1. Introduction and Preliminaries
For a convex mapping \(\prod:I\rightarrow \mathbb{R}\) on a real interval, for all \(f_1,f_2\in I\) and \(t\in[0,1]\), the inequality
\begin{equation}\label{p1}
\prod\left( \frac{f_1+f_2}{2}\right) \leq
\frac{1}{f_2-f_1}\int^{f_2}_{f_1}\prod(u)du\leq\frac{\prod(f_1)+\prod(f_2)}{2},
\end{equation}
(1)
is known as the Hermite-Hadamard inequality [
1]. The inequality (1) has been established for several generalized convex functions [
2,
3,
4,
5,
6,
7,
8,
9]. Dragomir [
10] and Sarikaya [
11] calculated Hermite-Hadamard inequality for co-ordinated convex functions. They define co-ordinated convex function as:
Definition 1. [10]
A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2}
\rightarrow \mathbb{R}\) is called co-ordinate convex on \(\Delta\) with \(f_1 < f_2\) and \(g_1 < g_2\), if the partial functions
\begin{equation}
\prod_{y}:[f_1,f_2] \rightarrow \mathbb{R}, \prod_{y}(u)=\prod(u,y), and
\prod_{x}:[g_1,g_2] \rightarrow \mathbb{R}, \prod_{x}(v)=\prod(x,v),
\end{equation}
are convex for all \(x\in[f_1,f_2]\) and \(y\in [g_1,g_2]\).
Sarikaya [11] define the co-ordinated convex function as:
Definition 2. [11]
A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2}\rightarrow \mathbb{R}\) is called coordinate convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\), if
\begin{align*}
\begin{split}
&\prod(t_1x+(1-t_1)z,t_2 y+(1-t_2)w)
\\
&\leq t_1t_2\prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2
\prod(z,y)+(1-t_1)(1-t_2)\prod(z,w),
\end{split}
\end{align*}
holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).
Every convex function is co-ordinated convex but not conversely [10].
Theorem 3. [10] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow
\mathbb{R}\) be convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\). Then
\begin{align}
\prod\left(\frac{f_1+f_2}{2},\frac{g_1+g_2}{2}\right)
&\leq\frac{1}{2}\Bigg[\frac{1}{f_2-f_1}\int_{f_1}^{f_2}\prod\left( x,\frac{g_1+g_2}{2}\right) dx
+\frac{1}{g_2-g_1}\int_{g_1}^{g_2}\prod\left( \frac{f_1+f_2}{2},y\right) dy\Bigg]\notag
\\
&\leq\frac{1}{(f_2-f_1)(g_2-g_1)}\int_{g_1}^{g_2}\int_{f_1}^{f_2}\prod(x,y)dxdy\notag\end{align}\begin{align}
&\leq\frac{1}{4}\Bigg[\frac{1}{f_2-f_1} \int_{f_1}^{f_2}\prod(x,g_1)dx+\frac{1}{f_2-f_1}\int_{f_1}^{f_2}\prod(x,d)dx
\notag\\
&\hspace{0.5cm}+\frac{1}{g_2-g_1}\int_{g_1}^{g_2} \prod(f_1,y)dy+\frac{1}{g_2-g_1}\int_{g_1}^{g_2}\prod(f_2,y)dy\Bigg]
\notag\\
&\leq\frac{\prod(f_1,g_2)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}.
\end{align}
(2)
Definition 4. [12]
A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2}
\rightarrow \mathbb{R}\) is called harmonically convex on \(\Delta\) with
\(f_1< f_2\) and \(g_1< g_2\), if
\begin{equation*}
\prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right)
\leq t_1t_2
\prod(x,y)+(1-t_1)(1-t_2)\prod(z,w),
\end{equation*}
holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).
Definition 5. [12]
A function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty)
\rightarrow \mathbb{R}\) is called coordinated harmonically convex on \(\Delta\) with
\(f_1< f_2\) and \(g_1< g_2\), if
\begin{align*}
\begin{split}
&\prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right)
\\
&\leq t_1t_2
\prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2
\prod(z,y)+(1-t_1)(1-t_2)\prod(z,w),
\end{split}
\end{align*}
holds for all \(t_1,t_2\in [0,1]\) and \((x,y),(z,w)\in\Delta\).
Note that, a function \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty)
\rightarrow \mathbb{R}\) is called coordinated harmonically convex on \(\Delta\) with
\(f_1< f_2\) and \(g_1< g_2\), if the partial functions
\begin{equation}
\prod_{y}:[f_1,f_2] \rightarrow \mathbb{R}, \prod_{y}(u)=\prod(u,y),
\prod_{x}:[g_1,g_2] \rightarrow \mathbb{R}, \prod_{x}(v)=\prod(x,v),
\end{equation}
are harmonically convex for all \(x\in[f_1,f_2]\) and \(y\in [g_1,g_2]\), (for more detail, see [
9,
12]).
Theorem 6. [12]
Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow
\mathbb{R}\) be co-ordinated harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\). Then
\begin{align}
\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
&\leq \frac{(f_1f_2)(g_1g_2)}{(f_2-f_1)(g_2-g_1)}\int_{f_1}^{f_2}\int_{g_1}^{g_2}\frac{\prod(x,y)}{x^{2}y^{2}}dydx
\notag\\
&\leq\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}.
