Skip to main content

Approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex functions and related integral inequalities

Abstract

The aim of this study is to introduce the notion of two-dimensional approximately harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex functions. We show that the new class covers many new and known extensions of harmonic convex functions. We formulate several new refinements of Hermite–Hadamard like inequalities involving two-dimensional approximately harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex functions. We discuss in detail the special cases that can be deduced from the main results of the paper.

Introduction and preliminaries

A function \(f:I\subset \mathbb{R}\to \mathbb{R}\) is said to be convex, if

$$ f\bigl((1-t)x+ty\bigr)\leq (1-t)f(x)+tf(y),\quad \forall x,y\in I,t\in [0,1]. $$

In recent years the classical concepts of convex functions have been extended and generalized in different directions using innovative and novel ideas. Işcan [1] introduced the class of harmonic convex functions.

A function \(f:I\subset (0,\infty )\to \mathbb{R}\) is said to be harmonically convex if

$$\begin{aligned} f \biggl(\frac{xy}{(1-t)x+ty} \biggr)\leq tf(x)+(1-t)f(y),\quad \forall x,y\in I,t\in [0,1]. \end{aligned}$$

Noor et al. [2] generalized the notion of harmonic convex functions and gave the definition of harmonically h-convex functions. This class contains several other classes of harmonic convex functions as well. In [3], the authors introduced the definition of p-harmonic convex function.

A function \(f:I\subset (0,\infty )\to \mathbb{R}\) is said to be p-harmonically convex if

$$\begin{aligned} f \biggl(\frac{x^{p}y^{p}}{(1-t)x^{p}+ty^{p}} \biggr)^{\frac{1}{p}} \leq tf(x)+(1-t)f(y),\quad \forall x,y\in I,t\in [0,1] \end{aligned}$$

holds, where I is a p-harmonic convex set.

Noor et al. [4] also extended the class of harmonic convex functions on coordinates and introduced the class of coordinated harmonic convex functions.

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be two-dimensional harmonically convex function on Ω if

$$\begin{aligned} &f \biggl(\frac{xy}{tx+(1-t)y},\frac{uw}{ru+(1-r)w} \biggr) \\ &\quad\leq \operatorname{tr} f(y,w)+t(1-r) f(y,u)+(1-t)r f(x,w)+(1-t) (1-r) f(x,u), \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\). Recently, Awan et al. [5] gave the definition of approximately harmonic h-convex functions depending on a metric function \(d:X\times X \to \mathbb{R}\) where \((X,\Vert \cdot\Vert )\) is a real normed space. Let \(h:(0,1)\to \mathbb{R}\) and Θ be a harmonic convex subset of X. A function \(f:\Theta \to \mathbb{R}\) is an approximately harmonic h-convex function if

$$\begin{aligned} f \biggl(\frac{xy}{(1-t)x+ty} \biggr)\leq h(t)f(x)+h(1-t)f(y)+d(x,y), \quad\forall x,y \in \Theta,t\in [0,1]. \end{aligned}$$

For more and recent details on convexity and its generalizations, see [618].

Theory of convexity also played a significant role in the development of theory of inequalities. Many famously known results in inequalities theory can be obtained using the convexity property of the functions. Hermite–Hadamard double inequality is one of the most intensively studied results involving convex functions. This result provides us a necessary and sufficient condition for a function to be convex. For interesting details on Hermite–Hadamard inequality and its generalizations, see [15, 19].

The main motive of this article is to extend the notion of approximately harmonic h-convex functions on two dimensions and derive some new corresponding Hermite–Hadamard like inequalities.

New notions

In this section, we define the class of two-dimensional approximately harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex functions. We also discuss that for suitable choices we get several other new classes of harmonic convexity.

Definition 1

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional approximately harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u) \\ &\qquad{} +h_{1}(1-t)h_{2}(r )f(x,w)+h_{1}(1-t)h_{2}(1-r) f(x,u)+\Delta (x,y)+ \Delta (u,w), \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\).

We now discuss some special cases of Definition 1.

I. If we take \(\Delta (x,y)=\epsilon (\| x^{p_{1}}-y^{p_{1}} \|)^{\gamma }\) and \(\Delta (u,w)=\epsilon (\| u^{p_{2}}-w^{p_{2}} \|)^{\gamma }\) for some \(\epsilon \in \mathbb{R}\) and \(\gamma >1\) in Definition 1, we have a new definition of γ-paraharmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function of higher order.

Definition 2

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional γ-paraharmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function of higher order if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u) \\ &\qquad{} +h_{1}(1-t)h_{2}(r )f(x,w)+h_{1}(1-t)h_{2}(1-r) f(x,u)+ \epsilon \bigl( \bigl\Vert x^{p_{1}}-y^{p_{1}} \bigr\Vert \bigr)^{\gamma }\\ &\qquad{}+\epsilon \bigl( \bigl\Vert u^{p_{2}}-w^{p_{2}} \bigr\Vert \bigr)^{\gamma }, \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\).

II. If we take \(\Delta (x,y)=\epsilon (\| x^{p_{1}}-y^{p_{1}} \|)\) and \(\Delta (u,w)=\epsilon (\|u^{p_{2}}-w^{p_{2}}\|)\) for some \(\epsilon \in \mathbb{R}\) and in Definition 1, we have a new definition of ϵ-paraharmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function.

Definition 3

Consider the rectangle \(\Omega =[a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f:\Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional ϵ-paraharmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u) \\ &\qquad{} +h_{1}(1-t)h_{2}(r )f(x,w)+h_{1}(1-t)h_{2}(1-r) f(x,u) \\ &\qquad{} +\epsilon \bigl( \bigl\Vert x^{p_{1}}-y^{p_{1}} \bigr\Vert \bigr)+\epsilon \bigl( \bigl\Vert u^{p_{2}}-w^{p_{2}} \bigr\Vert \bigr), \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\).

III. If we take \(\Delta (x,y)=-\mu (t^{\sigma }(1-t)+t(1-t)^{\sigma } ) ( \Vert \frac{1}{y^{p_{1}}}-\frac{1}{x^{p_{1}}} \Vert )^{\sigma }\) and

\(\Delta (u,w)=-\mu (r^{\sigma }(1-r)+r(1-r)^{\sigma } ) ( \Vert \frac{1}{w^{p_{2}}}-\frac{1}{u^{p_{2}}} \Vert )^{\sigma }\) for some \(\mu >0\) and \(\sigma >0\) in Definition 1, we have a new definition of two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function of higher order.

Definition 4

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function of higher order if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u) \\ &\qquad{} +h_{1}(1-t)h_{2}(r )f(x,w)+h_{1}(1-t)h_{2}(1-r) f(x,u) \\ &\qquad{} -\mu \bigl(t^{\sigma }(1-t)+t(1-t)^{\sigma } \bigr) \biggl( \biggl\Vert \frac{1}{y^{p_{1}}}-\frac{1}{x^{p_{1}}} \biggr\Vert \biggr)^{ \sigma }\\ &\qquad{}- \mu \bigl(r^{\sigma }(1-r)+r(1-r)^{\sigma } \bigr) \biggl( \biggl\Vert \frac{1}{w^{p_{2}}}-\frac{1}{u^{p_{2}}} \biggr\Vert \biggr)^{ \sigma }, \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d], \sigma >0\), and \(t,r \in [0,1]\).

IV. If we take \(\sigma =2\) in Definition 4, then \(\Delta (x,y)=-\mu t(1-t) ( \Vert \frac{1}{y^{p_{1}}}- \frac{1}{x^{p_{1}}} \Vert )^{2}\) and \(\Delta (u,w)=-\mu r(1-r) ( \Vert \frac{1}{w^{p_{2}}}- \frac{1}{u^{p_{2}}} \Vert )^{2}\) for some \(\mu >0\) in Definition 1, then

Definition 5

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u)\\ &\qquad{}+h_{1}(1-t)h_{2}(r )f(x,w)+h_{1}(1-t)h_{2}(1-r) f(x,u) \\ &\qquad{} -\mu t(1-t) \biggl( \biggl\Vert \frac{1}{y^{p_{1}}}- \frac{1}{x^{p_{1}}} \biggr\Vert \biggr)^{2}-\mu r(1-r) \biggl( \biggl\Vert \frac{1}{w^{p_{2}}}- \frac{1}{u^{p_{2}}} \biggr\Vert \biggr)^{2}, \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\).

