New version of midpoint-type inequalities for co-ordinated convex functions via generalized conformable integrals

Abstract

In the current research, some midpoint-type inequalities are generalized for co-ordinated convex functions with the help of generalized conformable fractional integrals. Moreover, some findings of this paper include results based on Riemann–Liouville fractional integrals and Riemann integrals.

1 Introduction

Convex functions are a fundamental and widely-used mathematical concept in various fields of analysis and optimization. A function is considered convex if the line segment connecting any two points on its graph lies either below or on the graph itself, indicating a curve that is upward-curving. Convex functions have notable properties, including the fact that the slope between any two points is either increasing or constant, making them valuable in optimization problems to find minimum or maximum values. The definition known for convex functions is as follows:

Definition 1

[12] Let I be convex set on $\mathbb{R}$. The function $\chi :I\rightarrow \mathbb{R}$ is said to be convex on I if it satisfies the following inequality:

$$ \chi \bigl(t\lambda _{1}+(1-t) \lambda _{2}\bigr) \leq t\chi (\lambda _{1})+(1-t) \chi (\lambda _{2}) $$

(1.1)

for all $\lambda _{1},\lambda _{2}\in I$ and $t\in {}[ 0,1]$. The mapping χ is a concave on I if the inequality (1.1) holds in reversed direction for all $t\in {}[ 0,1]$ and $\lambda _{1},\lambda _{2}\in I$.

To define convexity on co-ordinates let us first consider a bidimensional interval $\Delta := [ \lambda _{1},\lambda _{2} ] \times [ \mu _{1},\mu _{2} ]$ in $\mathbb{R} ^{2}$ with $\lambda _{1}<\lambda _{2}$ and $\mu _{1}<\mu _{2}$. A formal definition for co-ordinated convex function may be stated as follows:

Definition 2

[11] A function $\chi :\Delta \rightarrow \mathbb{R}$ will be called coordinated convex on Δ for all $(x_{1},x_{2}),(y_{1},y_{2})\in \Delta$ and $t,s\in {}[ 0,1]$ if it satisfies the following inequality:

$$\begin{aligned}& \chi \bigl(tx_{1}+(1-t)x_{2},sy_{1}+(1-s)y_{2} \bigr) \\& \quad \leq ts\chi (x_{1},y_{1})+t(1-s)\chi (x_{1},y_{2})+s(1-t)\chi (x_{2},y_{1})+(1-t) (1-s) \chi (x_{2},y_{2}). \end{aligned}$$

It is clear that all convex functions are convex on co-ordinates. However, not every function that is a convex function in coordinates has to be convex (see, [11]).

In the realm of inequalities, one prominent result is the Hermite–Hadamard inequality, which holds for convex functions. This inequality gives upper and lower bounds for the average value of a convex function over an interval. It serves as a powerful tool in various mathematical analyzes and has applications in diverse scientific fields (see, e.g., [12], [25, p.137]). Hermite–Hadamard inequality is stated that if $\chi :I\rightarrow \mathbb{R}$ is a convex function on the interval I of real numbers and $\lambda _{1},\lambda _{2}\in I$ with $\lambda _{1}<\lambda _{2}$, then

$$ \chi \biggl( \frac{\lambda _{1}+\lambda _{2}}{2} \biggr) \leq \frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{ \lambda _{2}}\chi (\delta )\,d\delta \leq \frac{\chi ( \lambda _{1} ) +\chi (\lambda _{2})}{2}. $$

(1.2)

If χ is concave, the inequality that is stated above is provided reversely. The references may be seen for the examples of Hermite–Hadamard’s inequality for some convex function on the co-ordinates in mathematics literature [3–5, 7, 8, 10, 22, 23, 28, 31]. Recently, this inequality has been expanded by many researchers. The left side of the Hermite–Hadamard inequality, namely the midpoint type inequality, has been the focus of many studies. Midpoint type inequalities for convex functions were first derived by Kırmacıin [21]. In [32], Sarikaya et al. generalized the inequalities (1.2) for fractional integrals. Iqbal et al. proved corresponding midpoint type inequalities for Riemann–Liouville fractional integrals in [15].

In [11], Dragomir proved the Hermite–Hadamard inequality, which formed the basis of this article and is valid for co-ordinated convex functions on the rectangle from the plane $\mathbb{R} ^{2}$.

Theorem 1

Suppose that $\chi :\Delta \rightarrow \mathbb{R}$ is co-ordinated convex, then we have the following inequalities:

$$\begin{aligned} \chi \biggl( \frac{\lambda _{1}+\lambda _{2}}{2}, \frac{\mu _{1}+\mu _{2}}{2} \biggr) \leq & \frac{1}{2} \biggl[ \frac{1}{\lambda _{2}-\lambda _{1}}\int _{\lambda _{1}}^{\lambda _{2}}\chi \biggl( \delta , \frac{\mu _{1}+\mu _{2}}{2} \biggr)\,d\delta \\ & {}+ \frac{1}{\mu _{2}-\mu _{1}} \int _{\mu _{1}}^{\mu _{2}} \chi \biggl( \frac{\lambda _{1}+\lambda _{2}}{2},\rho \biggr)\,d\rho \biggr] \\ \leq &\frac{1}{(\lambda _{2}-\lambda _{1})(\mu _{2}-\mu _{1})}\int _{\lambda _{1}}^{\lambda _{2}} \int _{\mu _{1}}^{ \mu _{2}}\chi (\delta ,\rho )\,d\rho \,d\delta \\ \leq &\frac{1}{4} \biggl[ \frac{1}{\lambda _{2}-\lambda _{1}}\int _{\lambda _{1}}^{\lambda _{2}}\chi (\delta ,\mu _{1})\,d\delta +\frac{1}{\lambda _{2}-\lambda _{1}} \int _{\lambda _{1}}^{ \lambda _{2}}\chi (\delta ,\mu _{2})\,d\delta \\ & {}+ \frac{1}{\mu _{2}-\mu _{1}} \int _{\mu _{1}}^{\mu _{2}} \chi (\lambda _{1},\rho )\,d\rho +\frac{1}{\mu _{2}-\mu _{1}} \int _{\mu _{1}}^{\mu _{2}}\chi (\lambda _{2},\rho )\,d\rho \biggr] \\ \leq & \frac{\chi (\lambda _{1},\mu _{1})+\chi (\lambda _{1},\mu _{2})+\chi (\lambda _{2},\mu _{1})+\chi (\lambda _{2},\mu _{2})}{4}. \end{aligned}$$

(1.3)

The inequalities in (1.3) hold in reverse direction if the mapping χ is a co-ordinated concave mapping.

