill & +\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^{d}h(\varkappa ,\gamma )\left[g(\varkappa ,\varsigma +d-\gamma )\right.\hfill \\ \hfill & +\left.g(\sigma +\rho -\varkappa ,\gamma )+g(\sigma +\rho -\varkappa ,\varsigma +d-\gamma )\right]d\gamma d\varkappa .\hfill \end{array}$$

Therefore, the first inequality of (18) is proven.

Since $ϝ$ is co-ordinated $(g,h)$-convex on $[\sigma ,\rho ]\times [\varsigma ,d]$, we have

$$\begin{array}{cc}\hfill & ϝ(\xi \sigma +(1-\xi )\rho ,\eta \varsigma +(1-\eta )d)\hfill \\ \hfill & \le {\xi }^2{\eta }^2ϝ(\sigma ,\varsigma )+{\xi }^2{(1-\eta )}^2ϝ(\sigma ,d)+{(1-\xi )}^2{\eta }^2ϝ(\rho ,\varsigma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(\rho ,d)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\sigma ,\varsigma ,d)+\xi \eta (1-\xi )(1-\eta )M_2(\sigma ,\rho ,\varsigma ,d)+\xi {\eta }^2(1-\xi )M_3(\sigma ,\rho ,\varsigma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\sigma ,\rho ,\varsigma ,d)+\xi (1-\xi ){(1-\eta )}^2{M}_5(\sigma ,\rho ,d)+\eta {(1-\xi )}^2(1-\eta )M_6(\rho ,\varsigma ,d)\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\sigma ,\varsigma )+{\xi }^2{(1-\eta )}^2ϝ(\sigma ,d)+{(1-\xi )}^2{\eta }^2ϝ(\rho ,\varsigma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(\rho ,d)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )K(\sigma ,\varsigma ,d)+\xi \eta (1-\xi )(1-\eta )L(\sigma ,\rho ,\varsigma ,d)+\xi {\eta }^2(1-\xi )M(\sigma ,\rho ,\varsigma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )N(\sigma ,\rho ,\varsigma ,d)+\xi (1-\xi ){(1-\eta )}^{2}O(\sigma ,\rho ,d)+\eta {(1-\xi )}^2(1-\eta )P(\rho ,\varsigma ,d),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & ϝ(\xi \sigma +(1-\xi )\rho ,(1-\eta )\varsigma +\eta d)\hfill \\ \hfill & \le {\xi }^2{\eta }^2ϝ(\sigma ,d)+{\xi }^2{(1-\eta )}^2ϝ(\sigma ,\varsigma )+{(1-\xi )}^2{\eta }^2ϝ(\rho ,d)+{(1-\xi )}^2{(1-\eta )}^2ϝ(\rho ,\varsigma )\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\sigma ,d,\varsigma )+\xi \eta (1-\xi )(1-\eta )M_2(\sigma ,\rho ,d,\varsigma )+\xi {\eta }^2(1-\xi )M_3(\sigma ,\rho ,d)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\sigma ,\rho ,d,\varsigma )+\xi (1-\xi ){(1-\eta )}^2{M}_5(\sigma ,\rho ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )M_6(\rho ,d,\varsigma )\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\sigma ,d)+{\xi }^2{(1-\eta )}^2ϝ(\sigma ,\varsigma )+{(1-\xi )}^2{\eta }^2ϝ(\rho ,d)+{(1-\xi )}^2{(1-\eta )}^2ϝ(\rho ,\varsigma )\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )K(\sigma ,\varsigma ,d)+\xi \eta (1-\xi )(1-\eta )N(\sigma ,\rho ,\varsigma ,d)+\xi {\eta }^2(1-\xi )O(\sigma ,\rho ,d)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )L(\sigma ,\rho ,\varsigma ,d)+\xi (1-\xi ){(1-\eta )}^{2}M(\sigma ,\rho ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )P(\rho ,\varsigma ,d),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & ϝ((1-\xi )\sigma +\xi \rho ,\eta \varsigma +(1-\eta )d)\hfill \\ \hfill & \le {\xi }^2{\eta }^2ϝ(\rho ,\varsigma )+{\xi }^2{(1-\eta )}^2ϝ(\rho ,d)+{(1-\xi )}^2{\eta }^2ϝ(\sigma ,\varsigma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(\sigma ,d)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\rho ,\varsigma ,d)+\xi \eta (1-\xi )(1-\eta )M_2(\rho ,\sigma ,\varsigma ,d)+\xi {\eta }^2(1-\xi )M_3(\rho ,\sigma ,\varsigma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\rho ,\sigma ,\varsigma ,d)+\xi (1-\xi ){(1-\eta )}^2{M}_5(\rho ,\sigma ,d)+\eta {(1-\xi )}^2(1-\eta )M_6(\sigma ,\varsigma ,d)\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\rho ,\varsigma )+{\xi }^2{(1-\eta )}^2ϝ(\rho ,d)+{(1-\xi )}^2{\eta }^2ϝ(\sigma ,\varsigma )+{(1-\xi )}^2{(1-\eta )}^2ϝ(\sigma ,d)\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )P(\rho ,\varsigma ,d)+\xi \eta (1-\xi )(1-\eta )N(\sigma ,\rho ,\varsigma ,d)+\xi {\eta }^2(1-\xi )M(\sigma ,\rho ,\varsigma )\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )L(\sigma ,\rho ,\varsigma ,d)+\xi (1-\xi ){(1-\eta )}^{2}O(\sigma ,\rho ,d)+\eta {(1-\xi )}^2(1-\eta )K(\sigma ,\varsigma ,d),\hfill \end{array}$$