\end{align}
(3)
Definition 7. [13]
Let \(\prod \in L[f_1,f_2]\). The right-hand side and left-hand side Riemann-
Liouville fractional integrals \(J^{\alpha}_{f_1+}\prod\) and
\(J^{\alpha}_{f_2-}\prod\) of order \(\alpha
> 0\) with \(f_2 > f_1\geq 0\) are defined by
\begin{equation*}
J^{\alpha}_{f_1+}\prod(x)=\frac{1}{\Gamma(\alpha)}\int_{f_1}^{x}(x-t)^{\alpha-1}\prod(t)dt,\
x>f_1,
\end{equation*}
and
\begin{equation*}
J^{\alpha}_{f_2-}\prod(x)=\frac{1}{\Gamma(\alpha)}\int_{x}^{f_2}(t-x)^{\alpha-1}\prod(t)dt,\
x< f_2,
\end{equation*}
respectively, where \(\Gamma(\alpha)\) is the Gamma function defined
by \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt\).
Theorem 8. [14]
Let \(\prod:I\subseteq (0,\infty)\rightarrow \mathbb{R}\) be a function such that \(\prod\in L_1(f_1,f_2)\) where \(f_1,f_2\in I\) with \(f_1< f_2\). If \(\prod\) is harmonocally convex function on \([f_1,f_2]\), then following inequality for fractional integral hold:
\begin{align}\label{e1}
\begin{split}
&\prod\left(\frac{2f_1f_2}{f_1+f_2}\right)\leq\frac{\Gamma(\alpha+1)}{2}
\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}
\left[J^{\alpha}_{1/f_1-}\left(\prod\circ \Omega\right) \left( \frac{1}{f_2}\right) +J^{\alpha}_{1/f_2+}\left(\prod\circ \Omega\right)\left( \frac{1}{f_1}\right) \right]
\leq \frac{\prod(f_1)+\prod(f_2)}{2},
\end{split}
\end{align}
(4)
where \(\alpha>0\) and \(\Omega(x)=\frac{1}{x}\).
Definition 9. [11]
Let \(\prod\in L_{1}([f_1,f_2]\times [g_1,g_2])\). The Riemann-Liouville integrals \(J^{\alpha,\beta}_{f_1+,g_1+}\), \(J^{\alpha,\beta}_{f_1+,g_2-}\), \(J^{\alpha,\beta}_{f_2-,g_1+}\) and \(J^{\alpha,\beta}_{f_2-,g_2-}\) of order \(\alpha,\beta>0\) with \(f_1,g_1\geq 0\) are defined by
\begin{equation*}
J^{\alpha,\beta}_{f_1+,g_1+}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{f_1}^{x}\int_{g_1}^{y}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\
x>f_1 \ y>g_1,
\end{equation*}
\begin{equation*}
J^{\alpha,\beta}_{f_1+,g_2-}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{f_1}^{x}\int_{y}^{g_2}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\
x>f_1 \ y< g_2,
\end{equation*}
\begin{equation*}
J^{\alpha,\beta}_{f_2-,g_1+}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{f_2}\int_{g_1}^{y}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\
xg_1,
\end{equation*}
and
\begin{equation*}
J^{\alpha,\beta}_{f_2-,g_2-}\prod(x,y)=\frac{1}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{f_2}\int_{y}^{g_2}(x-t)^{\alpha-1}(y-s)^{\beta-1}\prod(t,s)dsdt,\
x< f_2 \ y< g_2,
\end{equation*}
respectively. Here \(\Gamma\) is the Gamma function.
Theorem 10. [11] Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq \mathbb{R}^{2} \rightarrow
\mathbb{R}\) be convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\prod\in L_{1}(\Delta)\). Then
\begin{align}
&\prod\left(\frac{f_1+f_2}{2},\frac{g_1+g_2}{2}\right)
\leq \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4(f_2-f_1)^{\alpha}(g_2-g_1)^{\beta}}
\notag\\
&\hspace{0.5cm}\times \left[J^{\alpha,\beta}_{f_1+,g_1+}\prod(f_2,g_2)+J^{\alpha,\beta}_{f_1+,g_2-}\prod(f_2,g_1)+J^{\alpha,\beta}_{f_2-,g_1+}\prod(f_1,g_2)+J^{\alpha,\beta}_{f_2-,g_2-}\prod(f_1,g_1)\right]
\notag\\
&\leq\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}.
\end{align}
(5)
In this paper, we gave integral results for co-ordinated harmonically convex functions via fractional integrals.
2. Main Results
In this section, our aim is to prove some Hermite-Hadamard type ineqalities for co-ordinated harmonically convex functions in fractional integrals.
Theorem 11.
Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow
\mathbb{R}\) be harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\prod\in L_{1}(\Delta)\). Then
\begin{align}\label{t1e1}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
\leq \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega )\left( \frac{1}{f_2},\frac{1}{g_2}\right) +J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left( \frac{1}{f_2},\frac{1}{g_1}\right)
\notag\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}\left(\prod\circ \Omega\right)\left( \frac{1}{f_1},\frac{1}{g_2}\right) +J^{\alpha,\beta}_{1/f_2+,1/g_2+}\left(\prod\circ \Omega\right)\left( \frac{1}{f_1},\frac{1}{g_1}\right) \Bigg]
\notag\\
&\leq\frac{\prod(f_1,g_2)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4},
\end{align}
(6)
where \(\Omega(x,y)=\left(\frac{1}{x},\frac{1}{y} \right) \) for all \((x,y)\in ([\frac{1}{f_2},\frac{1}{f_1}],[\frac{1}{g_2},\frac{1}{g_1}])\).