The fractional calculus [16, 19, 24, 26, 27] is defined as any random real number or derivative and integral calculus in complex order. As a result of having various uses in other branches besides mathematics it is an updated study area. These definitions are the most notable definitions of Caputo, Riemann–Liouville, Grünwald–Letnikov play an important role in many fields such as physics, biology, and engineering. However, it is known that these definitions have some difficulties despite their availability. For instance, unless derivative of order in Riemann–Liouville fractional derivative definition is a natural number, derivative of fixed function is not 0. Likewise, the function f must be differentiable in Caputo fractional derivatives. Moreover, many definitions of fractional derivatives do not provide the quotient formula, the product of two functions, and the chain rule. In order to overcome these and similar difficulties, conformable fractional derivative was defined by Khalil et al. in [17]. Khalil et al. described the higher order ($\alpha >1$) fractional derivative and the fractional integral of order ($0<\alpha \leq 1$). They also proved important theorems such as the product rule, the fractional mean value theorem. They solved conformable fractional differential equations for fractional exponential functions (see, [2, 13, 17, 33]). Thus, conformable fractional integrals became an important field of study for many researchers. For some papers on conformable fractional integrals, please see [1, 6, 14, 17, 18, 29, 30].

The definitions and mathematical underpinnings of conformable fractional calculus principles that are used later in this study are provided below:

Definition 3

[19] For $\xi \in L_{1}[\eta _{1},\eta _{2}]$, the Riemann–Liouville integrals of order $\alpha >0$ are given by

$$ J_{\eta _{1}+}^{\alpha }\xi (\delta )=\frac{1}{\Gamma (\alpha )} \int _{ \eta _{1}}^{\delta } ( \delta -t ) ^{\alpha -1}\xi (t)\,dt,\quad \delta >\eta _{1} $$

(1.4)

and

$$ J_{\eta _{2}-}^{\alpha }\xi (\delta )=\frac{1}{\Gamma (\alpha )} \int _{ \delta }^{\eta _{2}} ( t-\delta ) ^{\alpha -1}\xi (t)\,dt,\quad \delta < \eta _{2}, $$

(1.5)

respectively. Here Γ is the Gamma function. The Riemann–Liouville integrals will be equal to the classical Riemann integrals for the condition $\alpha =1$.

Definition 4

[28] Let $\xi \in L_{1}( [ \eta _{1},\eta _{2} ] \times [ \vartheta _{1},\vartheta _{2} ] )$. The Riemann–Liouville integrals $J_{\eta _{1}+,\vartheta _{1}^{+}}^{\alpha ,\beta }$, $J_{\eta _{1}+, \vartheta _{2}^{-}}^{\alpha ,\beta }$, $J_{\eta _{2}^{-},\vartheta _{1}^{+}}^{ \alpha ,\beta }$ and $J_{\eta _{2}^{-},\vartheta _{2}^{-}}^{\alpha ,\beta }$ of order $\alpha ,\beta >0$ with $\eta _{1},\vartheta _{1}\geq 0$ are defined by

$$\begin{aligned}& J_{\eta _{1}+,\vartheta _{1}^{+}}^{\alpha ,\beta }\xi (\delta ,\rho ) \\& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\eta _{1}}^{\delta } \int _{ \vartheta _{1}}^{\rho } ( \delta -t ) ^{\alpha -1} ( \rho -s ) ^{\beta -1}\xi (t,s)\,ds\,dt,\quad \delta >\eta _{1}, \rho >\vartheta _{1}, \end{aligned}$$

(1.6)

$$\begin{aligned}& J_{\eta _{1}+,\vartheta _{2}^{-}}^{\alpha ,\beta }\xi (\delta ,\rho ) \\& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\eta _{1}}^{\delta } \int _{ \rho }^{\vartheta _{2}} ( \delta -t ) ^{\alpha -1} ( s- \rho ) ^{\beta -1}\xi (t,s)\,ds\,dt, \quad \delta >\eta _{1},\rho < \vartheta _{2}, \end{aligned}$$

(1.7)

$$\begin{aligned}& J_{\eta _{2}-,\vartheta _{1}^{+}}^{\alpha ,\beta }\xi (\delta ,\rho ) \\& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\delta }^{\eta _{2}} \int _{ \vartheta _{1}}^{\rho } ( t-\delta ) ^{\alpha -1} ( \rho -s ) ^{\beta -1}\xi (t,s)\,ds\,dt, \delta < \eta _{2}, \quad \rho >\vartheta _{1}, \end{aligned}$$

(1.8)

and

$$\begin{aligned}& J_{\eta _{2}-,\vartheta _{2}^{-}}^{\alpha ,\beta }\xi (\delta ,\rho ) \\& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\delta }^{\eta _{2}} \int _{ \rho }^{\vartheta _{2}} ( t-\delta ) ^{\alpha -1} ( s- \rho ) ^{\beta -1}\xi (t,s)\,ds\,dt, \quad \delta < \eta _{2}, \rho < \vartheta _{2}, \end{aligned}$$

(1.9)

respectively.

Definition 5

[16] For $\xi \in L_{1}[\eta _{1},\eta _{2}]$, the fractional conformable integral operator ${}^{\beta }I_{\eta _{1}+}^{\alpha }\xi$ and ${}^{\beta }I_{\eta _{2}-}^{\alpha }\xi$ of order $\beta >0$ and $\alpha \in (0,1]$ are presented by

$$ ^{\beta }\mathcal{I}_{\eta _{1}+}^{\alpha }\xi (\delta )= \frac{1}{\Gamma (\beta )} \int _{\eta _{1}}^{\delta } \biggl( \frac{(\delta -\eta _{1})^{\alpha }-(t-\eta _{1})^{\alpha }}{\alpha } \biggr) ^{\beta -1}\frac{\xi (t)}{(t-\eta _{1})^{1-\alpha }}\,dt,\quad t>\eta _{1} $$

(1.10)

and

$$ ^{\beta }\mathcal{I}_{\eta _{2}-}^{\alpha }\xi (\delta )= \frac{1}{\Gamma (\beta )} \int _{\delta }^{\eta _{2}} \biggl( \frac{(\eta _{2}-\delta )^{\alpha }-(\eta _{2}-t)^{\alpha }}{\alpha } \biggr) ^{\beta -1}\frac{\xi (t)}{(\eta _{2}-t)^{1-\alpha }}\,dt, \quad t< \eta _{2}, $$

(1.11)

respectively.