V. If we take \(\Delta (x,y)=\mu t(1-t) (\frac{1}{y^{p_{1}}}- \frac{1}{x^{p_{1}}} )^{2}\) and \(\Delta (u,w)=\mu r(1-r) (\frac{1}{w^{p_{2}}}- \frac{1}{u^{p_{2}}} )^{2}\) for some \(\mu >0\) in Definition 1, then

Definition 6

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional harmonically relaxed \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u)+h_{1}(1-t)h_{2}(r )f(x,w)\\ &\qquad{}+h_{1}(1-t)h_{2}(1-r) f(x,u) \\ & \qquad{}+\mu t(1-t) \biggl(\frac{1}{y^{p_{1}}}-\frac{1}{x^{p_{1}}} \biggr)^{2}+ \mu r(1-r) \biggl(\frac{1}{w^{p_{2}}}-\frac{1}{u^{p_{2}}} \biggr)^{2}, \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\).

VI. If we take \(\Delta (x,y)= -t(1-t) ( \frac{x^{p_{1}}y^{p_{1}}}{x^{p_{1}}-y^{p_{1}}} )^{2}\) and \(\Delta (u,w)= -r(1-r) ( \frac{u^{p_{2}}w^{p_{2}}}{u^{p_{2}}-w^{p_{2}}} )^{2}\) in Definition 1, we have a new definition of two-dimensional strongly F harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function.

Definition 7

Consider the rectangle \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\). A function \(f: \Omega \rightarrow \mathbb{R}\) is said to be a two-dimensional strongly F harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function if

$$\begin{aligned} &f \biggl( \biggl[\frac{x^{p_{1}}y^{p_{1}}}{tx^{p_{1}}+(1-t)y^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{u^{p_{2}}w^{p_{2}}}{ru^{p_{2}}+(1-r)w^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(y,w)+h_{1}(t)h_{2}(1-r) f(y,u)\\ &\qquad{}+h_{1}(1-t)h_{2}(r )f(x,w)+h_{1}(1-t)h_{2}(1-r) f(x,u) \\ &\qquad{}-t(1-t) \biggl(\frac{x^{p_{1}}y^{p_{1}}}{x^{p_{1}}-y^{p_{1}}} \biggr)^{2}- r(1-r) \biggl( \frac{u^{p_{2}}w^{p_{2}}}{u^{p_{2}}-w^{p_{2}}} \biggr)^{2}, \end{aligned}$$

whenever \(x,y \in [a,b], u,w \in [c,d]\), and \(t,r \in [0,1]\).

Main results

In this section, we discuss our main results.

Theorem 1

Let \(f: \Omega \rightarrow \mathbb{R}\) be an integrable function. If f is an approximately two-dimensional harmonically \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, then

$$\begin{aligned} &\frac{1}{4h_{1}({\frac{1}{2}})h_{2}({\frac{1}{2}})} \biggl[f \biggl( \biggl[\frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}- \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \int _{a}^{b} \frac{\Delta (x, ((a^{p_{1}})^{-1}+(b^{p_{1}})^{-1}-(x^{p_{1}})^{-1})^{-1})}{x^{1+p_{1}}} \,\mathrm{d}x \\ &\qquad{} -\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \int _{c}^{d} \frac{\Delta (u, ((c^{p_{2}})^{-1}+(d^{p_{2}})^{-1}-(u^{p_{2}})^{-1})^{-1})}{u^{1+p_{2}}} \,\mathrm{d}u \biggr] \\ &\quad\leq p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x \\ &\quad\leq \bigl[f(a,c)+f(a,d)+f(b,c)+f(b,d) \bigr] \int _{0}^{1} \int _{0}^{1}h_{1}(t)h_{2}(r) \,\mathrm{d}t\,\mathrm{d}r+\Delta (a,b)+ \Delta (c,d). \end{aligned}$$

Proof

Since f is an approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, we have

$$\begin{aligned} &f \biggl( \biggl[\frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1} \biggl(\frac{1}{2} \biggr)h_{2} \biggl( \frac{1}{2} \biggr) \\ &\qquad{}\times \biggl[f \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{ta^{p_{1}}+(1-t)b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rc^{p_{2}}+(1-r)d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}+f \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{ta^{p_{1}}+(1-t)b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\qquad{} +f \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rc^{p_{2}}+(1-r)d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}+f \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr] \biggr) \\ & \qquad{}+\Delta \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{ta^{p_{1}}+(1-t)b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}} \biggr) \\ &\qquad{}+\Delta \biggl( \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rc^{p_{2}}+(1-r)d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) ]. \end{aligned}$$

Integrating the above inequality with respect to \((t,r)\) on \([0,1]\times [0,1]\), we have

$$\begin{aligned} &\frac{1}{4h_{1} (\frac{1}{2} )h_{2} (\frac{1}{2} )} \biggl[f \biggl( \biggl[\frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}- \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \int _{a}^{b} \frac{\Delta (x,((a^{p_{1}})^{-1}+(b^{p_{1}})^{-1}-(x^{p_{1}})^{-1})^{-1})}{x^{1+p_{1}}} \,\mathrm{d}x \\ &\qquad{} -\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \int _{c}^{d} \frac{\Delta (u,((c^{p_{2}})^{-1}+(d^{p_{2}})^{-1}-(u^{p_{2}})^{-1})^{-1})}{u^{1+p_{2}}} \,\mathrm{d}u \biggr] \\ &\quad\leq p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x. \end{aligned}$$

Also

$$\begin{aligned} &f \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{ta^{p_{1}}+(1-t)b^{p_{1}}} \biggr]^{\frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rc^{p_{2}}+(1-r)d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \\ &\quad\leq h_{1}(t)h_{2}(r) f(b,d)+h_{1}(t)h_{2}(1-r) f(b,c)+h_{1}(1-t)h_{2}(r )f(a,d) \\ &\qquad{} +h_{1}(1-t)h_{2}(1-r) f(a,c)+\Delta (a,b)+\Delta (c,d). \end{aligned}$$

Integrating both sides of the above inequality with respect to \((t,r)\) on \([0,1]\times [0,1]\), we have

$$\begin{aligned} &p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x \\ &\quad\leq \bigl(f(a,c)+f(a,d)+f(b,c)+f(b,d) \bigr) \int _{0}^{1} \int _{0}^{1}h_{1}(t)h_{2}(r) \,\mathrm{d}t\,\mathrm{d}r+\Delta (a,b)+ \Delta (c,d). \end{aligned}$$

This completes the proof. □

We now discuss some special cases of Theorem 1.

I. If \(h_{1}(t)=t\) and \(h_{2}(r)=r\) in Theorem 1, then

Corollary 1

Under the assumptions of Theorem 1, if f is an approximately two-dimensional harmonic \((p_{1}, p_{2})\)-convex function, then

$$\begin{aligned} &f \biggl( \biggl[\frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}- \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \int _{a}^{b} \frac{\Delta (x, ((a^{p_{1}})^{-1}+(b^{p_{1}})^{-1}-(x^{p_{1}})^{-1})^{-1})}{x^{1+p_{1}}} \,\mathrm{d}x \\ &\qquad{}-\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \int _{c}^{d} \frac{\Delta (u, ((c^{p_{2}})^{-1}+(d^{p_{2}})^{-1}-(u^{p_{2}})^{-1})^{-1})}{u^{1+p_{2}}} \,\mathrm{d}u \\ &\quad\leq p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x \\ &\quad\leq \frac{ [f(a,c)+f(a,d)+f(b,c)+f(b,d) ]}{4}+\Delta (a,b)+ \Delta (c,d). \end{aligned}$$