(23)

and

$$\begin{array}{cc}\hfill & ϝ((1-\xi )\sigma +\xi \rho ,(1-\eta )\varsigma +\eta d)\hfill \\ \hfill & \le {\xi }^2{\eta }^2ϝ(\rho ,d)+{\xi }^2{(1-\eta )}^2ϝ(\rho ,\varsigma )+{(1-\xi )}^2{\eta }^2ϝ(\sigma ,d)+{(1-\xi )}^2{(1-\eta )}^2ϝ(\sigma ,\varsigma )\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\rho ,d,\varsigma )+\xi \eta (1-\xi )(1-\eta )M_2(\rho ,\sigma ,d,\varsigma )+\xi {\eta }^2(1-\xi )M_3(\rho ,\sigma ,d)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )M_4(\rho ,\sigma ,d,\varsigma )+\xi (1-\xi ){(1-\eta )}^2{M}_5(\rho ,\sigma ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )M_6(\sigma ,d,\varsigma )\hfill \\ \hfill & ={\xi }^2{\eta }^2ϝ(\rho ,d)+{\xi }^2{(1-\eta )}^2ϝ(\rho ,\varsigma )+{(1-\xi )}^2{\eta }^2ϝ(\sigma ,d)+{(1-\xi )}^2{(1-\eta )}^2ϝ(\sigma ,\varsigma )\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )P(\rho ,\varsigma ,d)+\xi \eta (1-\xi )(1-\eta )L(\sigma ,\rho ,\varsigma ,d)+\xi {\eta }^2(1-\xi )O(\sigma ,\rho ,d)\hfill \\ \hfill & +\xi \eta (1-\xi )(1-\eta )N(\sigma ,\rho ,\varsigma ,d)+\xi (1-\xi ){(1-\eta )}^{2}M(\sigma ,\rho ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )K(\sigma ,\varsigma ,d).\hfill \end{array}$$

(24)

Combining (21)–(24), we get

$$\begin{array}{cc}\hfill & ϝ(\xi \sigma +(1-\xi )\rho ,\eta \varsigma +(1-\eta )d)+ϝ(\xi \sigma +(1-\xi )\rho ,(1-\eta )\varsigma +\eta d)\hfill \\ \hfill & +ϝ((1-\xi )\sigma +\xi \rho ,\eta \varsigma +(1-\eta )d)+ϝ((1-\xi )\sigma +\xi \rho ,(1-\eta )\varsigma +\eta d)\hfill \\ \hfill & \le (1-2\xi +2{\xi }^2)(1-2\eta +2{\eta }^2)\left[ϝ(\sigma ,\varsigma )+ϝ(\sigma ,d)+ϝ(\rho ,\varsigma )+ϝ(\rho ,d)\right]\hfill \\ \hfill & +2\eta (1-\eta )(1-2\xi +2{\xi }^2)\left[K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)\right]\hfill \\ \hfill & +2\xi (1-\xi )(1-2\eta +2{\eta }^2)\left[M(\sigma ,\rho ,\varsigma )+O(\sigma ,\rho ,d)\right]\hfill \\ \hfill & +4\xi \eta (1-\xi )(1-\eta )\left[K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)\right].\hfill \end{array}$$

(25)

Integrating both sides of (25) over $[0,1]\times [0,1]$ with respect to $\xi$ and $\eta$, we obtain

$$\begin{array}{cc}\hfill \frac{4}{(\rho -\sigma )(d-\varsigma )}& {\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(\varkappa ,\gamma )d\gamma d\varkappa ={\int }_0^1{\int }_0^1ϝ(\xi \sigma +(1-\xi )\rho ,\eta \varsigma +(1-\eta )d)d\eta d\xi \hfill \\ \hfill & +{\int }_0^1{\int }_0^1ϝ(\xi \sigma +(1-\xi )\rho ,(1-\eta )\varsigma +\eta d)d\eta d\xi \hfill \\ \hfill & +{\int }_0^1{\int }_0^1ϝ((1-\xi )\sigma +\xi \rho ,\eta \varsigma +(1-\eta )d)d\eta d\xi \hfill \\ \hfill & +{\int }_0^1{\int }_0^1ϝ((1-\xi )\sigma +\xi \rho ,(1-\eta )\varsigma +\eta d)d\eta d\xi \hfill \\ \hfill & \le \left[ϝ(\sigma ,\varsigma )+ϝ(\sigma ,d)+ϝ(\rho ,\varsigma )+ϝ(\rho ,d)\right]\hfill \\ \hfill & \times {\int }_0^1{\int }_0^1(1-2\xi +2{\xi }^2)(1-2\eta +2{\eta }^2)d\eta d\xi \hfill \\ \hfill & +\left[K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)\right]{\int }_0^1{\int }_0^{1}2\eta (1-\eta )(1-2\xi +2{\xi }^2)d\eta d\xi \hfill \\ \hfill & +\left[M(\sigma ,\rho ,\varsigma )+O(\sigma ,\rho ,d)\right]{\int }_0^1{\int }_0^{1}2\xi (1-\xi )(1-2\eta +2{\eta }^2)d\eta d\xi \hfill \\ \hfill & +\left[K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)\right]{\int }_0^1{\int }_0^{1}4\xi \eta (1-\xi )(1-\eta )d\eta d\xi \hfill \\ \hfill & =\frac{4}{9}\left[ϝ(\sigma ,\varsigma )+ϝ(\sigma ,d)+ϝ(\rho ,\varsigma )+ϝ(\rho ,d)\right]+\frac{2}{9}[K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)\hfill \\ \hfill & +M(\sigma ,\rho ,\varsigma )+O(\sigma ,\rho ,d)]+\frac{1}{9}\left[K(\sigma ,\varsigma ,d)+P(\rho ,\varsigma ,d)\right].\hfill \end{array}$$

Multiplying the above inequality by $\frac{1}{4}$, the second inequality of (18) is derived. □

Remark 3.

If we consider $h(\varkappa ,\gamma )=ϝ(\varkappa ,\gamma )$ and $g(\varkappa ,\gamma )=1$, then the inequalities (18) reduces to the inequalities

$$ϝ\left(\frac{\sigma +\rho }{2},\frac{\varsigma +d}{2}\right)\le \frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(\varkappa ,\gamma )d\gamma d\varkappa \le \frac{ϝ(\sigma ,\varsigma )+ϝ(\sigma ,d)+ϝ(\rho ,\varsigma )+ϝ(\rho ,d)}{4}.$$

This inequality is presented in Theorem 2.