Definition 6

[9] Let $\xi \in L_{1}([\eta _{1},\eta _{2}]\times [ \vartheta _{1}, \vartheta _{2} ] )$ and let $\gamma _{1}\neq 0$, $\gamma _{2}\neq 0$, $\alpha ,\beta \in \mathbf{C}$, $\operatorname{Re} (\alpha )>0$ and $\operatorname{Re} (\beta )>0$. The generalized conformable integral of order α, β of $\xi ( \delta ,\rho )$ are defined by

$$\begin{aligned}& \bigl( ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{ \alpha ,\beta }\xi \bigr) (\delta ,\rho ) \end{aligned}$$

(1.12)

$$\begin{aligned}& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\eta _{1}}^{\delta } \int _{\vartheta _{1}}^{\rho } \biggl( \frac{(\delta -\eta _{1})^{\gamma _{1}}-(t-\eta _{1})^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\& \qquad {} \times \biggl( \frac{(\rho -\vartheta _{1})^{\gamma _{2}}-(s-\vartheta _{1})^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1} \frac{\xi (t,s)}{(t-\eta _{1})^{1-\gamma _{1}}(s-\vartheta _{1})^{1-\gamma _{2}}}\,ds\,dt, \\& \bigl( ^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{ \alpha ,\beta }\xi \bigr) (\delta ,\rho ) \end{aligned}$$

(1.13)

$$\begin{aligned}& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\delta }^{\eta _{2}} \int _{\vartheta _{1}}^{\rho } \biggl( \frac{(\eta _{2}-\delta )^{\gamma _{1}}-(\eta _{2}-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\& \qquad {} \times \biggl( \frac{(\rho -\vartheta _{1})^{\gamma _{2}}-(s-\vartheta _{1})^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1} \frac{\xi (t,s)}{(\eta _{2}-t)^{1-\gamma _{1}}(s-\vartheta _{1})^{1-\gamma _{2}}}\,ds\,dt, \\& \bigl( ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \bigr) (\delta ,\rho ) \\& \quad = \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\eta _{1}}^{\delta } \int _{\rho }^{\vartheta _{2}} \biggl( \frac{(\delta -\eta _{1})^{\gamma _{1}}-(t-\eta _{1})^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\& \qquad {}\times \biggl( \frac{(\vartheta _{2}-\rho )^{\gamma _{2}}-(\vartheta _{2}-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1} \frac{\xi (t,s)}{(t-\eta _{1})^{1-\gamma _{1}}(\vartheta _{2}-s)^{1-\gamma _{2}}}\,ds\,dt, \end{aligned}$$

(1.14)

and

$$\begin{aligned} \bigl( ^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \bigr) (\delta ,\rho ) =& \frac{1}{\Gamma (\alpha )\Gamma (\beta )} \int _{\delta }^{\eta _{2}} \int _{\rho }^{\vartheta _{2}} \biggl( \frac{(\eta _{2}-\delta )^{\gamma _{1}}-(\eta _{2}-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\ &{}\times \biggl( \frac{(\vartheta _{2}-\rho )^{\gamma _{2}}-(\vartheta _{2}-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1} \frac{\xi (t,s)}{(\eta _{2}-t)^{1-\gamma _{1}}(\vartheta _{2}-s)^{1-\gamma _{2}}}\,ds\,dt. \end{aligned}$$

(1.15)

Remark 1

[9] If we choose $\gamma _{1}=\gamma _{2}=1$ in (1.12)–(1.15), then we have the fractional integrals (1.6)–(1.9), respectively.

Remark 2

[9] If we consider $\alpha =1$ and $\beta =1$ in (1.12)–(1.15), then we have

$$\begin{aligned}& \bigl( I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{1,1}\xi \bigr) ( \delta ,\rho )= \int _{\eta _{1}}^{\delta } \int _{\vartheta _{1}}^{ \rho } \frac{\xi (t,s)}{(t-\eta _{1})^{1-\gamma _{1}}(s-\vartheta _{1})^{1-\gamma _{2}}}\,ds\,dt, \end{aligned}$$

(1.16)

$$\begin{aligned}& \bigl( I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{1,1}\xi \bigr) ( \delta ,\rho )= \int _{\delta }^{\eta _{2}} \int _{\vartheta _{1}}^{ \rho } \frac{\xi (t,s)}{(\eta _{2}-t)^{1-\gamma _{1}}(s-\vartheta _{1})^{1-\gamma _{2}}}\,ds\,dt, \end{aligned}$$

(1.17)

$$\begin{aligned}& \bigl( I_{\eta _{1}^{+},\vartheta _{2}^{-}}^{1,1}\xi \bigr) ( \delta ,\rho )= \int _{\eta _{1}}^{\delta } \int _{\rho }^{\vartheta _{2}} \frac{\xi (t,s)}{(t-\eta _{1})^{1-\gamma _{1}}(\vartheta _{2}-s)^{1-\gamma _{2}}}\,ds\,dt, \end{aligned}$$

(1.18)

and

$$ \bigl( I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{1,1}\xi \bigr) ( \delta ,\rho )= \int _{\delta }^{\eta _{2}} \int _{\rho }^{\vartheta _{2}} \frac{\xi (t,s)}{(\eta _{2}-t)^{1-\gamma _{1}}(\vartheta _{2}-s)^{1-\gamma _{2}}}\,ds\,dt. $$

(1.19)

Theorem 2

[20] Assume that $\xi :[\eta _{1},\eta _{2}]\times [ \vartheta _{1},\vartheta _{2} ] \rightarrow \mathbb{R}$ is a co-ordinated convex function and let $\gamma _{1}\neq 0$, $\gamma _{2}\neq 0$, $\gamma _{1},\gamma _{2}\in (0,1]$, $\operatorname{Re} (\alpha )>0$ and $\operatorname{Re} (\beta )>0$. The following inequalities hold for generalized conformable fractional integrals.

$$\begin{aligned}& \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \quad \leq \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \biggl[ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{ \alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+}, \vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{ \alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] \\& \quad \leq \frac{\xi ( \eta _{1},\vartheta _{1} ) +\xi ( \eta _{1},\vartheta _{2} ) +\xi ( \eta _{2},\vartheta _{1} ) +\xi ( \eta _{2},\vartheta _{2} ) }{4}. \end{aligned}$$