II. If \(h_{1}(t)=t^{s_{1}}\) and \(h(r)=r^{s_{2}}\) in Theorem 1, then

Corollary 2

Under the assumptions of Theorem 1, if f is a Breckner type approximately two-dimensional harmonic \((p_{1},s_{1})\)-\((p_{2},s_{2})\)-convex function, then

$$\begin{aligned} &\frac{2^{s_{1}+s_{2}}}{4} \biggl[f \biggl( \biggl[ \frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}- \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \int _{a}^{b} \frac{\Delta (x, ((a^{p_{1}})^{-1}+(b^{p_{1}})^{-1}-(x^{p_{1}})^{-1})^{-1})}{x^{1+p_{1}}} \,\mathrm{d}x \\ &\qquad{} -\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \int _{c}^{d} \frac{\Delta (u, ((c^{p_{2}})^{-1}+(d^{p_{2}})^{-1}-(u^{p_{2}})^{-1})^{-1})}{u^{1+p_{2}}} \,\mathrm{d}u \biggr] \\ &\quad\leq p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x \\ &\quad\leq \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{(s_{1}+1)(s_{2}+1)}+\Delta (a,b)+ \Delta (c,d). \end{aligned}$$

III. If \(h_{1}(t)=t^{-s_{1}}\) and \(h_{2}(r)=r^{-s_{2}}\) in Theorem 1, then

Corollary 3

Under the assumptions of Theorem 1, if f is a Godunova–Levin type approximately two-dimensional harmonic \((p_{1},s_{1})\)-\((p_{2},s_{2})\)-convex function, then

$$\begin{aligned} &\frac{1}{4.2^{s_{1}+s_{2}}} \biggl[f \biggl( \biggl[ \frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}- \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \int _{a}^{b} \frac{\Delta (x, ((a^{p_{1}})^{-1}+(b^{p_{1}})^{-1}-(x^{p_{1}})^{-1})^{-1})}{x^{1+p_{1}}} \,\mathrm{d}x \\ & \qquad{}-\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \int _{c}^{d} \frac{\Delta (u, ((c^{p_{2}})^{-1}+(d^{p_{2}})^{-1}-(u^{p_{2}})^{-1})^{-1})}{u^{1+p_{2}}} \,\mathrm{d}u \biggr] \\ &\quad\leq p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x \\ &\quad\leq \frac{f(a,c)+f(a,d)+f(b,c)+f(b,d)}{(1-s_{1})(1-s_{2})}+\Delta (a,b)+ \Delta (c,d). \end{aligned}$$

IV. If \(h_{1}(t)=1\) and \(h_{2}(r)=1\) in Theorem 1, then

Corollary 4

Under the assumptions of Theorem 1, if f is an approximately two-dimensional harmonic \((p_{1}, P)\)-\((p_{2},P)\)-convex function, then

$$\begin{aligned} &\frac{1}{4} \biggl[f \biggl( \biggl[ \frac{2a^{p_{1}}b^{p_{1}}}{a^{p_{1}}+b^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{2c^{p_{2}}d^{p_{2}}}{c^{p_{2}}+d^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}- \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \int _{a}^{b} \frac{\Delta (x, ((a^{p_{1}})^{-1}+(b^{p_{1}})^{-1}-(x^{p_{1}})^{-1})^{-1})}{x^{1+p_{1}}} \,\mathrm{d}x \\ &\qquad{} -\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \int _{c}^{d} \frac{\Delta (u, ((c^{p_{2}})^{-1}+(d^{p_{2}})^{-1}-(u^{p_{2}})^{-1})^{-1})}{u^{1+p_{2}}} \,\mathrm{d}u \biggr] \\ &\quad\leq p_{1}p_{2} \biggl(\frac{a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggr) \biggl( \frac{c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggr) \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x \\ &\quad\leq \bigl[f(a,c)+f(a,d)+f(b,c)+f(b,d)\bigr]+\Delta (a,b)+\Delta (c,d). \end{aligned}$$

Now we introduce a generalized identity that will play a significant role in the development of our next results.

Lemma 1

Let \(f:\Omega \rightarrow \mathbb{R}\) be a partial differential function on \(\Omega =[a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\) with \(a^{p_{1}}< b^{p_{1}}\) and \(c^{p_{2}}< d^{p_{2}}\). If \(\frac{\partial ^{2}f}{\partial t\, \partial r}\in L_{1}(\Omega )\), then

$$\begin{aligned} &\Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \\ &\quad= \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \int _{0}^{1} \biggl((1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr)\\ &\qquad{}\times \biggl((1-2r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr) \\ &\qquad{} \times \frac{\partial ^{2}f}{\partial r\, \partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\, \mathrm{d}r\,\mathrm{d}t, \end{aligned}$$

where

$$\begin{aligned} &\Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \\ &\quad=\frac{f(a,c)+f(b,c)+f(a,d)+f(b,d)}{4} -\frac{1}{2} \biggl[ \frac{p_{1}a^{p_{1}}b^{p_{1}}}{b^{p_{1}}-a^{p_{1}}} \biggl[ \int _{a}^{b}\frac{f(x,c)}{x^{1+p_{1}}}\,\mathrm{d}x+ \int _{a}^{b} \frac{f(x,d)}{x^{1+p_{1}}}\,\mathrm{d}x \biggr] \\ &\qquad{}+\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggl[ \int _{c}^{d}\frac{f(a,u)}{u^{1+p_{2}}}\,\mathrm{d}u+ \int _{c}^{d} \frac{f(b,u)}{u^{1+p_{2}}}\,\mathrm{d}u \biggr] \biggr] \\ &\qquad{}+ \frac{p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}}{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})} \int _{a}^{b} \int _{c}^{d} \frac{f(x,u)}{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}u\, \mathrm{d}x. \end{aligned}$$

Proof

It suffices to note that we can write

$$\begin{aligned} &\Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \\ &\quad= \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \biggl((1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ &\qquad{}\times \biggl[ \int _{0}^{1} \biggl((1-2r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr)\\ &\qquad{}\times\frac{\partial ^{2}f}{\partial r\, \partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\,\mathrm{d}r \biggr]\,\mathrm{d}t. \end{aligned}$$

Now, integrating by parts, we have

$$\begin{aligned} I_{1}={}& \int _{0}^{1} \biggl( (1-2r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr) \\ &{}\times\frac{\partial ^{2}f}{\partial r\, \partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\,\mathrm{d}r \\ = {}&\frac{p_{2}c^{p_{2}}d^{p_{2}}}{ ( d^{p_{2}}-c^{p_{2}} ) } \frac{\partial f}{\partial t} \biggl( \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} ]^{ \frac{1}{p_{1}}},c \biggr) + \frac{p_{2}c^{p_{2}}d^{p_{2}}}{ ( d^{p_{2}}-c^{p_{2}} ) } \frac{\partial f}{\partial t} \biggl( \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} ]^{ \frac{1}{p_{1}}},d \biggr) \\ &{} - \frac{2p_{2}c^{p_{2}}d^{p_{2}}}{ ( d^{p_{2}}-c^{p_{2}} ) } \int _{0}^{1} \frac{\partial f}{\partial t} \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \,\mathrm{d}r, \\ I_{2}={}& \int _{0}^{1} \biggl((1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \frac{\partial f}{\partial t} \biggl( \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} ]^{ \frac{1}{p_{1}}},c \biggr) \,\mathrm{d}t \\ = {}&\frac{p_{1}a^{p}_{1}b^{p}_{1}}{ ( b^{p_{1}}-a^{p_{1}} ) } \bigl\{ f ( a,c ) +f ( b,c ) \bigr\} - \frac{2p_{1}^{2}(a^{p}_{1}b^{p}_{1})^{2}}{ ( b^{p_{1}}-a^{p_{1}} ) ^{2}} \int _{a}^{b}\frac{f ( x,c ) }{x^{1+p_{1}}}\,\mathrm{d}x, \\ I_{3}={}& \int _{0}^{1} \biggl((1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \frac{\partial f}{\partial t} \biggl( \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} ]^{ \frac{1}{p_{1}}},d \biggr) \,\mathrm{d}t \\ = {}&\frac{p_{1}a^{p}_{1}b^{p}_{1}}{ ( b^{p_{1}}-a^{p_{1}} ) } \bigl\{ f ( a,d ) +f ( b,d ) \bigr\} - \frac{2p_{1}^{2}(a^{p}_{1}b^{p}_{1})^{2}}{ ( b^{p_{1}}-a^{p_{1}} ) ^{2}} \int _{a}^{b}\frac{f ( x,d ) }{x^{1+p_{1}}}\,\mathrm{d}x, \\ I_{4}={}& \int _{0}^{1} \biggl\{ \int _{0}^{1} \biggl( (1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr)\\ &{}\times\frac{\partial f}{\partial t} \biggl( \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \,\mathrm{d}t \biggr\} \,\mathrm{d}r \\ ={}&\frac{\{f(a,c)+f(b,c)+f(a,d)+f(b,d)\}}{4}-\frac{1}{2} \biggl[ \frac{p_{1}a^{p}_{1}b^{p}_{1}}{b^{p_{1}}-a^{p_{1}}} \biggl\{ \int _{a}^{b}\frac{f(x,c)}{x^{1+p_{1}}} \,\mathrm{d}x+ \int _{a}^{b}\frac{f(x,d)}{x^{1+p_{1}}}\,\mathrm{d}x \biggr\} \\ &{} +\frac{p_{2}c^{p_{2}}d^{p_{2}}}{d^{p_{2}}-c^{p_{2}}} \biggl\{ \int _{c}^{d}\frac{f(a,u)}{u^{1+p_{2}}} \,\mathrm{d}u+ \int _{c}^{d}\frac{f(b,u)}{u^{1+p_{2}}}\,\mathrm{d}u \biggr\} \biggr]\\ &{}+ \frac{p_{1}p_{2}(a^{p}_{1}b^{p}_{1}a^{p}_{1}b^{p}_{1})}{ ( b^{p_{1}}-a^{p_{1}} ) ( d^{p_{2}}-c^{p_{2}} ) } \int _{a}^{b} \int _{c}^{d} \frac{f ( x,u ) }{x^{1+p_{1}}u^{1+p_{2}}}\,\mathrm{d}x \, \mathrm{d}u. \end{aligned}$$