4. Ostrowski-Type Inequalities

In this position, we derive some new Ostrowski-type inequalities for $\left(g,h\right)$-convex functions. To establish the inequalities of the current section, we receive help from a lemma.

Lemma 1

([22]). Let $ϝ:△\to \mathbb{R}$ be a twice-partial differentiable function on ${△}^{\circ }$. If ${\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}\in L(△)$, then

$$\begin{array}{cc}ϝ(\varkappa ,\gamma )\hfill & +\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A\hfill \\ \hfill & =\frac{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta {\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )d\eta d\xi \hfill \\ \hfill & -\frac{{(\varkappa -\sigma )}^2{(d-\gamma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta {\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )d)d\eta d\xi \hfill \\ \hfill & -\frac{{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta {\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )\varsigma )d\eta d\xi \hfill \\ \hfill & +\frac{{(\rho -\varkappa )}^2{(d-\gamma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta {\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )d)d\eta d\xi ,\hfill \end{array}$$

(26)

for all $(\varkappa ,\gamma )\in △$, where A is defined in (7).

Theorem 9.

Under the assumptions of Lemma 1, if $\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|$ is co-ordinated $(g,h)$-convex on △, then

$$\begin{array}{cc}\hfill & |ϝ(\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\le \frac{1}{144(\rho -\sigma )(d-\varsigma )}\hfill \\ \hfill & \times \left\{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|\right.\right.\hfill \\ \hfill & +\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\sigma ,\gamma ,\varsigma )+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,\varsigma )+O(\varkappa ,\sigma ,\varsigma )+P(\sigma ,\gamma ,\varsigma )\right]\hfill \\ \hfill & +{(\varkappa -\sigma )}^2{(d-\gamma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|\right.\hfill \\ \hfill & +\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\sigma ,\gamma ,d)+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,d)+O(\varkappa ,\sigma ,d)+P(\sigma ,\gamma ,d)\right]\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|\right.\hfill \\ \hfill & +\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,\varsigma )+O(\varkappa ,\rho ,\varsigma )+P(\rho ,\gamma ,\varsigma )\right]\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(d-\gamma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|\right.\hfill \\ \hfill & +\left.\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,d)+O(\varkappa ,\rho ,d)+P(\rho ,\gamma ,d)\right]\right\},\hfill \end{array}$$

(27)

where A is defined as in (7), and $K,L,M,N,O,P$ are defined as in Theorem 8.

Proof. 

The conclusion of Lemma 1 yields

$$\begin{array}{cc}|ϝ\hfill & (\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\hfill \\ \hfill & \le \frac{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & +\frac{{(\varkappa -\sigma )}^2{(d-\gamma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & +\frac{{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & +\frac{{(\rho -\varkappa )}^2{(d-\gamma )}^2}{(\rho -\sigma )(d-\varsigma )}{\int }_0^1{\int }_0^1\xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi .\hfill \end{array}$$

(28)

By the co-ordinated $(g,h)$-convexity of $\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|$, we obtain

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & \le {\int }_0^1{\int }_0^1\xi \eta \left[{\xi }^2{\eta }^2\right|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )|+{\xi }^2{(1-\eta )}^2|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )|\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+{(1-\xi )}^2{(1-\eta )}^2\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,\varsigma )+\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,\sigma ,\gamma ,\varsigma )\hfill \\ \hfill & +\xi {\eta }^2(1-\xi )M_3(\varkappa ,\sigma ,\gamma )+\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,\sigma ,\gamma ,\varsigma )\hfill \\ \hfill & +\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,\sigma ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )M_6(\sigma ,\gamma ,\varsigma )]d\eta d\xi \hfill \\ \hfill & =\frac{1}{16}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+\frac{1}{144}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,\varsigma )+\frac{1}{144}L(\varkappa ,\sigma ,\gamma ,\varsigma )+\frac{1}{48}M(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +\frac{1}{144}N(\varkappa ,\sigma ,\gamma ,\varsigma )+\frac{1}{144}O(\varkappa ,\sigma ,\varsigma )+\frac{1}{144}P(\sigma ,\gamma ,\varsigma ).\hfill \end{array}$$

(29)

Similarly, we have

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & \le \frac{1}{16}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+\frac{1}{144}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,d)+\frac{1}{144}L(\varkappa ,\sigma ,\gamma ,d)+\frac{1}{48}M(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +\frac{1}{144}N(\varkappa ,\sigma ,\gamma ,d)+\frac{1}{144}O(\varkappa ,\sigma ,d)+\frac{1}{144}P(\sigma ,\gamma ,d),\hfill \end{array}$$

(30)

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & \le \frac{1}{16}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|+\frac{1}{144}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,\varsigma )+\frac{1}{144}L(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{48}M(\varkappa ,\rho ,\gamma )\hfill \\ \hfill & +\frac{1}{144}N(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{144}O(\varkappa ,\rho ,\varsigma )+\frac{1}{144}P(\rho ,\gamma ,\varsigma ),\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & \le \frac{1}{16}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|+\frac{1}{48}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|+\frac{1}{144}\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,d)+\frac{1}{144}L(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{48}M(\varkappa ,\rho ,\gamma )\hfill \\ \hfill & +\frac{1}{144}N(\varkappa ,\rho ,\gamma ,d)+\frac{1}{144}O(\varkappa ,\rho ,d)+\frac{1}{144}P(\rho ,\gamma ,d).\hfill \end{array}$$

(32)

Substituting the inequalities (29)–(32) in the inequality (28), we obtain

$$\begin{array}{cc}\hfill |ϝ& (\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\le \frac{1}{144(\rho -\sigma )(d-\varsigma )}\hfill \\ \hfill & \times \left\{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|\right.\right.\hfill \\ \hfill & +\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\sigma ,\gamma ,\varsigma )+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,\varsigma )+O(\varkappa ,\sigma ,\varsigma )+P(\sigma ,\gamma ,\varsigma )\right]\hfill \\ \hfill & +{(\varkappa -\sigma )}^2{(d-\gamma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|\right.\hfill \\ \hfill & +\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\sigma ,\gamma ,d)+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,d)+O(\varkappa ,\sigma ,d)+P(\sigma ,\gamma ,d)\right]\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|\right.\hfill \\ \hfill & +\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,\varsigma )+O(\varkappa ,\rho ,\varsigma )+P(\rho ,\gamma ,\varsigma )\right]\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(d-\gamma )}^2\left[9\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|+3\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|+\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|\right.\hfill \\ \hfill & +\left.\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,d)+O(\varkappa ,\rho ,d)+P(\rho ,\gamma ,d)\right]\right\}.\hfill \end{array}$$

The proof is completed. □

Remark 4.