Proof.
Let \((x,y),(z,w)\in \Delta\) and \(t_1,t_2 \in [0,1]\). Since \(\prod\) is co-ordinated harmonically convex on \(\Delta\), we have
\begin{align}\label{t1e2}
&\prod\left(\frac{xz}{t_1x+(1-t_1)z},\frac{yw}{t_2 y+(1-t_2)w}\right)
\notag\\
&\leq t_1t_2
\prod(x,y)+t_1(1-t_2)\prod(x,w)+(1-t_1)t_2
\prod(z,y)+(1-t_1)(1-t_2)\prod(z,w).
\end{align}
(7)
By taking \(x=\frac{f_1f_2}{t_1f_1+(1-t_1)f_2}\), \(z=\frac{f_1f_2}{t_1f_2+(1-t_1)f_1}\), \(y=\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\), \(w=\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\) and \(t_1=t_2=\frac{1}{2}\) in (7), we get
\begin{align}\label{t1e3}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
\notag\\
&\leq \frac{1}{4}\Bigg[\prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) +\prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right)
\notag\\
&+
\prod\left( \frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right) +\prod\left( \frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) \Bigg].
\end{align}
(8)
Multiplying both sides of (8) by \(t_1^{\alpha-1}t_2^{\beta-1}\) and then integrating with respect to \((t_1,t_2)\) over \([0,1]\times [0,1]\), we get
\begin{align}
\frac{1}{\alpha\beta}\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
&\leq \frac{1}{4}\Bigg[\int_{0}^{1}\int_{0}^{1}\bigg\lbrace \prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right)
\notag\\
&\hspace{0.5cm}+\prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right) \bigg\rbrace t_1^{\alpha-1}t_2^{\beta-1} dt_1dt_2
\notag\\
&\hspace{0.5cm}+
\int_{0}^{1}\int_{0}^{1}\bigg\lbrace \prod\left( \frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right)
\notag\\
&\hspace{0.5cm}+\prod\left( \frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right)\bigg\rbrace t_1^{\alpha-1}t_2^{\beta-1}dt_1dt_2 \Bigg].
\end{align}
(9)
Applying change of variable, we find
\begin{align}\label{t1e4}
\begin{split}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
\leq \frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\\
&\times \Bigg[\int_{1/g_2}^{1/g_1}\int_{1/f_2}^{1/f_1}\bigg\lbrace \left(\frac{1}{f_1}-x \right)^{\alpha-1}\left(\frac{1}{g_1}-y \right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right)
+\left(\frac{1}{f_1}-x \right)^{\alpha-1}\left(y-\frac{1}{g_2} \right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right) \bigg\rbrace dxdy
\\
&+
\int_{1/g_2}^{1/g_1}\int_{1/f_2}^{1/f_1}\bigg\lbrace \left(x-\frac{1}{f_2} \right)^{\alpha-1}\left(\frac{1}{g_1}-y \right)^{\beta-1}\prod\left( \frac{1}{x},\frac{1}{y}\right)
+\left(x-\frac{1}{f_2} \right)^{\alpha-1}\left(y-\frac{1}{g_2} \right)^{\beta-1}\prod\left( \frac{1}{x},\frac{1}{y}\right)\bigg\rbrace dxdy \Bigg].
\end{split}
\end{align}
(10)
Then by multiplying and dividing by \(\Gamma(\alpha)\Gamma(\beta)\) on right hand side of inequality (10), we get the first inequality of (6). For the second inequality of (6) we use the co-ordinated harmonically convexity of \(\prod\) as:
\begin{align*}
\begin{split}
&\prod\left( \frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right)
\\
&\leq t_1t_2
\prod(f_1,g_1)+t_1(1-t_2)\prod(f_1,g_2)+(1-t_1)t_2
\prod(f_2,g_1)+(1-t_1)(1-t_2)\prod(f_2,g_2),
\end{split}
\end{align*}
\begin{align*}
\begin{split}
&\prod\left(\frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right)
\\
&\leq t_1t_2
\prod(f_1,g_2)+t_1(1-t_2)\prod(f_1,g_1)+(1-t_1)t_2
\prod(f_2,g_2)+(1-t_1)(1-t_2)\prod(f_2,g_1),
\end{split}
\end{align*}
\begin{align*}
\begin{split}
&\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right)
\\
&\leq t_1t_2
\prod(f_2,g_1)+t_1(1-t_2)\prod(f_2,g_2)+(1-t_1)t_2
\prod(f_1,g_1)+(1-t_1)(1-t_2)\prod(f_1,g_2),
\end{split}
\end{align*}
and
\begin{align*}
\begin{split}
&\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right)
\\
&\leq t_1t_2
\prod(f_2,g_2)+t_1(1-t_2)\prod(f_2,g_1)+(1-t_1)t_2
\prod(f_1,g_2)+(1-t_1)(1-t_2)\prod(f_1,g_1).