Summing up integrals \(I_{1}\) to \(I_{4}\) and using the change of variable technique will give the required result. □

Remark 1

Letting \(p_{1}=1=p_{2}\) in the above lemma, we get the lemma proved in [4].

In order to obtain next results, we need gamma, beta, and hypergeometric functions. Gamma \(\Gamma (\cdot)\) and beta function \(B(\cdot,\cdot)\) are defined respectively as follows:

$$\begin{aligned} &\Gamma (x)= \int _{0}^{\infty }e^{-x}t^{x-1}\, \mathrm{d}t, \\ &\mathrm{B}(x,y)=\frac{\Gamma (x)\Gamma (y)}{\Gamma (x+y)}= \int _{0}^{1}t^{x-1}(1-t)^{y-1} \,\mathrm{d}t. \end{aligned}$$

The integral form of the hypergeometric function is

$$ _{2}F_{1}(x,y;c;z)=\frac{1}{\mathrm{B}(y,c-y)}\int _{0}^{1}t^{y-1}(1-t)^{c-y-1}(1-zt)^{-x} \,\mathrm{d}t $$

for \(|z|<1,c>y>0\). Now, using Lemma 1, we prove the next results of the article.

Theorem 2

Let \(f:\Omega \rightarrow \mathbb{R}\) be a partial differentiable function on \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\) with \(a^{p_{1}}< b^{p_{1}}\) and \(c^{p_{2}}< d^{p_{2}}\) and \(\frac{\partial ^{2}f}{\partial t\, \partial r}\in L_{1}(\Omega )\). If \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is an approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ &\qquad{} +\vartheta _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \vartheta _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ & \qquad{}+\vartheta _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert + \vartheta _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \vartheta _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where

$$\begin{aligned} &\vartheta _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert h_{1}(t) \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert h_{2}(r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r, \\ &\vartheta _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert h_{1}(1-t) \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert h_{2}(r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r, \\ &\vartheta _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert h_{1}(t) \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert h_{2}(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r, \\ &\vartheta _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert h_{1}(1-t) \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert h_{2}(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r, \\ &\vartheta _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad=\Delta (a,b) \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad=\Delta (a,b) \biggl[ \bigl(b^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},2,3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr)\\ &\qquad{}- \bigl(b^{p_{1}}\bigr)^{1+ \frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}+\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \biggr] \\ &\qquad{}\times\biggl[ \bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},2,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},1,2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr)\\ &\qquad{}+ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3, \frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \biggr], \end{aligned}$$

and

$$\begin{aligned} &\vartheta _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad=\Delta (c,d) \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad=\Delta (c,d) \biggl[ \bigl(b^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},2,3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr)- \bigl(b^{p_{1}}\bigr)^{1+ \frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}+\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \biggr] \biggl[ \bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},2,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},1,2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr)\\ &\qquad{}+ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3, \frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \biggr]. \end{aligned}$$

Proof

Using Lemma 1 and the fact that \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is an approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad= \biggl\vert \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \int _{0}^{1} \biggl((1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ &\qquad{}\times\biggl((1-2r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr) \\ &\qquad{} \times \frac{\partial ^{2}f}{\partial r \,\partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\, \mathrm{d}r\,\mathrm{d}t \biggr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr] \\ &\qquad{} \times \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \biggr\vert \,\mathrm{d}r\,\mathrm{d}t \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr] \\ &\qquad{} \times \biggl[h_{1}(t)h_{2}(r) \biggl\vert \frac{\partial ^{2}f}{\partial r\, \partial t}(a,c) \biggr\vert +h_{1}(1-t)h_{2}(r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert \\ &\qquad{}+h_{1}(t)h_{2}(1-r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ &\qquad{} +h_{1}(1-t)h_{2}(1-r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert +\Delta (a,b)+ \Delta (c,d) \biggr]\, \mathrm{d}r\,\mathrm{d}t. \end{aligned}$$

This completes the proof. □

Corollary 5

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\le M\) is a bounded function for \(M>0\) on Ω, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{M(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\qquad{} +\vartheta _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)+\vartheta _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\qquad{} +\vartheta _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)+\frac{1}{M}\vartheta _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+\frac{1}{M}\vartheta _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where \(\vartheta _{1}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) to \(\vartheta _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 2.

We now discuss some special cases of Theorem 2.

I. If we take \(h_{1}(t)=t\) and \(h_{2}(r)=r\) in Theorem 2, then

Corollary 6

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is an approximately two-dimensional harmonic \((p_{1},p_{2})\)-convex function, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}^{*} \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ & \qquad{}+\vartheta _{2}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \vartheta _{3}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ &\qquad{} +\vartheta _{4}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert + \vartheta _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \vartheta _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where

$$\begin{aligned} &\vartheta _{1}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert t \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert r \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,4,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr)+\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},3,4,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \biggl[ \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,4,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},3,4,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr)- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{2}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert (1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert r \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr)\\ &\qquad{} -\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,4,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,4,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr)- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,4,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},3,4,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr)- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{3}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert t \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert (1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\, \mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,4,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr)+\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},3,4,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr)\\ &\qquad{} - \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,4,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{}+\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,4,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr)- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{4}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert (1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert (1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\, \mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr)\\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,4,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,4,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr)- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ & \qquad{}\times \biggl[\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr)\\ &\qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{12} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,4,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{}+\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{3} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,4,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr)- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \end{aligned}$$

\(\vartheta _{5}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) and \(\vartheta _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 2.