Set $h(\varkappa ,\gamma )=\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|$, $g(\varkappa ,\gamma )=1$ and $\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|\le M$ for all $(\varkappa ,\gamma )\in △$. In that case, the inequality (28) reduces to the inequality (6).

Theorem 10.

Under the assumptions of Lemma 1, if ${\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|}^q$ is co-ordinated $(g,h)$-convex on △ for $\frac{1}{p}+\frac{1}{q}=1$ and $p,q>1$, then

$$\begin{array}{cc}|ϝ\hfill & (\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\le \frac{1}{(\rho -\sigma )(d-\varsigma ){(p+1)}^{2/p}}{\left(\frac{1}{36}\right)}^{1/q}\hfill \\ \hfill & \times \left\{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\right.\right.\hfill \\ \hfill & +{\left.2K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\sigma ,\gamma ,\varsigma )+2M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,\varsigma )+2O(\varkappa ,\sigma ,\varsigma )+2P(\sigma ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\varkappa -\sigma )}^2{(d-\gamma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|}^q\right.\hfill \\ \hfill & +{\left.2K(\varkappa ,\gamma ,d)+L(\varkappa ,\sigma ,\gamma ,d)+2M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,d)+2O(\varkappa ,\sigma ,d)+2P(\sigma ,\gamma ,d)\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +{\left.2K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\rho ,\gamma ,\varsigma )+2M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,\varsigma )+2O(\varkappa ,\rho ,\varsigma )+2P(\rho ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(d-\gamma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|}^q\right.\hfill \\ \hfill & +\left.{\left.2K(\varkappa ,\gamma ,d)+L(\varkappa ,\rho ,\gamma ,\varsigma )+2M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,d)+2O(\varkappa ,\rho ,d)+2P(\rho ,\gamma ,d)\right]}^{1/q}\right\},\hfill \end{array}$$

(33)

where A is defined as in (7) and $K,L,M,N,O,P$ are defined as in Theorem 8.

Proof. 

By the Hölder inequality for double integrals and co-ordinated $(g,h)$-convexity of ${\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|}^q$, with the help of Lemma 1, we have

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & \le {\left({\int }_0^1{\int }_0^1{\xi }^p{\eta }^{p}d\eta d\xi \right)}^{1/p}{\left({\int }_0^1{\int }_0^1{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )\right|}^{q}d\eta d\xi \right)}^{1/q}\hfill \\ \hfill & \le \frac{1}{{(p+1)}^{2/p}}\left({\int }_0^1{\int }_0^1{\xi }^2{\eta }^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+{\xi }^2{(1-\eta )}^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+{(1-\xi )}^2{(1-\eta )}^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & +{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,\varsigma )+\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,\sigma ,\gamma ,\varsigma )+\xi {\eta }^2(1-\xi )M_3(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +{\left.\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,\sigma ,\gamma ,\varsigma )+\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,\sigma ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )M_6(\sigma ,\gamma ,\varsigma )d\eta d\xi \right)}^{1/q}\hfill \\ \hfill & =\frac{1}{{(p+1)}^{2/p}}\left(\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +\frac{1}{18}K(\varkappa ,\gamma ,\varsigma )+\frac{1}{36}L(\varkappa ,\sigma ,\gamma ,\varsigma )+\frac{1}{18}M(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +{\left.\frac{1}{36}N(\varkappa ,\sigma ,\gamma ,\varsigma )+\frac{1}{18}O(\varkappa ,\sigma ,\varsigma )+\frac{1}{18}P(\sigma ,\gamma ,\varsigma )\right)}^{1/q}.\hfill \end{array}$$

Similarly, we obtain

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & \le \frac{1}{{(p+1)}^{2/p}}(\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+\frac{1}{9}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q\hfill \\ \hfill & +\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d){|}^q+\frac{1}{18}K(\varkappa ,\gamma ,d)+\frac{1}{36}L(\varkappa ,\sigma ,\gamma ,d)+\frac{1}{18}M(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +\frac{1}{36}N(\varkappa ,\sigma ,\gamma ,d)+\frac{1}{18}O(\varkappa ,\sigma ,d)+\frac{1}{18}P(\sigma ,\gamma ,d){)}^{1/q},\hfill \end{array}$$

(35)

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & \le \frac{1}{{(p+1)}^{2/p}}(\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma ){|}^q+\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma ){|}^q+\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma ){|}^q\hfill \\ \hfill & +\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma ){|}^q+\frac{1}{18}K(\varkappa ,\gamma ,\varsigma )+\frac{1}{36}L(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{18}M(\varkappa ,\rho ,\gamma )\hfill \\ \hfill & +\frac{1}{36}N(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{18}O(\varkappa ,\rho ,\varsigma )+\frac{1}{18}P(\rho ,\gamma ,\varsigma ){)}^{1/q},\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & \le \frac{1}{{(p+1)}^{2/p}}(\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma ){|}^q+\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d){|}^q+\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma ){|}^q\hfill \\ \hfill & +\frac{1}{9}|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d){|}^q+\frac{1}{18}K(\varkappa ,\gamma ,d)+\frac{1}{36}L(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{18}M(\varkappa ,\rho ,\gamma )\hfill \\ \hfill & +\frac{1}{36}N(\varkappa ,\rho ,\gamma ,d)+\frac{1}{18}O(\varkappa ,\rho ,d)+\frac{1}{18}P(\rho ,\gamma ,d){)}^{1/q}.\hfill \end{array}$$