(1.20)

2 Midpoint type inequalities for co-ordinated convex functions

Lemma 1

Let $\xi :\triangle :=[\eta _{1},\eta _{2}]\times [ \vartheta _{1}, \vartheta _{2} ] \subset \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a partial differentiable mapping on $( \eta _{1},\eta _{2} ) ]\times ( \vartheta _{1}, \vartheta _{2} )$. If $\frac {\partial ^{2}\xi (t,s)}{\partial t\partial s}\in L_{1}( \Delta )$, then the following identity holds:

$$\begin{aligned}& \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1) \gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\& \qquad {}\times \biggl[ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1} \gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A \\& \quad = \frac{\gamma _{1}^{\alpha }\gamma _{2}^{\beta } ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16} \\& \qquad {}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2}\vartheta _{1}+ \frac{1-s}{2}\vartheta _{2} \biggr)\,ds\,dt \\& \qquad {} - \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1-s}{2}\vartheta _{1}+ \frac{1+s}{2}\vartheta _{2} \biggr)\,ds\,dt \\& \qquad {} - \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1+s}{2}\vartheta _{1}+ \frac{1-s}{2}\vartheta _{2} \biggr)\,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {} \times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1-s}{2}\vartheta _{1}+ \frac{1+s}{2}\vartheta _{2} \biggr)\,ds\,dt \biggr\} , \end{aligned}$$

(2.1)

where

$$\begin{aligned} A =& \frac{2^{\gamma _{2}\beta -1} \gamma _{2}^{\beta } \Gamma (\beta +1)}{ ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\ &{}\times \biggl[ ^{\gamma _{2}}I_{\vartheta _{1}^{+}}^{\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{2}}I_{\vartheta _{2}^{-}}^{\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] \\ & {}+. \frac{2^{\gamma _{1}\alpha -1} \gamma _{1}^{\alpha } \Gamma (\alpha +1)}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha }} \biggl[ ^{ \gamma _{1}}I_{\eta _{1}^{+}}^{\alpha }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1}}I_{ \eta _{2}^{-}}^{\alpha }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] . \end{aligned}$$

(2.2)

Proof

By integration by parts, we get

$$\begin{aligned} I_{1} =& \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \frac {\partial ^{2}\xi }{\partial t\partial s} \\ &{}\times \biggl( \frac{1+t}{2} \eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2} \vartheta _{2} \biggr)\,ds\,dt \\ =& \int _{0}^{1} \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \\ &{}\times \biggl\{ \frac{-2}{(\eta _{2}-\eta _{1})} \frac {\partial \xi }{\partial s} \biggl( \frac{1+t}{2}\eta _{1}+ \frac{1-t}{2}\eta _{2}, \frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \bigg\vert _{0}^{1} \\ & {}+ \int _{0}^{1} \frac{2\alpha }{(\eta _{2}-\eta _{1})} \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1}(1-t)^{ \gamma _{1}-1}\frac {\partial \xi }{\partial s} \\ & {}\times \biggl( \frac{1+t}{2} \eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2} \vartheta _{2} \biggr)\,dt \biggr\} \,ds \\ =& \int _{0}^{1} \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \biggl\{ \biggl( \frac{2}{\eta _{2}-\eta _{1}} \biggr) \frac{1}{\gamma _{1}^{\alpha }}\frac {\partial \xi }{\partial s} \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{1+s}{2}\vartheta _{1}+ \frac{1-s}{2}\vartheta _{2} \biggr) \\ &{}- \frac{2\alpha }{(\eta _{2}-\eta _{1})} \int _{0}^{1} \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1}(1-t)^{ \gamma _{1}-1}\frac {\partial \xi }{\partial s} \\ &{}\times \biggl( \frac{1+t}{2} \eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr)\,dt \biggr\} \,ds \\ =& \frac{2}{ ( \eta _{2}-\eta _{1} ) \gamma _{1}^{\alpha }}\int _{0}^{1} \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \frac {\partial \xi }{\partial s} \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{1+s}{2}\vartheta _{1}+ \frac{1-s}{2} \vartheta _{2} \biggr)\,ds \\ &{}-\frac{2\alpha }{(\eta _{2}-\eta _{1})} \biggl[ \int _{0}^{1} \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1}(1-t)^{ \gamma _{1}-1} \\ &{}\times \biggl\{ \int _{0}^{1} \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\ &{}\times \frac {\partial \xi }{\partial s} \biggl( \frac{1+t}{2}\eta _{1}+ \frac{1-t}{2}\eta _{2}, \frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr)\,ds \biggr\} \,dt \biggr] \\ =&\frac{2}{ ( \eta _{2}-\eta _{1} ) } \biggl( \frac{1}{\gamma _{1}} \biggr) ^{\alpha } \biggl[ \biggl( \frac{1}{\gamma _{2}} \biggr) ^{ \beta } \frac{2}{ ( \vartheta _{2}-\vartheta _{1} ) }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{2\beta }{(\vartheta _{2}-\vartheta _{1})} \int _{0}^{1} \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1}(1-s)^{ \gamma _{2}-1}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr)\,ds \biggr] \\ &{}-\frac{2\alpha }{(\eta _{2}-\eta _{1})} \int _{0}^{1} \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\ &{}\times (1-t)^{ \gamma _{1}-1} \biggl\{ \biggl( \frac{1}{\gamma _{2}} \biggr) ^{\beta } \frac{2}{ ( \vartheta _{2}-\vartheta _{1} ) }\xi \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{2\beta }{(\vartheta _{2}-\vartheta _{1})} \int _{0}^{1} \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1} \\ &{}\times (1-s)^{ \gamma _{2}-1}\xi \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2}, \frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr)\,ds \biggr\} \,dt \\ =&\frac{4}{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{4\beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \biggl( \frac{1}{\gamma _{1}} \biggr) ^{\alpha } \int _{0}^{1} \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1} \\ &{}\times (1-s)^{ \gamma _{2}-1}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2} \vartheta _{2} \biggr)\,ds \\ &{}- \frac{4\alpha }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \biggl( \frac{1}{\gamma _{2}} \biggr) ^{\beta } \int _{0}^{1} \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\ &{}\times (1-t)^{ \gamma _{1}-1}\xi \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr)\,dt \\ & {}+ \frac{4\alpha \beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \biggl[ \int _{0}^{1} \int _{0}^{1} \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha -1} \\ &{}\times (1-t)^{ \gamma _{1}-1} \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta -1}(1-s)^{\gamma _{2}-1} \\ &{}\times \xi \biggl( \frac{1+t}{2}\eta _{1}+ \frac{1-t}{2} \eta _{2},\frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr)\,ds\,dt \biggr] . \end{aligned}$$