II. If we take \(h_{1}(t)=t^{s_{1}}\) and \(h_{2}(r)=r^{s_{2}}\) in Theorem 2, then

Corollary 7

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is a Breckner type approximately two-dimensional harmonic \((p_{1},s_{1})\)-\((p_{2},s_{2})\)-convex function, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}^{**} \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ & \qquad{}+\vartheta _{2}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \vartheta _{3}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ &\qquad{} +\vartheta _{4}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert + \vartheta _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{} + \vartheta _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where

$$\begin{aligned} &\vartheta _{1}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert t^{s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert r^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{s_{1}}(s_{1}+1)(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+3,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+2,s_{1}+3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr)\\ &\qquad{}- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{s_{2}}(s_{2}+1)(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+3,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+2,s_{2}+3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr)\\ &\qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(1-s_{1})} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{2}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert (1-t)^{s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert r^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[\frac{4(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+2,s_{1}+3,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{}- \frac{2(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+2,1-\frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{s_{1}}(s_{1}+1)(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+3,\frac{1}{2} \biggl(1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \biggr) \\ &\qquad{} +\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,s_{1}+3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr)\\ &\qquad{} - \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,s_{1}+2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{s_{2}}(s_{2}+1)(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+3,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{} +\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+2,s_{2}+3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{3}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert t^{s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert (1-r)^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{s_{1}}(s_{1}+1)(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+3,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+2,s_{1}+3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr)\\ &\qquad{}- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(1-s_{1})} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\frac{4(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+2,s_{2}+3,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{}- \frac{2(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+2,1-\frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ & \qquad{}+ \frac{(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{s_{2}}(s_{2}+1)(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+3,\frac{1}{2} \biggl(1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \biggr) \\ &\qquad{} +\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,s_{2}+3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,s_{2}+2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{4}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert (1-t)^{s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert (1-r)^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[\frac{4(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+2,s_{1}+3,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr)\\ &\qquad{} - \frac{2(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+2,1-\frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{s_{1}}(s_{1}+1)(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},s_{1}+1,s_{1}+3,\frac{1}{2} \biggl(1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \biggr) \\ &\qquad{} +\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)(s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,s_{1}+3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,s_{1}+2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\frac{4(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+2,s_{2}+3,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{}- \frac{2(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+2,1-\frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ & \qquad{}+ \frac{(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{s_{2}}(s_{2}+1)(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},s_{2}+1,s_{2}+3,\frac{1}{2} \biggl(1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \biggr) \\ &\qquad{} +\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)(s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,s_{2}+3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,s_{2}+2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \end{aligned}$$

\(\vartheta _{5}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) and \(\vartheta _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 2.

III. If we take \(h_{1}(t)=t^{-s_{1}}\) and \(h_{2}(r)=r^{-s_{2}}\) in Theorem 2, then

Corollary 8

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is a Godunova–Levin type approximately two-dimensional harmonic \((p_{1},s_{1})\)-\((p_{2},s_{2})\)-convex function, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}^{\prime } \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ & \qquad{}+\vartheta _{2}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \vartheta _{3}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ &\qquad{} +\vartheta _{4}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert + \vartheta _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \vartheta _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where

$$\begin{aligned} &\vartheta _{1}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert t^{-s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert r^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{-s_{1}}(-s_{1}+1)(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+2,-s_{1}+3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{-s_{2}}(-s_{2}+1)(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+3,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+2,-s_{2}+3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{2}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert (1-t)^{-s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert r^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[\frac{4(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+2,-s_{1}+3,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} - \frac{2(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+2,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{-s_{1}}(-s_{1}+1)(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+3,\frac{1}{2} \biggl(1-\frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \biggr) \\ &\qquad{} + \frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,-s_{1}+3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,-s_{1}+2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{-s_{2}}(-s_{2}+1)(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+3,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ & \qquad{}+\frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+2,-s_{2}+3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{3}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert t^{-s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times\biggl[ \vert 1-2r \vert (1-r)^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{-s_{1}}(-s_{1}+1)(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}+\frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+2,-s_{1}+3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\frac{4(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+2,-s_{2}+3,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ & \qquad{}- \frac{2(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+2,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{} + \frac{(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{-s_{2}}(-s_{2}+1)(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+3,\frac{1}{2} \biggl(1-\frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \biggr) \\ &\qquad{} + \frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,-s_{2}+3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,-s_{2}+2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\vartheta _{4}^{\prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert (1-t)^{-s_{1}} \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert (1-r)^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[\frac{4(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+2,-s_{1}+3,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} - \frac{2(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+2,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{-s_{1}}(-s_{1}+1)(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},-s_{1}+1,-s_{1}+3,\frac{1}{2} \biggl(1-\frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \biggr) \\ &\qquad{} + \frac{ 2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)(-s_{1}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,-s_{1}+3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(-s_{1}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,-s_{1}+2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\frac{4(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+2,-s_{2}+3,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{} - \frac{2(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+2,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{} + \frac{(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{-s_{2}}(-s_{2}+1)(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},-s_{2}+1,-s_{2}+3,\frac{1}{2} \biggl(1-\frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \biggr) \\ &\qquad{} + \frac{ 2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)(-s_{2}+2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,-s_{2}+3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(-s_{2}+1)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,-s_{2}+2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \end{aligned}$$

\(\vartheta _{5}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) and \(\vartheta _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 2.

IV. If we take \(h(t)=1\) and \(h(r)=1\) in Theorem 2, then

Corollary 9

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is an approximately two-dimensional harmonic \((p_{1}, P)\)-\((p_{2},P)\)-convex function, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \vartheta ^{\prime \prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl[ \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ &\qquad{} + \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert + \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert +\Delta (a,b)+ \Delta (c,d) \biggr], \end{aligned}$$

where

$$\begin{aligned} &\vartheta ^{\prime \prime }\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr]\\ &\qquad{}\times \biggl[ \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[ \bigl(b^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr)- \bigl(b^{p_{1}}\bigr)^{1+ \frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{}+\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \biggr] \biggl[ \bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},2,3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{}-\bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},1,2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr)\\ &\qquad{}+ \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3, \frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \biggr] \end{aligned}$$

V. If we take \(\Delta (a,b)=-\mu (t^{\sigma }(1-t)+t(1-t)^{\sigma } ) ( \Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \Vert )^{\sigma }\) and \(\Delta (c,d)=-\mu (r^{\sigma }(1-r)+r(1-r)^{\sigma } ) ( \Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \Vert )^{\sigma }\) for some \(\mu >0\) in Theorem 2, then

Corollary 10

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is a two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function with \(\mu >0\) of higher order, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ &\qquad{} +\vartheta _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \vartheta _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ & \qquad{}+\vartheta _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert + \vartheta _{5}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \vartheta _{6}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where \(\vartheta _{1}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \vartheta _{2}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \vartheta _{3}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \vartheta _{4}(a^{p_{1}},b^{p_{1}}, c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 2and

$$\begin{aligned} &\vartheta _{5}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{\sigma }\\ &\qquad{} \times \int _{0}^{1} \int _{0}^{1} \biggl( \vert 1-2t \vert \bigl(t^{\sigma }(1-t)+t(1-t)^{\sigma }\bigr) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ &\qquad{} \times \biggl( \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr)\,\mathrm{d}t\,\mathrm{d}r \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{\sigma } \biggl[ \frac{2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},\sigma +2,\sigma +4,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} + \frac{2(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(\sigma +3)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},3,\sigma +4,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ & \qquad{}- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +1,\sigma +3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} - \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,\sigma +3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{\sigma }(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +1,\sigma +3,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ & \qquad{}- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{\sigma +1}(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +2,\sigma +4,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ & \qquad{}+ \frac{4(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +2,\sigma +4,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} - \frac{2(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +1,\sigma +3,1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{\sigma }(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +1,\sigma +3,\frac{1}{2} \biggl(1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \biggr) \\ & \qquad{}- \frac{(a^{p_{1}})^{1+\frac{1}{p_{1}}}}{2^{\sigma +1}(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},\sigma +2,\sigma +4,\frac{1}{2} \biggl(1- \frac{a^{p_{1}}}{b^{p_{1}}} \biggr) \biggr) \biggr] \\ & \qquad{}\times \biggl[ \bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},2,3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr)- \bigl(d^{p_{2}}\bigr)^{1+ \frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},1,2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{} +\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \biggr], \\ &\vartheta _{6}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{\sigma }\times \int _{0}^{1} \int _{0}^{1} \biggl( \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ &\qquad{}\times \biggl( \vert 1-2r \vert \bigl(r^{\sigma }(1-r)+r(1-r)^{\sigma } \bigr) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr)\,\mathrm{d}t\, \mathrm{d}r \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{\sigma } \biggl[ \bigl(b^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},2,3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} - \bigl(b^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) + \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +2,\sigma +4,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{} + \frac{2(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(\sigma +3)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},3,\sigma +4,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ & \qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +1,\sigma +3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{} - \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,\sigma +3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ & \qquad{}+ \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{\sigma }(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +1,\sigma +3,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ & \qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{\sigma +1}(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +2,\sigma +4,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{} + \frac{4(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +2,\sigma +4,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{} - \frac{2(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +1,\sigma +3,1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \\ &\qquad{} + \frac{(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{\sigma }(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +1,\sigma +3,\frac{1}{2} \biggl(1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \biggr) \\ & \qquad{}- \frac{(c^{p_{2}})^{1+\frac{1}{p_{2}}}}{2^{\sigma +1}(\sigma +2)(\sigma +3)} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},\sigma +2,\sigma +4,\frac{1}{2} \biggl(1- \frac{c^{p_{2}}}{d^{p_{2}}} \biggr) \biggr) \biggr]. \end{aligned}$$