(37)

If we substitute the inequalities (35)–(37) in the inequality (28), we obtain

$$\begin{array}{cc}\hfill |ϝ& (\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\le \frac{1}{(\rho -\sigma )(d-\varsigma ){(p+1)}^{2/p}}{\left(\frac{1}{36}\right)}^{1/q}\hfill \\ \hfill & \times \left\{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\right.\right.\hfill \\ \hfill & +{\left.2K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\sigma ,\gamma ,\varsigma )+2M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,\varsigma )+2O(\varkappa ,\sigma ,\varsigma )+2P(\sigma ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\varkappa -\sigma )}^2{(d-\gamma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|}^q\right.\hfill \\ \hfill & +{\left.2K(\varkappa ,\gamma ,d)+L(\varkappa ,\sigma ,\gamma ,d)+2M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,d)+2O(\varkappa ,\sigma ,d)+2P(\sigma ,\gamma ,d)\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +{\left.2K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\rho ,\gamma ,\varsigma )+2M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,\varsigma )+2O(\varkappa ,\rho ,\varsigma )+2P(\rho ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(d-\gamma )}^2\left[4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+4{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|}^q\right.\hfill \\ \hfill & +\left.{\left.2K(\varkappa ,\gamma ,d)+L(\varkappa ,\rho ,\gamma ,\varsigma )+2M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,d)+2O(\varkappa ,\rho ,d)+2P(\rho ,\gamma ,d)\right]}^{1/q}\right\}.\hfill \end{array}$$

The proof is ended. □

Remark 5.

Set $h(\varkappa ,\gamma )={\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q$, $g(\varkappa ,\gamma )=1$ and $\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|\le M$ for all $(\varkappa ,\gamma )\in △$. Then, the inequality (33) reduces to the inequality (8).

Theorem 11.

Under the assumptions of Lemma 1, if ${\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|}^q$ is co-ordinated $(g,h)$-convex on △ for $q\ge 1$, then

$$\begin{array}{cc}|ϝ\hfill & (\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\le \frac{1}{4(\rho -\sigma )(d-\varsigma )}{\left(\frac{1}{36}\right)}^{1/q}\hfill \\ \hfill & \times \left\{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\right.\right.\hfill \\ \hfill & +{\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\sigma ,\gamma ,\varsigma )+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,\varsigma )+O(\varkappa ,\sigma ,\varsigma )+P(\sigma ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\varkappa -\sigma )}^2{(d-\gamma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|}^q\right.\hfill \\ \hfill & +{\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\sigma ,\gamma ,d)+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,d)+O(\varkappa ,\sigma ,d)+P(\sigma ,\gamma ,d)\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +{\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,\varsigma )+O(\varkappa ,\rho ,\varsigma )+P(\rho ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(d-\gamma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|}^q\right.\hfill \\ \hfill & +\left.{\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,d)+O(\varkappa ,\rho ,d)+P(\rho ,\gamma ,d)\right]}^{1/q}\right\},\hfill \end{array}$$

(38)

where A is defined as in (7) and $K,L,M,N,O,P$ are defined as in Theorem 8.

Proof. 

By vitue of the power mean inequality in relation to double integrals and by the co-ordinated $(g,h)$-convexity of ${\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|}^q$, with the help of Lemma 1, we have

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & \le {\left({\int }_0^1{\int }_0^1\xi \eta d\eta d\xi \right)}^{1-1/q}{\left({\int }_0^1{\int }_0^1\xi \eta {\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )\varsigma )\right|}^{q}d\eta d\xi \right)}^{1/q}\hfill \\ \hfill & \le {\left(\frac{1}{4}\right)}^{1-1/q}\left({\int }_0^1{\int }_0^1\xi \eta \left[{\xi }^2{\eta }^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+{\xi }^2{(1-\eta )}^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q\right.\right.\hfill \\ \hfill & +{(1-\xi )}^2{\eta }^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+{(1-\xi )}^2{(1-\eta )}^2{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\hfill \\ \hfill & +{\xi }^2\eta (1-\eta )M_1(\varkappa ,\gamma ,\varsigma )+\xi \eta (1-\xi )(1-\eta )M_2(\varkappa ,\sigma ,\gamma ,\varsigma )+\xi {\eta }^2(1-\xi )M_3(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +{\left.\left.\xi \eta (1-\xi )(1-\eta )M_4(\varkappa ,\sigma ,\gamma ,\varsigma )+\xi (1-\xi ){(1-\eta )}^2{M}_5(\varkappa ,\sigma ,\varsigma )+\eta {(1-\xi )}^2(1-\eta )M_6(\sigma ,\gamma ,\varsigma )\right]d\eta d\xi \right)}^{1/q}\hfill \\ \hfill & ={\left(\frac{1}{4}\right)}^{1-1/q}\left(\frac{1}{16}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+\frac{1}{144}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,\varsigma )+\frac{1}{144}L(\varkappa ,\sigma ,\gamma ,\varsigma )+\frac{1}{48}M(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +{\left.\frac{1}{144}N(\varkappa ,\sigma ,\gamma ,\varsigma )+\frac{1}{144}O(\varkappa ,\sigma ,\varsigma )+\frac{1}{144}P(\sigma ,\gamma ,\varsigma )\right)}^{1/q}.\hfill \end{array}$$

(39)