\end{split}
\end{align*}
Then by adding above inequalities, we get
\begin{align}\label{t1e5}
&\prod\left(\frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right)
+\prod\left(\frac{f_1f_2}{t_1f_1+(1-t_1)f_2},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right)
\notag\\
&+\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_1+(1-t_2)g_2}\right) +\prod\left(\frac{f_1f_2}{t_1f_2+(1-t_1)f_1},\frac{g_1g_2}{t_2 g_2+(1-t_2)g_1}\right)
\notag\\
&\leq \prod(f_1,g_1)+\prod(f_2,g_1)+\prod(f_1,g_2)+\prod(f_2,g_2).
\end{align}
(11)
Thus by multiplying (11) by \(t_1^{\alpha-1}t_2^{\beta-1}\) and then integrating with respect to \((t_1,t_2)\) over \([0,1]\times [0,1]\), we get the second inequality of (6).
Hence the proof is completed.
Remark 1.
In Theorem 11, if one takes \(\alpha=\beta=1\) and using change of variable \(u=1/x\) and \(v=1/y\), then one has Theorem in [12].
Theorem 12.
Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2] \subseteq (0,\infty)\times (0,\infty)\rightarrow
\mathbb{R}\) be harmonically convex on \(\Delta\) with \(f_1< f_2\) and \(g_1< g_2\) and \(\varPsi\in L_{1}(\Delta)\). Then
\begin{align}\label{t2e1}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)\leq \frac{\Gamma(\alpha+1)}{4}\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}
\notag\\
&\hspace{0.5cm}\times\left[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},\frac{2g_1g_2}{g_1+g_2}\right) +J^{\alpha}_{1/c_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},\frac{2g_1g_2}{g_1+g_2}\right) \right] +\frac{\Gamma(\beta+1)}{4}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
& \hspace{0.5cm}\times \left[J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{g_1}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{g_2}\right) \right]
\notag\\
&\leq \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{2}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\times \hspace{0.5cm}\Bigg[J^{\alpha,\beta}_{f_1+,g_1+}(\prod\circ \Omega)\left( \frac{1}{f_2},\frac{1}{g_1}\right)
\notag\\
&\hspace{0.5cm} +J^{\alpha,\beta}_{f_1+,g_2-}(\prod\circ \Omega)\left( \frac{1}{f_2},\frac{1}{g_1}\right)+J^{\alpha,\beta}_{f_2-,g_1+}(\prod\circ \Omega)\left( \frac{1}{f_1},\frac{1}{g_2}\right) +J^{\alpha,\beta}_{f_2-,g_2-}(\prod\circ \Omega)\left( \frac{1}{f_1},\frac{1}{g_1}\right) \Bigg]
\notag\\
&\leq \frac{\Gamma(\alpha+1)}{4}\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\Bigg[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},g_2\right)
+J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},g_1\right)
\notag\\
&\hspace{0.5cm}+J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},g_1\right) +J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},g_1\right)\Bigg]
\notag\\
&\hspace{0.5cm}+\frac{\Gamma(\beta+1)}{4}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\alpha}\Big[J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( f_1,\frac{1}{g_2}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( f_2,\frac{1}{g_2}\right)
\notag\\
&\hspace{0.5cm}+J^{\alpha}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_1}\right) +J^{\alpha}_{1/g_2+}(\prod\circ \Omega_{2})\left( f_2,\frac{1}{g_1}\right)\Big]\notag\\
&\leq\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4},
\end{align}
(12)
where \(\Omega(x,y)=\left(\frac{1}{x},\frac{1}{y} \right) \), \(\Omega_{1}(x,y)=\left(\frac{1}{x},y \right) \) and \(\Omega_{2}(x,y)=\left(x,\frac{1}{y} \right) \) for all \((x,y)\in \left( [\frac{1}{f_2},\frac{1}{f_1}],[\frac{1}{g_2},\frac{1}{g_1}]\right) \).
Proof.
Since \(\prod\) is co-ordinated harmonically convex on \(\Delta\) then we have \(\prod_{\frac{1}{x}}:[f_1,f_2]\rightarrow \mathbb{R}\), \(\prod_{\frac{1}{x}}(y)=\prod(\frac{1}{x},y)\), is harmonically convex on \([g_1,g_2]\) for all \(x\in \left[ \frac{1}{f_2},\frac{1}{f_1}\right] \). Then from inequality (4), we have
\begin{align}\label{t2e2}
&\prod_{\frac{1}{x}}\left(\frac{2g_1g_2}{g_1+g_2}\right)
\leq\frac{\Gamma(\beta+1)}{2}
\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\left[J^{\beta}_{1/c-}(\prod_{\frac{1}{x}}\circ \Omega_{2}) \left( \frac{1}{g_2}\right) +J^{\beta}_{1/g_2+}(\prod_{\frac{1}{x}}\circ \Omega_{2})\left( \frac{1}{g_1}\right) \right]
\notag\\
&\leq \frac{\prod_{\frac{1}{x}}(g_1)+\prod_{\frac{1}{x}}(g_2)}{2}.