VI. If we take \(\sigma =2\) in Corollary 10, then

Corollary 11

Under the assumptions of Corollary 10, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\) is a two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function with \(\mu >0\), then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \biggl[\vartheta _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert \\ &\qquad{} +\vartheta _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert + \vartheta _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert \\ &\qquad{} +\vartheta _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert + \vartheta _{5}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \vartheta _{6}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr], \end{aligned}$$

where \(\vartheta _{1}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \vartheta _{2}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \vartheta _{3}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \vartheta _{4}(a^{p_{1}}, b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 2and

$$\begin{aligned} &\vartheta _{5}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{2} \times \int _{0}^{1} \int _{0}^{1} ( \vert 1-2t \vert \biggl(t(1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ & \qquad{}\times \biggl( \vert 1-2r \vert \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr)\,\mathrm{d}t\,\mathrm{d}r \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{2} \biggl[\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{4} {}_{2}F_{1} \biggl(1+\frac{1}{p_{1}},2,3,\frac{1}{2} \biggl(1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{}-\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{4} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},3,4,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \\ &\qquad{} +\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{16} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},4,5,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr)\\ &\qquad{}+\bigl(b^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},3,4,\biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}}\biggr) \biggr) \\ &\qquad{} -\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},4,5,\biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}}\biggr) \biggr)\\ &\qquad{}- \frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,3,\biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}}\biggr) \biggr) \biggr] \\ &\qquad{} \times \biggl[ \bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},2,3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr)- \bigl(d^{p_{2}}\bigr)^{1+ \frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+\frac{1}{p_{2}},1,2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ & \qquad{}+\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},1,3,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \biggr], \\ &\vartheta _{6}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{2} \times \int _{0}^{1} \int _{0}^{1} \biggl( \vert 1-2t \vert \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ &\qquad{} \times \biggl( \vert 1-2r \vert r(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr)\,\mathrm{d}t\,\mathrm{d}r \\ &\qquad{} -\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{2}\times \biggl[ \bigl(b^{p_{1}} \bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},2,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ & \qquad{}- \bigl(d^{p_{1}}\bigr)^{1+\frac{1}{p_{1}}} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} +\frac{(b^{p_{1}})^{1+\frac{1}{p_{1}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{1}},1,3,\frac{1}{2} \biggl(1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr) \biggr] \\ &\qquad{} \times \biggl[\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{4} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,3,\frac{1}{2} \biggl(1- \frac{d^{p_{2}}}{a^{p_{2}}} \biggr) \biggr) \\ &\qquad{}-\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{4} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},3,4,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr) \\ &\qquad{} +\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{16} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},4,5,\frac{1}{2} \biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr)\\ &\qquad{}+\bigl(d^{p_{2}}\bigr)^{1+\frac{1}{p_{2}}} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},3,4,\biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}}\biggr) \biggr) \\ & \qquad{}-\frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},4,5,\biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}}\biggr) \biggr)\\ &\qquad{}- \frac{(d^{p_{2}})^{1+\frac{1}{p_{2}}}}{2} {}_{2}F_{1} \biggl(1+ \frac{1}{p_{2}},2,3,\biggl(1-\frac{d^{p_{2}}}{c^{p_{2}}}\biggr) \biggr) \biggr]. \end{aligned}$$

Theorem 3

Let \(f:\Omega \rightarrow \mathbb{R}\) be a partial differentiable function on \(\Omega = [a,b]\times [c,d]\subset (0,\infty )\times (0,\infty )\) with \(a< b\) and \(c< d\) and \(\frac{\partial ^{2}f}{\partial t \partial r}\in L_{1}(\Omega )\). If \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is an approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), we have

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ & \qquad{}+\varphi _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} + \varphi _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} + \varphi _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \varphi _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} &\varphi _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[h_{1}(t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[h_{2}(r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r, \\ &\varphi _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[h_{1}(1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[h_{2}(r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r, \\ &\varphi _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[h_{1}(t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr]\\ &\qquad{}\times \biggl[h_{2}(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r, \\ &\varphi _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[h_{1}(1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[h_{2}(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r, \\ &\varphi _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=\Delta (a,b) \int _{0}^{1} \int _{0}^{1} \biggl[ \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[ \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad=\Delta (a,b) \biggl( \biggl[\bigl(b^{p_{1}}\bigr)^{q (1+\frac{1}{p_{1}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\bigl(d^{p_{2}}\bigr)^{q (1+\frac{1}{p_{2}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr), \end{aligned}$$

and

$$\begin{aligned} &\varphi _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \biggl[ \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad=\Delta (c,d) \biggl( \biggl[\bigl(b^{p_{1}}\bigr)^{q (1+\frac{1}{p_{1}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\bigl(d^{p_{2}}\bigr)^{q (1+\frac{1}{p_{2}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr). \end{aligned}$$

Proof

Using Lemma 1, well-known Hölder’s inequality, and the fact that \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is an approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, we have

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad= \biggl\vert \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \int _{0}^{1} \biggl((1-2t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{1+ \frac{1}{p_{1}}} \biggr) \\ &\qquad{}\times\biggl((1-2r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{1+ \frac{1}{p_{2}}} \biggr) \\ &\qquad{} \times \frac{\partial ^{2}f}{\partial r \,\partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr)\, \mathrm{d}r\,\mathrm{d}t \biggr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}} \int _{0}^{1} \int _{0}^{1} \bigl( \bigl\vert (1-2t) (1-2r) \bigr\vert ^{p} \,\mathrm{d}r\,\mathrm{d}t \bigr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \\ &\qquad{} \times \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t} \biggl( \biggl[\frac{a^{p_{1}}b^{p_{1}}}{tb^{p_{1}}+(1-t)a^{p_{1}}} \biggr]^{ \frac{1}{p_{1}}}, \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{rd^{p_{2}}+(1-r)c^{p_{2}}} \biggr]^{ \frac{1}{p_{2}}} \biggr) \biggr\vert ^{q}\,\mathrm{d}r\,\mathrm{d}t \biggr)^{\frac{1}{q}} \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \\ &\qquad{}\times\biggl( \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \\ & \qquad{}\times \biggl[h_{1}(t)h_{2}(r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q}+h_{1}(1-t)h_{2}(r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} \\ &\qquad{}+h_{1}(t)h_{2}(1-r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ &\qquad{} +h_{1}(1-t)h_{2}(1-r) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} +\Delta (a,b)+ \Delta (c,d) \biggr]\,\mathrm{d}r\,\mathrm{d}t \biggr)^{\frac{1}{q}}. \end{aligned}$$

This completes the proof. □

Corollary 12

Under the assumptions of Theorem 2, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |\le M\) is a bounded function for \(M>0\) on Ω, then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{M(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\qquad{} +\varphi _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) + \varphi _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \\ &\qquad{} +\varphi _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) + \frac{1}{M^{q}}\varphi _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+\frac{1}{M^{q}}\varphi _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where \(\varphi _{1}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) to \(\varphi _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 3.

We now discuss some special cases of Theorem 3.