In a similar way, one can obtain

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\sigma ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & \le {\left(\frac{1}{4}\right)}^{1-1/q}\left(\frac{1}{16}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+\frac{1}{144}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|}^q\right.\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,d)+\frac{1}{144}L(\varkappa ,\sigma ,\gamma ,d)+\frac{1}{48}M(\varkappa ,\sigma ,\gamma )\hfill \\ \hfill & +{\left.\frac{1}{144}N(\varkappa ,\sigma ,\gamma ,d)+\frac{1}{144}O(\varkappa ,\sigma ,d)+\frac{1}{144}P(\sigma ,\gamma ,d)\right)}^{1/q},\hfill \end{array}$$

(40)

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )\varsigma )\right|d\eta d\xi \hfill \\ \hfill & \le {\left(\frac{1}{4}\right)}^{1-1/q}\left(\frac{1}{16}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+\frac{1}{144}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,\varsigma )+\frac{1}{144}L(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{48}M(\varkappa ,\rho ,\gamma )\hfill \\ \hfill & +{\left.\frac{1}{144}N(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{144}O(\varkappa ,\rho ,\varsigma )+\frac{1}{144}P(\rho ,\gamma ,\varsigma )\right)}^{1/q},\hfill \end{array}$$

(41)

and

$$\begin{array}{cc}{\int }_0^1{\int }_0^1\hfill & \xi \eta \left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\xi \varkappa +(1-\xi )\rho ,\eta \gamma +(1-\eta )d)\right|d\eta d\xi \hfill \\ \hfill & \le {\left(\frac{1}{4}\right)}^{1-1/q}\left(\frac{1}{16}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+\frac{1}{48}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+\frac{1}{144}{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|}^q\right.\hfill \\ \hfill & +\frac{1}{48}K(\varkappa ,\gamma ,d)+\frac{1}{144}L(\varkappa ,\rho ,\gamma ,\varsigma )+\frac{1}{48}M(\varkappa ,\rho ,\gamma )\hfill \\ \hfill & +{\left.\frac{1}{144}N(\varkappa ,\rho ,\gamma ,d)+\frac{1}{144}O(\varkappa ,\rho ,d)+\frac{1}{144}P(\rho ,\gamma ,d)\right)}^{1/q}.\hfill \end{array}$$

(42)

By substituting the inequalities (39)–(42) in the inequality (28), we obtain

$$\begin{array}{cc}\hfill |ϝ& (\varkappa ,\gamma )+\frac{1}{(\rho -\sigma )(d-\varsigma )}{\int }_{\sigma }^{\rho }{\int }_{\varsigma }^dϝ(u,v)dvdu-A|\le \frac{1}{4(\rho -\sigma )(d-\varsigma )}{\left(\frac{1}{36}\right)}^{1/q}\hfill \\ \hfill & \times \left\{{(\varkappa -\sigma )}^2{(\gamma -\varsigma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\varsigma )\right|}^q\right.\right.\hfill \\ \hfill & +{\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\sigma ,\gamma ,\varsigma )+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,\varsigma )+O(\varkappa ,\sigma ,\varsigma )+P(\sigma ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\varkappa -\sigma )}^2{(d-\gamma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\sigma ,d)\right|}^q\right.\hfill \\ \hfill & +{\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\sigma ,\gamma ,d)+3M(\varkappa ,\sigma ,\gamma )+N(\varkappa ,\sigma ,\gamma ,d)+O(\varkappa ,\sigma ,d)+P(\sigma ,\gamma ,d)\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(\gamma -\varsigma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\varsigma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\varsigma )\right|}^q\right.\hfill \\ \hfill & +{\left.3K(\varkappa ,\gamma ,\varsigma )+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,\varsigma )+O(\varkappa ,\rho ,\varsigma )+P(\rho ,\gamma ,\varsigma )\right]}^{1/q}\hfill \\ \hfill & +{(\rho -\varkappa )}^2{(d-\gamma )}^2\left[9{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,d)\right|}^q+3{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,\gamma )\right|}^q+{\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\rho ,d)\right|}^q\right.\hfill \\ \hfill & +\left.{\left.3K(\varkappa ,\gamma ,d)+L(\varkappa ,\rho ,\gamma ,\varsigma )+3M(\varkappa ,\rho ,\gamma )+N(\varkappa ,\rho ,\gamma ,d)+O(\varkappa ,\rho ,d)+P(\rho ,\gamma ,d)\right]}^{1/q}\right\}.\hfill \end{array}$$

The proof is completed. □

Remark 6.

Set $h(\varkappa ,\gamma )={\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|}^q$, $g(\varkappa ,\gamma )=1$ and $\left|{\displaystyle \frac{{\partial }^2}{\partial \xi \partial \eta }}ϝ(\varkappa ,\gamma )\right|\le M$ for all $(\varkappa ,\gamma )\in △$. In that case, the inequality (38) reduces to (9).

5. Examples

In the section of examples, we give some examples to demonstrate and confirm the consistency of our main findings.

Example 2.

Let $g,h,ϝ:[0,1]\times [0,1]\to \mathbb{R}$ be given by $g(\varkappa ,\gamma )=\varkappa \gamma$, $h(\varkappa ,\gamma )={\mathrm{e}}^{-\varkappa -\gamma }$ and $ϝ(\varkappa ,\gamma )=\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }$. By Example 1, ϝ is co-ordinated $(g,h)$-convex. Applying Theorem 8, the first inequality of (18) is

$$\begin{array}{cc}\hfill 0.038126\dots & =\frac{1}{\mathrm{e}}-2\left(\frac{-2}{\mathrm{e}}+1\right)\left(\frac{1}{\mathrm{e}}\right)-\left(\frac{1}{{\mathrm{e}}^2}\right)\hfill \\ \hfill & =4ϝ\left(\frac{1}{2},\frac{1}{2}\right)-{\int }_0^1{\int }_0^1{\mathrm{e}}^{-\varkappa -\gamma }\left[\varkappa (1-\gamma )+(1-\varkappa )\gamma +(1-\varkappa )(1-\gamma )\right]d\gamma d\varkappa \hfill \\ \hfill & \le {\int }_0^1{\int }_0^1\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }d\gamma d\varkappa =\frac{4}{{\mathrm{e}}^2}-\frac{4}{\mathrm{e}}+1=0.069823\dots .\hfill \end{array}$$