\end{align}
(13)
In other words,
\begin{align}\label{t2e3}
&\prod\left(\frac{1}{x},\frac{2g_1g_2}{g_1+g_2}\right)\leq\frac{\beta}{2}
\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}\left[\int_{1/g_2}^{1/g_1}\left(y-\frac{1}{g_2}\right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right) {\text d}y\right.\notag\\
&\hspace{0.5cm}\left.+\int_{1/g_2}^{1/g_1}\left(\frac{1}{g_1}-y \right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right){\text d}y\right]
\leq \frac{\prod\left( \frac{1}{x},g_1\right) +\prod\left( \frac{1}{x},g_2\right) }{2},
\end{align}
(14)
for all \(x\in\left[ \frac{1}{f_2},\frac{1}{f_1}\right] \). Now by multiplying (14) by \(\frac{\alpha(x-1/f_2)^{\alpha-1}}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\) and \(\frac{\alpha(1/f_1-x)^{\alpha-1}}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\), and then integrating with respect to \(x\) over \([1/f_2,1/f_1]\), respectively, we find
\begin{align}
\label{t2e4}
&\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\alpha-1}\prod\left(\frac{1}{x},\frac{2g_1g_2}{g_1+g_2}\right){\text d}x
\leq\frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}
\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\Bigg[\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( x-\frac{1}{f_2}\right) ^{\alpha-1}\left(y-\frac{1}{g_2}\right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x
\notag\\
&\hspace{0.5cm}+\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( x-\frac{1}{f_2}\right) ^{\alpha-1}\left(\frac{1}{g_1}-y \right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x\Bigg]
\notag\\
&\leq \frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\Bigg[\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x
+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x\Bigg],
\end{align}
(15)
and
\begin{align}
\label{t2e5}
&\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left(\frac{1}{x},\frac{2g_1g_2}{g_1+g_2}\right){\text d}x
\leq\frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}
\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\notag\\
&\hspace{0.5cm}\times
\Bigg[\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\left(y-\frac{1}{g_2}\right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x
\notag\\
&\hspace{0.5cm}+\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\left(\frac{1}{g_1}-y \right)^{\beta-1} \prod\left( \frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x\Bigg]
\notag\\
&\leq \frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\Bigg[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x
+\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x\Bigg].
\end{align}
(16)
Again by similar arguments for \(\prod_{\frac{1}{y}}:[f_1,f_2]\rightarrow \mathbb{R}\), \(\prod_{\frac{1}{y}}(x)=\prod(x,\frac{1}{y})\), we get
\begin{align*}
&\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{1}{y}\right){\text d}y\\
&\leq\frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}
\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\Bigg[\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( u-\frac{1}{f_2}\right) ^{\alpha-1}\left(y-\frac{1}{g_2}\right)^{\beta-1} \prod\left(\frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x
\notag\\
&\hspace{0.5cm}+\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\left(y-\frac{1}{g_2} \right)^{\beta-1} \prod\left(\frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x\Bigg]\end{align*}
\begin{align}
\label{t2e6}
&\leq \frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\Bigg[\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\alpha-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y\Bigg] ,
\end{align}
(17)
and
\begin{align}
\label{t2e7}
&\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{1}{y}\right){\text d}y
\notag\\&\leq\frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}
\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\Bigg[\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( x-\frac{1}{f_2}\right) ^{\alpha-1}\left(\frac{1}{g_1}-y\right)^{\beta-1} \prod\left(\frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x
\notag\\
&\hspace{0.5cm}+\int_{1/f_2}^{1/f_1}\int_{1/g_2}^{1/g_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\left(\frac{1}{g_1}-y \right)^{\beta-1} \prod\left(\frac{1}{x},\frac{1}{y}\right){\text d}y{\text d}x\Bigg]
\notag\\
&\leq \frac{\alpha\beta}{4}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\Bigg[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\alpha-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y
+\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y\Bigg].
\end{align}
(18)
By adding inequalities (15)-(18), we have
\begin{align}
\label{t2e8}
\begin{split}
&\frac{\Gamma(\alpha+1)}{4}\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},\frac{2g_1g_2}{g_1+g_2}\right) +J^{\alpha}_{1/c_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},\frac{2g_1g_2}{g_1+g_2}\right) \right]
\\
& \hspace{0.5cm}+\frac{\Gamma(\beta+1)}{4}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\left[J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{g_1}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{g_2}\right) \right]
\\
&\leq \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{2}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\times \Bigg[J^{\alpha,\beta}_{f_1+,g_1+}(\prod\circ \Omega)\left( \frac{1}{f_2},\frac{1}{g_1}\right) +J^{\alpha,\beta}_{f_1+,g_2-}(\prod\circ \Omega)\left( \frac{1}{f_2},\frac{1}{g_1}\right)
\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{f_2-,g_1+}(\prod\circ \Omega)\left( \frac{1}{f_1},\frac{1}{g_2}\right) +J^{\alpha,\beta}_{f_2-,g_2-}(\prod\circ \Omega)\left( \frac{1}{f_1},\frac{1}{g_1}\right) \Bigg]
\\
&\leq \frac{\Gamma(\alpha+1)}{4}\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\Big[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},g_2\right)
+J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},g_1\right)
\\
&\hspace{0.5cm}+J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},g_1\right) +J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},g_1\right)\Big]
\\
&\hspace{0.5cm}+\frac{\Gamma(\beta+1)}{4}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\alpha}\Big[J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( f_1,\frac{1}{g_2}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( f_2,\frac{1}{g_2}\right)
\\
&\hspace{0.5cm}+J^{\alpha}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_1}\right) +J^{\alpha}_{1/g_2+}(\prod\circ \Omega_{2})\left( f_2,\frac{1}{g_1}\right)\Big].