I. If we take \(h_{1}(t)=t\) and \(h_{2}(r)=r\) in Theorem 3, then

Corollary 13

Under the assumptions of Theorem 3, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is an approximately two-dimensional harmonic \((p_{1},p_{2})\)-convex function, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}^{*} \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ & \qquad{}+\varphi _{2}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} +\varphi _{3}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{4}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} +\varphi _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \varphi _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} &\varphi _{1}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[t \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \biggl[r \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),2,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr]\\ &\qquad{}\times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{2}} \biggr),2,3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{2}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[(1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr]\\ &\qquad{}\times \biggl[r \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr]\\ &\qquad{}\times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{2}} \biggr),2,3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{3}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[t \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),2,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr]\\ &\qquad{}\times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{2}} \biggr),1,3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{4}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[(1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr]\\ &\qquad{}\times \biggl[(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,3,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr]\\ &\qquad{}\times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{2} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{2}} \biggr),1,3,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \end{aligned}$$

\(\varphi _{5}(a,b,c,d;\Omega )\) and \(\varphi _{6}(a,b,c,d;\Omega )\) are given in Theorem 3.

II. If we take \(h_{1}(t)=t^{s_{1}}\) and \(h_{2}(r)=r^{s_{2}}\) in Theorem 3, then

Corollary 14

Under the assumptions of Theorem 3, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is a Breckner type approximately two-dimensional harmonic \((p_{1},s_{1})\)-\((p_{2},s_{2})\)-convex function, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}^{**} \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{2}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} +\varphi _{3}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{4}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} +\varphi _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \varphi _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} &\varphi _{1}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[t^{s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \biggl[r^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{s_{1}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),s_{1}+1,s_{1}+2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ & \qquad{}\times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{s_{2}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),s_{2}+1,s_{2}+2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{2}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[(1-t)^{s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[r^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{s_{1}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,s_{1}+2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{s_{2}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),s_{2}+1,s_{2}+2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{3}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[t^{s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[(1-r)^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{s_{1}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),s_{1}+1,s_{1}+2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{s_{2}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,s_{2}+2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{4}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[(1-t)^{s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr]\\ &\qquad{}\times \biggl[(1-r)^{s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{s_{1}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,s_{1}+2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{s_{2}+1} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,s_{2}+2,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \end{aligned}$$

\(\varphi _{5}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}};\Omega )\) and \(\varphi _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}};\Omega )\) are given in Theorem 3.

III. If we take \(h_{1}(t)=t^{-s_{1}}\) and \(h_{2}(r)=r^{-s_{2}}\) in Theorem 3, then

Corollary 15

Under the assumptions of Theorem 3, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is a Godunova–Levin type approximately two-dimensional harmonic \((p_{1},s_{1})\)-\((p_{2},s_{2})\)-convex function, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), we have

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}^{***} \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{2}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} +\varphi _{3}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{4}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} +\varphi _{5}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \varphi _{6}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} &\varphi _{1}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[t^{-s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \biggl[r^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{1-s_{1}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1-s_{1},2-s_{1},1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{1-s_{2}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1-s_{2},2-s_{2},1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{2}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[(1-t)^{-s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr]\\ &\qquad{}\times \biggl[r^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{1-s_{1}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,2-s_{1},1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{1-s_{2}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1-s_{2},2-s_{2},1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{3}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[t^{-s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[(1-r)^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{1-s_{1}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1-s_{1},2-s_{1},1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ & \qquad{}\times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{1-s_{2}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2-s_{2},1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \\ &\varphi _{4}^{***}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[(1-t)^{-s_{1}} \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{}\times\biggl[(1-r)^{-s_{2}} \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\, \mathrm{d}r \\ &\quad= \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{1-s_{1}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,2-s_{1},1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{1-s_{2}} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2-s_{2},1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr], \end{aligned}$$

\(\varphi _{5}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}};\Omega )\) and \(\varphi _{6}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}};\Omega )\) are given in Theorem 3.

IV. If we take \(h_{1}(t)=1\) and \(h_{2}(r)=1\) in Theorem 3, then

Corollary 16

Under the assumptions of Theorem 3, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is an approximately two-dimensional harmonic \((p_{1},P)\)-\((p_{2},P)\)-convex function, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}}\bigl[ \varphi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\bigr]^{ \frac{1}{q}} \biggl[ \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ &\qquad + \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q}+ \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q}+ \Delta (a,b)+\Delta (c,d) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where

$$\begin{aligned} &\varphi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad= \int _{0}^{1} \int _{0}^{1} \biggl[ \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \biggl[ \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \\ &\quad= \biggl( \biggl[\bigl(b^{p_{1}}\bigr)^{q (1+\frac{1}{p_{1}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),1,2,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{}\times \biggl[\bigl(d^{p_{2}}\bigr)^{q (1+\frac{1}{p_{2}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr). \end{aligned}$$

V. If we take \(\Delta (a,b)=-\mu (t^{\sigma }(1-t)+t(1-t)^{\sigma } ) ( \Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \Vert )^{\sigma }\) and \(\Delta (c,d)=-\mu (r^{\sigma }(1-r)+r(1-r)^{\sigma } ) ( \Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \Vert )^{\sigma }\) for some \(\mu >0\) in Theorem 3, then

Corollary 17

Under the assumptions of Theorem 3, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is a two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function of higher order, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} + \varphi _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} + \varphi _{5}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \varphi _{6}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where \(\varphi _{1}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \varphi _{2}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \varphi _{3}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \varphi _{4}(a^{p_{1}},b^{p_{1}}, c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 3, and

$$\begin{aligned} &\varphi _{5}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{\sigma } \biggl( \int _{0}^{1} \int _{0}^{1} \biggl[ \bigl(t^{\sigma }(1-t)+t(1-t)^{\sigma } \bigr) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+ \frac{1}{p_{1}} )} \biggr] \\ &\qquad{} \times \biggl[ \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\, \mathrm{d}t\,\mathrm{d}r\biggr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{\sigma } \biggl( \biggl[ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr), \sigma +1,\sigma +3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} + \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{(\sigma +2)(\sigma +1)} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{1}} \biggr),2,\sigma +3,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[\bigl(d^{p_{2}}\bigr)^{q (1+\frac{1}{p_{2}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr), \\ &\varphi _{6}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{\sigma } \biggl( \int _{0}^{1} \int _{0}^{1} \biggl[ \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+\frac{1}{p_{1}} )} \biggr] \\ &\qquad{} \times \biggl[ \bigl(r^{\sigma }(1-r)+r(1-r)^{\sigma } \bigr) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+\frac{1}{p_{2}} )} \biggr]\,\mathrm{d}t\,\mathrm{d}r \biggr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{\sigma } \biggl( \biggl[\bigl(b^{p_{1}} \bigr)^{q (1+\frac{1}{p_{1}} )} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{1}} \biggr),1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{(\sigma +1)(\sigma +2)} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),\sigma +1,\sigma +3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \\ &\qquad{} + \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{(\sigma +2)(\sigma +1)} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{2}} \biggr),2,\sigma +3,1- \frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr). \end{aligned}$$