Additionally, the second inequality of (18) is

$$\begin{array}{cc}\hfill & 0.069823\dots ={\int }_0^1{\int }_0^1\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }d\gamma d\varkappa \le \frac{1}{9}[ϝ(0,0)+ϝ(0,1)+ϝ(1,0)+ϝ(1,1)\hfill \\ \hfill & +\frac{K(0,0,1)+P(1,0,1)+M(0,1,0)+O(0,1,1)}{2}+\frac{L(0,1,0,1)+N(0,1,0,1)}{4}]\hfill \\ \hfill & =\frac{1}{9}\left[\frac{1}{{\mathrm{e}}^2}+\frac{1}{\mathrm{e}}+\frac{1}{4}\right]=0.083690\dots .\hfill \end{array}$$

It is clear that

$$0.038126\dots \le 0.069823\dots \le 0.083690\dots .$$

This confirms the correctness of the result established in Theorem 8.

Example 3.

Let $g,h,ϝ:[0,1]\times [0,1]\to \mathbb{R}$ be formulated by $g(\varkappa ,\gamma )=(1-\varkappa )(1-\gamma )$, $h(\varkappa ,\gamma )={\mathrm{e}}^{-\varkappa -\gamma }$ and $ϝ(\varkappa ,\gamma )=\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }$ (see Figure 1). Then, ϝ is partially differentiable on $(0,1)\times (0,1)$, and its partial derivative is integrable and co-ordinated $(g,h)$-convex on $[0,1]\times [0,1]$. By utilizing Theorem 9, we have

$$\begin{array}{cc}\hfill & \left|\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }+\left(\frac{4}{{\mathrm{e}}^2}-\frac{4}{\mathrm{e}}+1\right)+\left(\varkappa {\mathrm{e}}^{-\varkappa }\right)\left(\frac{2}{\mathrm{e}}-1\right)+\left(\gamma {\mathrm{e}}^{-\gamma }\right)\left(\frac{2}{\mathrm{e}}-1\right)\right|\hfill \\ \hfill & =\left|\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }+{\int }_0^1{\int }_0^{1}uv{\mathrm{e}}^{-u-v}dvdu-{\int }_0^{1}u\gamma {\mathrm{e}}^{-u-\gamma }du-{\int }_0^1\varkappa v{\mathrm{e}}^{-\varkappa -v}dv\right|\hfill \\ \hfill & \le \frac{1}{144}\left\{{\varkappa }^2{\gamma }^2\left[9(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa -\gamma }+3(1-\varkappa ){\mathrm{e}}^{-\varkappa }+3(1-\gamma ){\mathrm{e}}^{-\gamma }+1\right.\right.\hfill \\ \hfill & +3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa }+3(1-\varkappa ){\mathrm{e}}^{-\varkappa -\gamma }+(1-\gamma ){\mathrm{e}}^{-\varkappa }+(1-\varkappa ){\mathrm{e}}^{-\gamma }\hfill \\ \hfill & +3(1-\gamma ){\mathrm{e}}^{-\varkappa -\gamma }+3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\gamma }+{\mathrm{e}}^{-\varkappa -\gamma }+(1-\varkappa )(1-\gamma )\hfill \\ \hfill & +\left.{\mathrm{e}}^{-\varkappa }+(1-\varkappa )+{\mathrm{e}}^{-\gamma }+(1-\gamma )\right]+{\varkappa }^2{(1-\gamma )}^2\left[9(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa -\gamma }+3(1-\gamma ){\mathrm{e}}^{-\gamma }\right.\hfill \\ \hfill & +3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa -1}+(1-\gamma ){\mathrm{e}}^{-\varkappa -1}+3(1-\gamma ){\mathrm{e}}^{-\varkappa -\gamma }+3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\gamma }\hfill \\ \hfill & \left.+(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-1}+(1-\gamma ){\mathrm{e}}^{-1}\right]+{(1-\varkappa )}^2{\gamma }^2\left[9(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa -\gamma }+3(1-\varkappa ){\mathrm{e}}^{-\varkappa }\right.\hfill \\ \hfill & +3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa }+3(1-\varkappa ){\mathrm{e}}^{-\varkappa -\gamma }+(1-\varkappa ){\mathrm{e}}^{-\gamma -1}+3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\gamma -1}\hfill \\ \hfill & +\left.(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\gamma -1}+(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-1}+(1-\varkappa ){\mathrm{e}}^{-1}\right]\hfill \\ \hfill & +{(1-\varkappa )}^2{(1-\gamma )}^2\left[9(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa -\gamma }+3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\varkappa -1}+(1-\gamma ){\mathrm{e}}^{-\varkappa -1}\right.\hfill \\ \hfill & \left.\left.+3(1-\varkappa )(1-\gamma ){\mathrm{e}}^{-\gamma -1}\right]\right\}.\hfill \end{array}$$

Example 4.

Let $g,h,ϝ:[0,1]\times [0,1]\to \mathbb{R}$ be defined by $g(\varkappa ,\gamma )={(1-\varkappa )}^2{(1-\gamma )}^2$, $h(\varkappa ,\gamma )={\mathrm{e}}^{-2\varkappa -2\gamma }$ and $ϝ(\varkappa ,\gamma )=\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }$ (see Figure 2). Then, ϝ is partially differentiable on $(0,1)\times (0,1)$, its partial derivative is integrable and ${\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|}^2$ is co-ordinated $(g,h)$-convex on ${[0,1]}^2$. Via Theorem 10 with $p=q=2$, we have