\end{split}
\end{align}
(19)
This completes the second and third inequality of (12). Now again using (4), we have
\begin{align}
\label{t2e9}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
\leq \frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\Bigg[\int_{1/f_2}^{1/f-1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},\frac{2g_1g_2}{g_1+g_2}\right) {\text d}x
\notag\\
&\hspace{0.5cm}+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},\frac{2g_1g_2}{g_1+g_2}\right) {\text d}x\Bigg],
\end{align}
(20)
\begin{align}
\label{t2e10}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)
\leq \frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\Bigg[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{y}\right) {\text d}y
\notag\\
&\hspace{0.5cm}+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{y}\right) {\text d}y\Bigg].
\end{align}
(21)
Adding (20) and (21), we get
\begin{align}
&\prod\left(\frac{2f_1f_2}{f_1+f_2},\frac{2g_1g_2}{g_1+g_2}\right)\notag\\
&\leq \frac{\Gamma(\alpha+1)}{4}\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}
\left[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left( \frac{1}{f_1},\frac{2g_1g_2}{g_1+g_2}\right) +J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left( \frac{1}{f_2},\frac{2g_1g_2}{g_1+g_2}\right) \right]
\notag\\
&\hspace{0.5cm}+\frac{\Gamma(\beta+1)}{4}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\times\left[J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{g_1}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left( \frac{2f_1f_2}{f_1+f_2},\frac{1}{g_2}\right) \right].
\end{align}
(22)
This completes the first inequality of (12). For the last inequality by using (4), we have
\begin{align*}
\begin{split}
&\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\left[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},g_1\right) {\text d}x\right]
\\
&\leq \frac{\prod(f_1,g_1)+\prod(f_2,g_1)}{2},\\
&\frac{\alpha}{2}\left( \frac{f_1f_2}{f_2-f_1}\right) ^{\alpha}\left[\int_{1/f_2}^{1/f_1}\left( \frac{1}{f_1}-x\right) ^{\alpha-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x+\int_{1/f_2}^{1/f_1}\left( x-\frac{1}{f_2}\right) ^{\beta-1}\prod\left( \frac{1}{x},g_2\right) {\text d}x\right]
\\
&\leq \frac{\prod(f_1,g_2)+\prod(f_2,g_2)}{2},\\
&\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\left[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_1,\frac{1}{y}\right) {\text d}y\right]
\\
&\leq \frac{\prod(f_1,g_1)+\prod(f_1,g_2)}{2},\\
&\frac{\beta}{2}\left( \frac{g_1g_2}{g_2-g_1}\right) ^{\beta}\left[\int_{1/g_2}^{1/g_1}\left( \frac{1}{g_1}-y\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y+\int_{1/g_2}^{1/g_1}\left( y-\frac{1}{g_2}\right) ^{\beta-1}\prod\left( f_2,\frac{1}{y}\right) {\text d}y\right]
\\
&\leq \frac{\prod(f_2,g_1)+\prod(f_2,g_2)}{2}.
\end{split}
\end{align*}
Thus by adding all above inequalities, we get the last inequality of (12). Hence the proof is completed.
Lemma 1.
Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow
\mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1 < f_2\) and \(0< g_1< g_2\). If \(\partial^{2} \prod/\partial t_1\partial t_2\in L_1(\Delta)\), then following holds:
\begin{align}
\label{L1e1}
&\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}
+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi
\notag\\
&=\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\Bigg[\int_{0}^{1}\int_{0}^{1}\frac{r_1^{\alpha}r_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1
\notag\\
&\hspace{0.5cm}-\int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2{\text d}t_1
-\int_{0}^{1}\int_{0}^{1}\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1\notag\\&+\int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1 \Bigg],
\end{align}
(23)
where
\begin{align}
\Xi&=\frac{\Gamma(\alpha+1)}{4}\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\bigg[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left(\frac{1}{f_1},g_2 \right)+J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left(\frac{1}{f_2},g_2 \right)+J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left(\frac{1}{f_1},g_1 \right)
\notag\\
&\hspace{0.5cm}+J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left(\frac{1}{f_2},g_1 \right) \bigg]
+\frac{\Gamma(\beta+1)}{4}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}\bigg[J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_2,\frac{1}{g_1}\right)
\notag\\
&\hspace{0.5cm}+J^{\beta}_{1/d_2+}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_1}\right) +J^{\beta}_{1/d_1-}(\prod\circ \Omega_{2})\left(f_2,\frac{1}{g_2}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_2}\right) \bigg],
\end{align}
(24)
and
\(A_{t_1}=t_1f_1+(1-t_1)f_2\), \(B_{t_2}=t_2 c+(1-t_2)d\). Also, \(g(x,y)=(\frac{1}{x},\frac{1}{y})\), \(g_{1}(x,y)=(\frac{1}{x},y)\), and \(g_{2}(x,y)=(x,\frac{1}{y})\) for all \((x,y)\in \Delta\).
Proof.