(2.3)

In (2.3), using the change of the variables, we can write

$$\begin{aligned} I_{1} =& \frac{4}{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{4\beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }} \biggl( \frac{2}{\vartheta _{2}-\vartheta _{1}} \biggr) ^{\gamma _{2}\beta }\Gamma (\beta ) \bigl( ^{\gamma _{2}}I_{ \vartheta _{1}^{+}}^{\beta } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{4\alpha }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})}\frac{1}{\gamma _{2}^{\beta }} \biggl( \frac{2}{\eta _{2}-\eta _{1}} \biggr) ^{\gamma _{1}\alpha }\Gamma (\alpha )^{\gamma _{1}}I_{\eta _{1}^{+}}^{ \alpha } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ & {}+ \frac{4\alpha \beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{2^{\gamma _{1}\alpha }2^{\gamma _{2}\beta }\Gamma (\alpha )\Gamma (\beta )}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \bigl( ^{ \gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha , \beta } \xi \bigr) \\ &{}\times \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) . \end{aligned}$$

(2.4)

Thus, similarly, by integration by parts it follows that

$$\begin{aligned}& I_{2} = \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \hphantom{I_{2} =} {}\times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2} \eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1-s}{2}\vartheta _{1}+\frac{1+s}{2} \vartheta _{2} \biggr)\,ds\,dt \\& \hphantom{I_{2} } = \frac{-4}{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \hphantom{I_{2} =} {}+ \frac{4\beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }} \biggl( \frac{2}{\vartheta _{2}-\vartheta _{1}} \biggr) ^{\gamma _{2}\beta }\Gamma (\beta ) \bigl( ^{\gamma _{2}}I_{ \vartheta _{2}^{-}}^{\beta } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \hphantom{I_{2} =} {}+ \frac{4\alpha }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})}\frac{1}{\gamma _{2}^{\beta }} \biggl( \frac{2}{\eta _{2}-\eta _{1}} \biggr) ^{\gamma _{1}\alpha }\Gamma (\alpha ) \bigl( ^{\gamma _{1}}I_{ \eta _{1}^{+}}^{\alpha } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \hphantom{I_{2} =} {}- \frac{4\alpha \beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{2^{\gamma _{1}\alpha }2^{\gamma _{2}\beta }\Gamma (\alpha )\Gamma (\beta )}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \bigl( ^{ \gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{2}^{-}}^{\alpha , \beta } \xi \bigr) \\& \hphantom{I_{2} =} {}\times \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) , \end{aligned}$$

(2.5)

$$\begin{aligned}& I_{3} = \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \hphantom{I_{3} =} {}\times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2} \eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2} \vartheta _{2} \biggr)\,ds\,dt \\& \hphantom{I_{3}} = \frac{-4}{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \hphantom{I_{3} =} {}+ \frac{4\beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }} \biggl( \frac{2}{\vartheta _{2}-\vartheta _{1}} \biggr) ^{\gamma _{2}\beta }\Gamma (\beta ) \bigl( ^{\gamma _{2}}I_{ \vartheta _{1}^{+}}^{\beta } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \hphantom{I_{3} =} {}+ \frac{4\alpha }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})}\frac{1}{\gamma _{2}^{\beta }} \biggl( \frac{2}{\eta _{2}-\eta _{1}} \biggr) ^{\gamma _{1}\alpha }\Gamma (\alpha ) \bigl( ^{\gamma _{1}}I_{ \eta _{2}^{-}}^{\alpha } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \hphantom{I_{3} =} {}- \frac{4\alpha \beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{2^{\gamma _{1}\alpha }2^{\gamma _{2}\beta }\Gamma (\alpha )\Gamma (\beta )}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \bigl( ^{ \gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha , \beta } \xi \bigr) \\& \hphantom{I_{3} =} {}\times \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) , \end{aligned}$$

(2.6)

and

$$\begin{aligned} I_{4} =& \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\ &{}\times \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2} \eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1-s}{2}\vartheta _{1}+\frac{1+s}{2} \vartheta _{2} \biggr)\,ds\,dt \\ =&\frac{4}{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{4\beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{1}{\gamma _{1}^{\alpha }} \biggl( \frac{2}{\vartheta _{2}-\vartheta _{1}} \biggr) ^{\gamma _{2}\beta }\Gamma (\beta ) \bigl( ^{\gamma _{2}}I_{ \vartheta _{2}^{-}}^{\beta } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ &{}- \frac{4\alpha }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})}\frac{1}{\gamma _{2}^{\beta }} \biggl( \frac{2}{\eta _{2}-\eta _{1}} \biggr) ^{\gamma _{1}\alpha }\Gamma (\alpha ) \bigl( ^{\gamma _{1}}I_{ \eta _{2}^{-}}^{\alpha } \xi \bigr) \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\ & {}+ \frac{4\alpha \beta }{(\eta _{2}-\eta _{1})(\vartheta _{2}-\vartheta _{1})} \frac{2^{\gamma _{1}\alpha }2^{\gamma _{2}\beta }\Gamma (\alpha )\Gamma (\beta )}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \bigl( ^{ \gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{\alpha , \beta } \xi \bigr) \\ &{}\times \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) . \end{aligned}$$

(2.7)

By the equalities (2.4)–(2.7), we obtain

$$\begin{aligned}& \frac{\gamma _{1}^{\alpha }\gamma _{2}^{\beta } ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16} [ I_{1}-I_{2}-I_{3}+I_{4} ] \\& \quad = \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\& \qquad {}\times \biggl[ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{ \alpha ,\beta }f \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +^{\gamma _{1}\gamma _{2}}I_{ \eta _{1}^{+},\vartheta _{2}^{-}}^{\alpha ,\beta }f \biggl( \frac{\eta _{1} +\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{ \alpha ,\beta }f \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-}, \vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A. \end{aligned}$$

This completes the proof. □

Next, we start to state the first theorem containing the midpoint type inequality for generalized conformable fractional integrals.

Theorem 3

Assume that the assumptions of Lemma 1hold. If $\vert \frac {\partial ^{2}\xi (t,s)}{\partial t\partial s} \vert$ is a co-ordinated convex function on Δ, then the following inequality holds.