$$\begin{array}{cc}\hfill & \left|\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }+\left(\frac{4}{{\mathrm{e}}^2}-\frac{4}{\mathrm{e}}+1\right)+\left(\varkappa {\mathrm{e}}^{-\varkappa }\right)\left(\frac{2}{\mathrm{e}}-1\right)+\left(\gamma {\mathrm{e}}^{-\gamma }\right)\left(\frac{2}{\mathrm{e}}-1\right)\right|\hfill \\ \hfill & =\left|\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }+{\int }_0^1{\int }_0^{1}uv{\mathrm{e}}^{-u-v}dvdu-{\int }_0^{1}u\gamma {\mathrm{e}}^{-u-\gamma }du-{\int }_0^1\varkappa v{\mathrm{e}}^{-\varkappa -v}dv\right|\hfill \\ \hfill & \le \frac{1}{18}\left\{{\varkappa }^2{\gamma }^2\left[4{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+4{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa }+4{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }+4\right.\right.\hfill \\ \hfill & +2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa }+2{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa }+{(1-\varkappa )}^2{\mathrm{e}}^{-2\gamma }\hfill \\ \hfill & +2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }+{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\varkappa )}^2{(1-\gamma )}^2\hfill \\ \hfill & +{\left.2{\mathrm{e}}^{-2\varkappa }+2{(1-\varkappa )}^2+2{\mathrm{e}}^{-2\gamma }+2{(1-\gamma )}^2\right]}^{1/2}+{\varkappa }^2{(1-\gamma )}^2\left[4{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+4{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }\right.\hfill \\ \hfill & +2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}+{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}+2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }\hfill \\ \hfill & {\left.+2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2}+2{(1-\gamma )}^2{\mathrm{e}}^{-2}\right]}^{1/2}+{(1-\varkappa )}^2{\gamma }^2\left[4{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+4{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa }\right.\hfill \\ \hfill & +2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa }+2{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\varkappa )}^2{\mathrm{e}}^{-2\gamma -2}+2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma -2}\hfill \\ \hfill & +{\left.{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma -2}+2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2}+2{(1-\varkappa )}^2{\mathrm{e}}^{-2}\right]}^{1/2}\hfill \\ \hfill & +{(1-\varkappa )}^2{(1-\gamma )}^2\left[4{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}+{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}\right.\hfill \\ \hfill & \left.{\left.+2{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma -2}\right]}^{1/2}\right\}.\hfill \end{array}$$

Example 5.

Consider the functions $g,h,ϝ:{[0,1]}^2\to \mathbb{R}$ formulated as $g(\varkappa ,\gamma )=(1-\varkappa )(1-\gamma )$, $h(\varkappa ,\gamma )={\mathrm{e}}^{-\varkappa -\gamma }$ and $ϝ(\varkappa ,\gamma )=\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }$ (see Figure 3). Then, ϝ is partially differentiable on $(0,1)\times (0,1)$, its partial derivative is integrable and ${\left|{\displaystyle \frac{{\partial }^2ϝ}{\partial \xi \partial \eta }}\right|}^2$ is co-ordinated $(g,h)$-convex on ${[0,1]}^2$. The utilization of Theorem 11 with $q=2$ gives

$$\begin{array}{cc}\hfill & \left|\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }+\left(\frac{4}{{\mathrm{e}}^2}-\frac{4}{\mathrm{e}}+1\right)+\left(\varkappa {\mathrm{e}}^{-\varkappa }\right)\left(\frac{2}{\mathrm{e}}-1\right)+\left(\gamma {\mathrm{e}}^{-\gamma }\right)\left(\frac{2}{\mathrm{e}}-1\right)\right|\hfill \\ \hfill & =\left|\varkappa \gamma {\mathrm{e}}^{-\varkappa -\gamma }+{\int }_0^1{\int }_0^{1}uv{\mathrm{e}}^{-u-v}dvdu-{\int }_0^{1}u\gamma {\mathrm{e}}^{-u-\gamma }du-{\int }_0^1\varkappa v{\mathrm{e}}^{-\varkappa -v}dv\right|\hfill \\ \hfill & \le \frac{1}{24}\left\{{\varkappa }^2{\gamma }^2\left[9{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+3{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa }+3{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }+1\right.\right.\hfill \\ \hfill & +3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa }+3{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa }+{(1-\varkappa )}^2{\mathrm{e}}^{-2\gamma }\hfill \\ \hfill & +3{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }+{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\varkappa )}^2{(1-\gamma )}^2\hfill \\ \hfill & +{\left.{\mathrm{e}}^{-2\varkappa }+{(1-\varkappa )}^2+{\mathrm{e}}^{-2\gamma }+{(1-\gamma )}^2\right]}^{1/2}+{\varkappa }^2{(1-\gamma )}^2\left[9{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+3{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }\right.\hfill \\ \hfill & +3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}+{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}+3{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma }\hfill \\ \hfill & {\left.+{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2}+{(1-\gamma )}^2{\mathrm{e}}^{-2}\right]}^{1/2}+{(1-\varkappa )}^2{\gamma }^2\left[9{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+3{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa }\right.\hfill \\ \hfill & +3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa }+3{(1-\varkappa )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+{(1-\varkappa )}^2{\mathrm{e}}^{-2\gamma -2}+3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma -2}\hfill \\ \hfill & +{\left.{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma -2}+{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2}+{(1-\varkappa )}^2{\mathrm{e}}^{-2}\right]}^{1/2}\hfill \\ \hfill & +{(1-\varkappa )}^2{(1-\gamma )}^2\left[9{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2\gamma }+3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}+{(1-\gamma )}^2{\mathrm{e}}^{-2\varkappa -2}\right.\hfill \\ \hfill & \left.{\left.+3{(1-\varkappa )}^2{(1-\gamma )}^2{\mathrm{e}}^{-2\gamma -2}\right]}^{1/2}\right\}.\hfill \end{array}$$

6. Conclusions

We gave a definition of newly defined co-ordinated $(g,h)$-convexity, which is a generalization of co-ordinated convexity. We also proved some of its significant properties. We established some new Hermite–Hadamard- and Ostrowski-type inequalities in relation to such co-ordinated $(g,h)$-convex functions. We established that the inequalities derived in this paper generalize the results given in earlier works. Lastly, we gave some examples to demonstrate and show the correctness of the main results. In the next works, one can extend similar theorems to other types of generalized convexity.