VI. If we take \(\sigma =2\) in Corollary 17, then

Corollary 18

Under the assumptions of Corollary 17, if \(|\frac{\partial ^{2}f}{\partial r \,\partial t} |^{q}\) is a two-dimensional harmonically strong \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, where \(\frac{1}{p}+\frac{1}{q}=1\) and \(q>1\), then

$$\begin{aligned} & \bigl\vert \Xi \bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}},x,y: \Omega \bigr) \bigr\vert \\ &\quad\leq \frac{(b^{p_{1}}-a^{p_{1}})(d^{p_{2}}-c^{p_{2}})}{4p_{1}p_{2}a^{p_{1}}b^{p_{1}}c^{p_{2}}d^{p_{2}}(p+1)^{\frac{2}{p}}} \biggl[\varphi _{1}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,c) \biggr\vert ^{q} \\ &\qquad{} +\varphi _{2}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,c) \biggr\vert ^{q} + \varphi _{3}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(a,d) \biggr\vert ^{q} \\ & \qquad{}+\varphi _{4}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggl\vert \frac{\partial ^{2}f}{\partial r \,\partial t}(b,d) \biggr\vert ^{q} + \varphi _{5}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr)\\ &\qquad{}+ \varphi _{6}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}: \Omega \bigr) \biggr]^{\frac{1}{q}}, \end{aligned}$$

where \(\varphi _{1}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \varphi _{2}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \varphi _{3}(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}:\Omega ), \varphi _{4}(a^{p_{1}},b^{p_{1}}, c^{p_{2}},d^{p_{2}}:\Omega )\) are given in Theorem 3, and

$$\begin{aligned} &\varphi _{5}^{**}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{2} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl[t(1-t) \biggl[ \frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+\frac{1}{p_{1}} )} \biggr] \\ &\qquad{} \times \biggl[ \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr]\, \mathrm{d}t\,\mathrm{d}r\biggr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{b^{p_{1}}}-\frac{1}{a^{p_{1}}} \biggr\Vert \biggr)^{2} \biggl( [ \frac{(b^{p_{1}})^{q (1+\frac{1}{p_{1}} )}}{6} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{1}} \biggr),2,4,1-\frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \\ &\qquad{} \times \biggl[\bigl(d^{p_{2}}\bigr)^{q (1+\frac{1}{p_{2}} )} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),1,2,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr), \\ &\varphi _{6}^{*}\bigl(a^{p_{1}},b^{p_{1}},c^{p_{2}},d^{p_{2}}; \Omega \bigr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{2} \biggl( \int _{0}^{1} \int _{0}^{1} \biggl[ \biggl[\frac{a^{p_{1}}b^{p_{1}}}{(tb^{p_{1}}+(1-t)a^{p_{1}})} \biggr]^{q (1+\frac{1}{p_{1}} )} \biggr] \\ & \qquad{}\times \biggl[r(1-r) \biggl[ \frac{c^{p_{2}}d^{p_{2}}}{(rd^{p_{2}}+(1-r)c^{p_{2}})} \biggr]^{q (1+ \frac{1}{p_{2}} )} \biggr] \,\mathrm{d}t\,\mathrm{d}r \biggr) \\ &\quad=-\mu \biggl( \biggl\Vert \frac{1}{d^{p_{2}}}-\frac{1}{c^{p_{2}}} \biggr\Vert \biggr)^{2} \biggl( \biggl[\bigl(b^{p_{1}} \bigr)^{q (1+\frac{1}{p_{1}} )} {}_{2}F_{1} \biggl(q \biggl(1+ \frac{1}{p_{1}} \biggr),1,2,1- \frac{b^{p_{1}}}{a^{p_{1}}} \biggr) \biggr] \\ &\qquad{} \times \biggl[ \frac{(d^{p_{2}})^{q (1+\frac{1}{p_{2}} )}}{6} {}_{2}F_{1} \biggl(q \biggl(1+\frac{1}{p_{2}} \biggr),2,4,1-\frac{d^{p_{2}}}{c^{p_{2}}} \biggr) \biggr] \biggr). \end{aligned}$$

Conclusion

In this paper, we defined a new interesting class of functions, two-dimensional approximately harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex function, and some Hermite–Hadamard type integral inequalities are provided as well on the coordinates. As a particular cases for \(p_{1},h_{1},p_{2},h_{2}\), we get several new classes of functions. For instance, \(p_{1}=1=p_{2}\) and \(h_{1}= h =h_{2}\), we get the results of the paper [20] for approximately harmonic h-convex function. These results can be applied in convex analysis, optimization, and different areas of pure and applied sciences. The authors hope that these results will serve as a motivation for future work in this fascinating area.

References

  1. 1.

    Işcan, İ.: Hermite–Hadamard type inequalities for harmonically convex functions. Hacet. J. Math. Stat. 43(6), 935–942 (2014)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Noor, M.A., Noor, K.I., Awan, M.U., Costache, S.: Some integral inequalities for harmonically h-convex functions. UPB Sci. Bull., Ser. A, Appl. Math. Phys. 77(1), 1–12 (2015)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Noor, M.A., Noor, K.I., Iftikhar, S.: Integral inequalities for differentiable p-harmonic convex functions. Filomat 31(20), 6575–6584 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Noor, M.A., Noor, K.I., Awan, M.U.: Integral inequalities for coordinated harmonically convex functions. Complex Var. Elliptic Equ. 60, 776–786 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Awan, M.U., Noor, M.A., Mihai, M.V., Noor, K.I., Akhtar, N.: On approximately harmonic h-convex functions depending on a given function. Filomat 33(12), 3783–3793 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Awan, M.U., Noor, M.A., Mihai, M.V., Noor, K.I.: Some fractional extensions of trapezium inequalities via coordinated harmonic convex functions. J. Nonlinear Sci. Appl. 10, 1714–1730 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Awan, M.U., Noor, M.A., Mihai, M.V., Noor, K.I., Khan, A.G.: Some new bounds for Simpson’s rule involving special functions via harmonic h-convexity. J. Nonlinear Sci. Appl. 10, 1755–1766 (2017)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Burai, P., Hazy, A.: On approximately h-convex functions. J. Convex Anal. 18(2), 447–454 (2011)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Cristescu, G., Noor, M.A., Awan, M.U.: Bounds of the second degree cumulative frontier gaps of functions with generalized convexity. Carpath. J. Math. 31(2), 173–180 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Cristescu, G., Lupsa, L.: Non-connected Convexities and Applications. Kluwer Academic, Dordrecht (2002)

    Google Scholar 

  11. 11.

    Dragomir, S.S., Pearce, C.E.M.: Selected Topics on Hermite–Hadamard Inequalities and Applications. Victoria University, Australia (2000)

    Google Scholar 

  12. 12.

    Niculescu, C.P., Persson, L.-E.: Convex Functions and Their Applications. A Contemporary Approach, 2nd edn. CMS Books in Mathematics, vol. 23. Springer, New York (2018)

    Google Scholar 

  13. 13.

    Mihai, M.V., Noor, M.A., Noor, K.I., Awan, M.U.: Some integral inequalities for harmonic h-convex functions involving hypergeometric functions. Appl. Math. Comput. 252, 257–262 (2015)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Varosanec, S.: On h-convexity. J. Math. Anal. Appl. 326, 303–311 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Zhang, Y., Du, T.S., Wang, H., Shen, Y.-J.: Different types of quantum integral inequalities via \((\alpha, m)\)-convexity. J. Inequal. Appl. 2018 264 (2018)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Liao, J.G., Wu, S.H., Du, T.S.: The Sugeno integral with respect to α-preinvex functions. Fuzzy Sets Syst. 379, 102–114 (2020)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Du, T.S., Awan, M.U., Kashuri, A., Zhao, S.S.: Some k-fractional extensions of the trapezium inequalities through generalized relative semi-(m; h)-preinvexity. Appl. Anal. 2019, 1–21 (2019). https://doi.org/10.1080/00036811.2019.1616083

    Article  Google Scholar 

  18. 18.

    Du, T.S., Wang, H., Khan, M.A., Zhang, Y.: Certain integral inequalities considering generalized m-convexity on fractal sets and their applications. Fractals 27(7), 1950117 (2019).

    MathSciNet  Article  Google Scholar 

  19. 19.

    Set, E.: New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals. Comput. Math. Appl. 63, 1147–1154 (2012)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Akhtar, N., Awan, M.U., Kashuri, A., Mihai, M.V., Noor, M.A., Noor, K.I.: Approximately two dimensional harmonic h-convex functions and related integral inequalities. (Submitted)

Download references

Acknowledgements

We thank the editor and referees for their careful reading and valuable suggestions to make the article reader-friendly.

Availability of data and materials

There is no additional data required for finding the results of this paper.

Funding

There is no funding available for the publication of this paper.

Author information

Affiliations

Authors

Contributions

All authors have equal contribution in this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Wei Gao.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Butt, S.I., Kashuri, A., Nadeem, M. et al. Approximately two-dimensional harmonic \((p_{1},h_{1})\)-\((p_{2},h_{2})\)-convex functions and related integral inequalities. J Inequal Appl 2020, 230 (2020). https://doi.org/10.1186/s13660-020-02495-6

Download citation

Keywords

  • Hermite–Hadamard inequality
  • Hölder inequality
  • Convexity