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\& \qquad {}\times \biggl[ ^{ \gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+}, \vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1} \gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A \biggr\vert \\& \quad \leq \frac{ ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16 } \biggl[ 1-\frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr] \biggl[ 1- \frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s}(\eta _{1},\vartheta _{1}) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s}(\eta _{1},\vartheta _{2}) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s}(\eta _{2},\vartheta _{1}) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s}(\eta _{2},\vartheta _{2}) \biggr\vert \biggr] , \end{aligned}$$

(2.8)

where A is defined by (2.2) and $B ( \cdot ,\cdot )$ refers to the Beta function.

Proof

From Lemma 1, we acquire

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\& \qquad {}\times \biggl[ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +^{\gamma _{1}\gamma _{2}}I_{ \eta _{1}^{+},\vartheta _{2}^{-}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ {}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-}, \vartheta _{1}^{+}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) {}^{\gamma _{1}\gamma _{2}}I_{ \eta _{2}^{-},\vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A \biggr\vert \\& \quad \leq \frac{\gamma _{1}^{\alpha }\gamma _{2}^{\beta } ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16} \\& \qquad {}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2}, \frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2}, \frac{1-s}{2} \vartheta _{1}+\frac{1+s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2}, \frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2}, \frac{1-s}{2} \vartheta _{1}+\frac{1+s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \biggr\} . \end{aligned}$$

(2.9)

Since $\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \vert$ is co-ordinated convex function on Δ, then one has:

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\& \qquad {}\times \biggl[ ^{ \gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+}, \vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1} \gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A \biggr\vert \\& \quad \leq \frac{\gamma _{1}^{\alpha }\gamma _{2}^{\beta } ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16} \\& \qquad {}\times \biggl\{ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl[ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert \\& \qquad {}+ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr]\,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl[ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert \\& \qquad {}+ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr]\,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl[ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert \\& \qquad {}+ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr]\,ds\,dt \\& \qquad {}+ \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \\& \qquad {}\times \biggl[ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert \\& \qquad {}+ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert + \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr]\,ds\,dt \biggr\} \\& \quad = \frac{\gamma _{1}^{\alpha }\gamma _{2}^{\beta } ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl[ \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr] \biggl[ \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr] \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr] \\& \quad = \frac{ ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16 } \biggl[ 1-\frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr] \biggl[ 1-\frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr] , \end{aligned}$$

which finishes the proof. □

Remark 3

In Theorem 3, if we choose $\gamma _{1}=1$ and $\gamma _{2}=1$, then the following inequality for Riemann–Liouville fractional integrals is achieved

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\alpha -1}2^{\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)}{ ( \eta _{2}-\eta _{1} ) ^{\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\beta }} \\& \qquad {}\times \biggl[ J_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ J_{\eta _{1}^{+},\vartheta _{2}^{-}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +J_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +J_{\eta _{2}^{-}, \vartheta _{2}^{-}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -D \biggr\vert \\& \quad \leq \frac{ ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16 } \biggl( \frac{\alpha }{\alpha +1} \biggr) \biggl( \frac{\beta }{\beta +1} \biggr) \\& \qquad {}\times \biggl[ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert + \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert \biggr] , \end{aligned}$$

(2.10)

where

$$\begin{aligned} D =& \frac{2^{\beta -1}\Gamma (\beta +1)}{ ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \biggl[ J_{\vartheta _{1}^{+}}^{\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +J_{\vartheta _{2}^{-}}^{\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] \\ & {}+. \frac{2^{\alpha -1}\Gamma (\alpha +1)}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha }} \biggl[ J_{\eta _{1}^{+}}^{\alpha }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +J_{\eta _{2}^{-}}^{\alpha }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] . \end{aligned}$$

(2.11)

The inequality (2.10) is the same of [10, Remark 5].

Remark 4

If we choose $\gamma _{1}=\gamma _{2}=\alpha =\beta =1$ in Theorem 3, then Theorem 3 reduces to [23, Theorem 2].

Theorem 4

Assume that the assumptions of Lemma 1hold. If $\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \vert ^{q}$, $q>1$, is a co-ordinated convex function on Δ, then the following inequality holds.

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha }\gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \biggl[ ^{ \gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+}, \vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1} \gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A \biggr\vert \\& \quad \leq \frac{ ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16 } \biggl[ \biggl( 16-\frac{16}{\gamma _{1}}B \biggl( \alpha p+1,\frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 16- \frac{16}{\gamma _{2}}B \biggl( \beta p+1,\frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{{\frac{1}{p}}} \\& \qquad {}\times \biggl[ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q}+ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q}+ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q}+ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr] ^{\frac{1}{q}}, \end{aligned}$$

(2.12)

where A is defined by (2.2), $B ( \cdot ,\cdot )$ refers to the Beta function and $\frac{1}{p}=1-\frac{1}{q}$.

Proof

By using the well-known Hölder’s inequality for double integrals, since $\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \vert ^{q}$ is convex functions on the co-ordinates on △, we get

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \quad \leq \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert ^{p} \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert ^{p}\,ds\,dt \biggr) ^{\frac{1}{p}} \\& \qquad {}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+ \frac{1-t}{2}\eta _{2},\frac{1+s}{2}\vartheta _{1}+\frac{1-s}{2} \vartheta _{2} \biggr) \biggr\vert ^{q}\,ds\,dt \biggr) ^{\frac{1}{q}} \\& \quad \leq \frac{1}{\gamma _{1}^{\alpha }}\frac{1}{\gamma _{2}^{\beta }} \biggl( \int _{0}^{1} \int _{0}^{1} \bigl( 1- \bigl( 1-(1-t)^{ \gamma _{1}} \bigr) ^{\alpha p} \bigr) \bigl( 1- \bigl( 1-(1-s)^{ \gamma _{2}} \bigr) ^{\beta p} \bigr)\,ds\,dt \biggr) ^{{\frac{1}{p}}} \\& \qquad {}\times \biggl\{ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q}+ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q}+ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q}\,ds\,dt \biggr\} ^{{\frac{1}{q}}} \\& \quad \leq \frac{1}{\gamma _{1}^{\alpha }}\frac{1}{\gamma _{2}^{\beta }} \biggl[ \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha p+1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1-\frac{1}{\gamma _{2}}B \biggl( \beta p+1, \frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{{\frac{1}{p}}} \\& \qquad {}\times \biggl( \frac{9}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q}+ \frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q}+\frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+\frac{1}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}}. \end{aligned}$$

(2.13)

Here, we take advantage of the fact that

$$ (\varpi -\sigma )^{j}\leq \varpi ^{j}-\sigma ^{j}, $$

for any $\varpi >\sigma \geq 0$ and $j\geq 1$.