By integration by parts and using the change of variable \(x=\frac{A_{t_1}}{f_1f_2}\) and \(y=\frac{B_{t_2}}{g_1g_2}\), we find that
\begin{align}
\label{L1e2}
\begin{split}
I_{1}&=\int_{0}^{1}\int_{0}^{1}\frac{t_1^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1
\\
&=
\int_{0}^{1}\frac{t_2^{\beta}}{B_{t_2}^{2}}\Bigg\{\frac{t_1^{\alpha}}{f_1f_2(f_2-f_1)}\frac{\partial \prod}{\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right)\Bigg|_{0}^{1}
– \frac{\alpha}{f_1f_2(f_2-f_1)}\int_{0}^{1} t_1^{\alpha-1}\frac{\partial \prod}{\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_1 \Bigg\}{\text d}t_2
\\
&=\frac{1}{f_1f_2(f_2-f_1)}\int_{0}^{1}\frac{t_2^{\beta}}{B_{t_2}^{2}}\frac{\partial \prod}{\partial t_2}\left(f_2,\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2
– \frac{\alpha}{f_1f_2(f_2-f_1)}\int_{0}^{1}t_1^{\alpha-1}\left\lbrace \int_{0}^{1} \frac{t_2^{\beta}}{B_{t_2}^{2}}\frac{\partial \prod}{\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2\right\rbrace {\text d}t_1
\\
&=\frac{1}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\prod(f_2,g_2)
-\frac{\beta}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\int_{0}^{1}t_2^{\beta-1}\prod\left(f_2,\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2
\\
&\hspace{0.5cm}-\frac{\alpha}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\int_{0}^{1}t_1^{\alpha-1}\prod\left(\frac{f_1f_2}{A_{t_1}},d\right){\text d}t_1
\\
&\hspace{0.5cm}+\frac{\alpha\beta}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\int_{0}^{1}t_1^{\alpha-1}t_2^{\beta-1}\prod\left(\frac{f_1f_2}{A_{r_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2
\\
&=\frac{1}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}
\times \bigg[\prod(f_2,g_2)-\Gamma(\beta+1)\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_2,\frac{1}{g_1}\right)
\\
&\hspace{0.5cm}-\Gamma(\alpha+1)\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left(\frac{1}{f_1},g_2 \right)+\Gamma(\alpha+1)\Gamma(\beta+1)
\\
&\hspace{0.5cm}\times\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\alpha,\beta}_{1/f_2+,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right) \bigg].
\end{split}
\end{align}
(25)
Similarly, we can have
\begin{align}
\label{L1e3}
I_{2}&=\int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2{\text d}t_1
\notag\\
&=\frac{1}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\bigg[-\prod(f_1,g_1)
+\Gamma(\beta+1)\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_1}\right)
\notag\\
&\hspace{0.5cm}+\Gamma(\alpha+1)\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left(\frac{1}{f_2},g_2 \right)-\Gamma(\alpha+1)\Gamma(\beta+1)
\notag\\
&\hspace{0.5cm}\times\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\alpha,\beta}_{1/f_2+,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right) \bigg].
\end{align}
(26)
\begin{align*}
I_{3}&=\int_{0}^{1}\int_{0}^{1}\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1
\end{align*}
\begin{align}
\label{L1e4}
&=\frac{1}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\bigg[-\prod(f_2,g_1)
+\Gamma(\beta+1)\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left(f_2,\frac{1}{g_2}\right)
\notag\\
&\hspace{0.5cm}+\Gamma(\alpha+1)\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left(\frac{1}{f_1},g_1 \right)-\Gamma(\alpha+1)\Gamma(\beta+1)
\notag\\
&\hspace{0.5cm}\times\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right) \bigg].
\end{align}
(27)
\begin{align}
\label{L1e5}
I_{4}&=\int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1
\notag\\
&=\frac{1}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\bigg[\prod(f_1,g_2)
-\Gamma(\beta+1)\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_2}\right)
\notag\\
&\hspace{0.5cm}-\Gamma(\alpha+1)\left( \frac{f_1f_2}{f_1-f_1}\right)^{\alpha}J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left(\frac{1}{f_2},g_1 \right)+\Gamma(\alpha+1)\Gamma(\beta+1)
\notag\\
&\hspace{0.5cm}\times\left( \frac{f_1f_2}{f_1-f_1}\right)^{\alpha}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right) \bigg].
\end{align}
(28)
Thus from equalities (25)-(28), we have
\begin{align}
\label{L1e6}
&I_{1}-I_{2}-I_{3}+I_{4}
=\frac{\prod(f_2,g_2)+\prod(f_1,g_1)+\prod(f_2,g_1)+\prod(f_1,g_2)}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}
-\frac{\Gamma(\beta+1)}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\left( \frac{g_1g_2}{g_2-g_1}\right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\bigg[J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_2,\frac{1}{g_1}\right) +J^{\beta}_{1/g_2+}(\prod\circ \Omega_{2})\left(f_1,\frac{1}{g_1}\right)
+J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\left(f_2,\frac{1}{g_2}\right) +J^{\beta}_{1/g_1-}(\prod\circ \Omega_{2})\notag\\
&\hspace{0.5cm}\times\left(f_1,\frac{1}{g_2}\right) \bigg]
-\frac{\Gamma(\alpha+1)}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}\left( \frac{f_1f_2}{f_2-f_1}\right)^{\alpha}
\bigg[J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left(\frac{1}{f_1},g_2 \right)+J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})
\notag\\
&\hspace{0.5cm}\times\left(\frac{1}{f_2},g_2 \right)+J^{\alpha}_{1/f_2+}(\prod\circ \Omega_{1})\left(\frac{1}{f_1},g_1 \right)+J^{\alpha}_{1/f_1-}(\prod\circ \Omega_{1})\left(\frac{1}{f_2},g_1 \right)\bigg]
+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}
\notag\\
&\hspace{0.5cm}\times\bigg[J^{\alpha,\beta}_{1/f_2+,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right) \bigg].