Similarly, we have

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \end{aligned}$$

(2.14)

$$\begin{aligned}& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1-s}{2} \vartheta _{1}+\frac{1+s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \} \\& \quad \leq \frac{1}{\gamma _{1}^{\alpha }}\frac{1}{\gamma _{2}^{\beta }} \biggl[ \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha p+1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1-\frac{1}{\gamma _{2}}B \biggl( \beta p+1, \frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{{\frac{1}{p}}} \\& \qquad {}\times \biggl( \frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q}+ \frac{9}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q}+\frac{1}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+\frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}}, \\& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \} \\& \quad \leq \frac{1}{\gamma _{1}^{\alpha }}\frac{1}{\gamma _{2}^{\beta }} \biggl[ \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha p+1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1-\frac{1}{\gamma _{2}}B \biggl( \beta p+1, \frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{{\frac{1}{p}}} \\& \qquad {}\times \biggl( \frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q}+ \frac{1}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q}+\frac{9}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+\frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}}, \end{aligned}$$

(2.15)

and

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1-s}{2} \vartheta _{1}+\frac{1+s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \quad \leq \frac{1}{\gamma _{1}^{\alpha }}\frac{1}{\gamma _{2}^{\beta }} \biggl[ \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha p+1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1-\frac{1}{\gamma _{2}}B \biggl( \beta p+1, \frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{{\frac{1}{p}}} \\& \qquad {}\times \biggl( \frac{1}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q}+ \frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q}+\frac{3}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+\frac{9}{16} \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}}. \end{aligned}$$

(2.16)

If we substitute the inequalities (2.13)–(2.16) in (2.9), we obtain the desired inequality (2.12). □

Remark 5

If we take $\gamma _{1}=1$ and $\gamma _{2}=1$ in Theorem 4, then the following inequality for Riemann–Liouville fractional integrals is achieved

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\alpha -1}2^{\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)}{ ( \eta _{2}-\eta _{1} ) ^{\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\beta }} \\& \qquad {}\times \biggl[ J_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ J_{\eta _{1}^{+},\vartheta _{2}^{-}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +J_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+J_{\eta _{2}^{-}, \vartheta _{2}^{-}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -D \biggr\vert \\& \quad \leq \frac{ ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{16} \biggl[ \biggl( \frac{16\alpha p}{\alpha p+1} \biggr) \biggl( \frac{16\beta p}{\beta p+1} \biggr) \biggr] ^{{ \frac{1}{p}}} \\& \qquad {}\times \biggl[ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q}+ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q}+ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q}+ \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr] ^{\frac{1}{q}}. \end{aligned}$$

(2.17)

Theorem 5

Assume that the assumptions of Lemma 1hold. If $\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \vert ^{q}$, $q\geq 1$, is a co-ordinated convex function on Δ, then we have the following inequality:

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\gamma _{1}\alpha -1}2^{\gamma _{2}\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)\gamma _{1}^{\alpha } \gamma _{2}^{\beta }}{ ( \eta _{2}-\eta _{1} ) ^{\gamma _{1}\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\gamma _{2}\beta }} \\& \qquad {}\times \biggl[ ^{ \gamma _{1}\gamma _{2}}I_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ ^{\gamma _{1}\gamma _{2}}I_{\eta _{1}^{+}, \vartheta _{2}^{-}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +{}^{\gamma _{1} \gamma _{2}}I_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +{}^{\gamma _{1}\gamma _{2}}I_{\eta _{2}^{-},\vartheta _{2}^{-}}^{ \alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -A \biggr\vert \\& \quad \leq \frac{ ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}-\vartheta _{1} ) }{\gamma _{1}\gamma _{2}} \biggl( \frac{1}{4} \biggr) ^{2+\frac{1}{q}} \biggl[ \biggl( 1-\frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1- \frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{1- \frac{1}{q}} \\& \qquad {}\times \bigg\{ \biggl( \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}} \\& \qquad {}+ \biggl( \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}} \\& \qquad {}+ \biggl( \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1,\frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{{\frac{1}{q}}}}. \end{aligned}$$

(2.18)

Here, A is defined as in (2.2).

Proof

By using power-mean inequality, we get

$$\begin{aligned} I_{9} =& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \\ &{}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\ \leq & \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \,ds\,dt \biggr) ^{1-\frac{1}{q}} \\ &{}\times \biggl( \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \\ & {} \times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2}, \frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert ^{q}\,ds\,dt \biggr) ^{\frac{1}{q}}. \end{aligned}$$

Taking into account co-ordinated convexity of $\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \vert ^{q}$, we acquire

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl\vert \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr\vert \biggl\vert \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr\vert \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \quad \leq \biggl( \int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr) \biggl( \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr)\,ds\,dt \biggr) ^{1-\frac{1}{q}} \\& \qquad {}\times \bigg( \int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr) \biggl( \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr) \\& \qquad {}\times \biggl\{ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q}+ \biggl( \frac{1+t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1+s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q}+ \biggl( \frac{1-t}{2} \biggr) \biggl( \frac{1-s}{2} \biggr) \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q}\,ds\,dt \biggr\} ^{{\frac{1}{q}}} \\& \quad = \biggl[ \frac{1}{\gamma _{1}^{\alpha }} \frac{1}{\gamma _{2}^{\beta }} \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1- \frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{1-\frac{1}{q}} \biggl\{ \frac{1}{4} \frac{1}{\gamma _{1}^{\alpha +1}} \frac{1}{\gamma _{2}^{\beta +1}} \\& \qquad {}\times \biggl( \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {} \times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) \biggr\} ^{\frac{1}{q}}. \end{aligned}$$

(2.19)

Similarly, we have

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr) \biggl( \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr) \end{aligned}$$

(2.20)

$$\begin{aligned}& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1+t}{2}\eta _{1}+\frac{1-t}{2}\eta _{2},\frac{1-s}{2} \vartheta _{1}+\frac{1+s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \quad \leq \biggl[ \frac{1}{\gamma _{1}^{\alpha }} \frac{1}{\gamma _{2}^{\beta }} \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1- \frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{1-\frac{1}{q}} \biggl\{ \frac{1}{4} \frac{1}{\gamma _{1}^{\alpha +1}} \frac{1}{\gamma _{2}^{\beta +1}} \\& \qquad {}\times \biggl( \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) \biggr\} ^{ \frac{1}{q}}, \\& \int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr) \biggl( \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr) \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} \biggl( \frac{1-t}{2}\eta _{1}+\frac{1+t}{2}\eta _{2},\frac{1+s}{2} \vartheta _{1}+\frac{1-s}{2}\vartheta _{2} \biggr) \biggr\vert \,ds\,dt \\& \quad \leq \biggl[ \frac{1}{\gamma _{1}^{\alpha }} \frac{1}{\gamma _{2}^{\beta }} \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1- \frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{1-\frac{1}{q}} \biggl\{ \frac{1}{4} \frac{1}{\gamma _{1}^{\alpha +1}} \frac{1}{\gamma _{2}^{\beta +1}} \\& \qquad {}\times \biggl( \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{2} ) \biggr\vert ^{q} \biggr) \biggr\} ^{ \frac{1}{q}} \end{aligned}$$

(2.21)

and

$$\begin{aligned}& \int _{0}^{1} \int _{0}^{1} \biggl( \frac{1}{\gamma _{1}^{\alpha }}- \biggl( \frac{1-(1-t)^{\gamma _{1}}}{\gamma _{1}} \biggr) ^{\alpha } \biggr) \biggl( \frac{1}{\gamma _{2}^{\beta }}- \biggl( \frac{1-(1-s)^{\gamma _{2}}}{\gamma _{2}} \biggr) ^{\beta } \biggr) \\& \qquad {}\times \biggl\vert \frac {\partial ^{2}f}{\partial t\partial s} \biggl( \frac{1-t}{2}a+ \frac{1+t}{2}b,\frac{1-s}{2}c+\frac{1+s}{2}\,d\biggr) \biggr\vert \,ds\,dt \\& \quad \leq \biggl[ \frac{1}{\gamma _{1}^{\alpha }} \frac{1}{\gamma _{2}^{\beta }} \biggl( 1- \frac{1}{\gamma _{1}}B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) \biggr) \biggl( 1- \frac{1}{\gamma _{2}}B \biggl( \beta +1,\frac{1}{\gamma _{2}} \biggr) \biggr) \biggr] ^{1-\frac{1}{q}} \biggl\{ \frac{1}{4} \frac{1}{\gamma _{1}^{\alpha +1}} \frac{1}{\gamma _{2}^{\beta +1}} \\& \qquad {}\times \biggl( \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{\gamma _{1}}{2}-B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1}, \vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1, \frac{2}{\gamma _{1}} \biggr) \biggr] \biggl[ \frac{\gamma _{2}}{2}-B \biggl( \beta +1,\frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2}, \vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3\gamma _{1}}{2}-2B \biggl( \alpha +1, \frac{1}{\gamma _{1}} \biggr) +B \biggl( \alpha +1,\frac{2}{\gamma _{1}} \biggr) \biggr] \\& \qquad {}\times \biggl[ \frac{3\gamma _{2}}{2}-2B \biggl( \beta +1, \frac{1}{\gamma _{2}} \biggr) +B \biggl( \beta +1, \frac{2}{\gamma _{2}} \biggr) \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) \biggr\} ^{\frac{1}{q}}. \end{aligned}$$

(2.22)

By considering (2.19)–(2.22) in (2.9), we obtain the required inequality (2.18). □

Remark 6

If we take $\gamma _{1}=1$ and $\gamma _{2}=1$ in Theorem 5, then the following inequality for Riemann–Liouville fractional integrals is achieved

$$\begin{aligned}& \biggl\vert \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) + \frac{2^{\alpha -1}2^{\beta -1}\Gamma (\alpha +1)\Gamma (\beta +1)}{ ( \eta _{2}-\eta _{1} ) ^{\alpha } ( \vartheta _{2}-\vartheta _{1} ) ^{\beta }} \\& \qquad {}\times \biggl[ J_{\eta _{1}^{+},\vartheta _{1}^{+}}^{\alpha ,\beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {}+ J_{\eta _{1}^{+},\vartheta _{2}^{-}}^{\alpha , \beta }\xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) +J_{\eta _{2}^{-},\vartheta _{1}^{+}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2}, \frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \\& \qquad {} +J_{\eta _{2}^{-}, \vartheta _{2}^{-}}^{\alpha ,\beta } \xi \biggl( \frac{\eta _{1}+\eta _{2}}{2},\frac{\vartheta _{1}+\vartheta _{2}}{2} \biggr) \biggr] -D \biggr\vert \\& \quad \leq ( \eta _{2}-\eta _{1} ) ( \vartheta _{2}- \vartheta _{1} ) \biggl( \frac{1}{4} \biggr) ^{2+\frac{1}{q}} \biggl[ \biggl( \frac{\alpha }{\alpha +1} \biggr) \biggl( \frac{\beta }{\beta +1} \biggr) \biggr] ^{1-\frac{1}{q}} \\& \qquad {}\times \bigg\{ \biggl( \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{\frac{1}{q}} \\& \qquad {}+ \biggl( \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}f}{\partial t\partial s}(b,d) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}} \\& \qquad {}+ \biggl( \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{\frac{1}{q}}} \\& \qquad {}+ \biggl( \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{1}{2}- \frac{1}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{1},\vartheta _{2} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{1}{2}- \frac{1}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{1} ) \biggr\vert ^{q} \\& \qquad {}+ \biggl[ \frac{3}{2}- \frac{2\alpha +3}{ ( \alpha +1 ) ( \alpha +2 ) } \biggr] \biggl[ \frac{3}{2}- \frac{2\beta +3}{ ( \beta +1 ) ( \beta +2 ) } \biggr] \biggl\vert \frac {\partial ^{2}\xi }{\partial t\partial s} ( \eta _{2},\vartheta _{2} ) \biggr\vert ^{q} \biggr) ^{{{\frac{1}{q}}}}. \end{aligned}$$

(2.23)

3 Conclusion

In this research, we acquired some inequality of midpoint type for co-ordinated convex functions by means of conformable fractional integrals. In the future studies, researchers can obtain some new inequalities with the aid of the different kinds of co-ordinated convex mappings or other types of fractional integral operators.