\end{align}
(29)
Multiplying both sides of equality (29) by \(\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\), we get the desired equality (23).
Theorem 13.
Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow
\mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1< f_2\) and \(0< g_1< g_2\). If \(\left| \partial^{2} \prod/\partial t_1\partial t_2\right| \) is a harmonically convex on the co-ordinates on \(\Delta\), then following holds:
\begin{align}
\label{tt1e1}
& \Bigg|\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}
+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
& \hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
& \hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi\Bigg|
\notag\\
& \leq\frac{f_1g_1(f_2-f_1)(g_2-g_1)}{4f_2g_2(\alpha+1)(\beta+1)(\alpha+2)(\beta+2)}\Bigg[\vartheta_{1} \left| \frac{\partial^{2} \prod}{\partial t_1\partial t_2} (f_1,g_1)\right|+\vartheta_{2}\left| \frac{\partial^{2} \prod}{\partial t_1\partial t_2} (f_1,g_2)\right|\notag\\
& \hspace{0.5cm}+\vartheta_{3}\left| \frac{\partial^{2} \prod}{\partial t_1\partial t_2} (f_2,g_1)\right|+\vartheta_{4}\left| \frac{\partial^{2} \prod}{\partial t_1\partial t_2} (f_2,g_2)\right| \Bigg] ,
\end{align}
(30)
where
\begin{align}
\vartheta_{1}& =(\alpha+1)(\beta+1)\ _{2}F_{1}\left( 2,\alpha+2;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+2;\beta+3;1-\frac{g_1}{g_2}\right)
\notag\\
& \hspace{0.5cm}+(\beta+1)\ _{2}F_{1}\left( 2,2;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+2;\beta+3;1-\frac{g_1}{g_2}\right)
+\ _{2}F_{1}\left( 2,\alpha+2;\alpha+3;1-\frac{f_1}{f_2}\right) \notag\\
& \hspace{0.5cm}\times \ _{2}F_{1}\left( 2,2;\beta+3;1-\frac{g_1}{g_2}\right)
+\ _{2}F_{1}\left( 2,2;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,2;\beta+3;1-\frac{g_1}{g_2}\right),\\
\end{align}
(31)
\begin{align}
\vartheta_{2}&=(\beta+1)\ _{2}F_{1}\left( 2,\alpha+1;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+2;\beta+3;1-\frac{g_1}{g_2}\right)
\notag\\
& \hspace{0.5cm}+(\alpha+1)(\beta+1)\ _{2}F_{1}\left( 2,1;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+2;\beta+3;1-\frac{g_1}{g_2}\right)
+\ _{2}F_{1}\left( 2,\alpha+1;\alpha+3;1-\frac{f_1}{f_2}\right)\notag\\
& \hspace{0.5cm}\times \ _{2}F_{1}\left( 2,2;\beta+3;1-\frac{g_1}{g_2}\right)
+\ _{2}F_{1}\left( 2,1;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,2;\beta+3;1-\frac{g_1}{g_2}\right),\\
\end{align}
(32)
\begin{align}
\vartheta_{3}& =(\alpha+1)\ _{2}F_{1}\left( 2,\alpha+2;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+1;\beta+3;1-\frac{g_1}{g_2}\right)
\notag\\
& \hspace{0.5cm}+(\beta+1)\ _{2}F_{1}\left( 2,2;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+1;\beta+3;1-\frac{g_1}{g_2}\right)
+(\beta+1)\ _{2}F_{1}\left( 2,\alpha+2;\alpha+3;1-\frac{f_1}{f_2}\right)\notag\\
& \hspace{0.5cm}\times \ _{2}F_{1}\left( 2,1;\beta+3;1-\frac{g_1}{g_2}\right)
+\ _{2}F_{1}\left( 2,2;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,1;\beta+3;1-\frac{g_1}{g_2}\right),\\
\end{align}
(33)
\begin{align}
\vartheta_{4}&=\ _{2}F_{1}\left( 2,\alpha+1;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+1;\beta+3;1-\frac{g_1}{g_2}\right)
\notag\\
& \hspace{0.5cm}+(\alpha+1)\ _{2}F_{1}\left( 2,1;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,\beta+1;\beta+3;1-\frac{g_1}{g_2}\right)
+(\beta+1)\ _{2}F_{1}\left( 2,\alpha+1;\alpha+3;1-\frac{f_1}{f_2}\right)\notag\\
& \hspace{0.5cm}\times \ _{2}F_{1}\left( 2,1;\beta+3;1-\frac{g_1}{g_2}\right)
+(\alpha+1)(\beta+1)\ _{2}F_{1}\left( 2,1;\alpha+3;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2,1;\beta+3;1-\frac{g_1}{g_2}\right) .
\end{align}
(34)
Proof.
Using Lemma 1, we have
\begin{align}
\label{tt1e2}
& \frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
& \hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
& \hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi
\notag\\
& =\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\Bigg[\int_{0}^{1}\int_{0}^{1}\frac{r_1^{\alpha}r_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1
\notag\\
& \hspace{0.5cm}+\int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2{\text d}t_1
+\int_{0}^{1}\int_{0}^{1}\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1
\notag\\
& \hspace{0.5cm}+\int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\frac{\partial^{2}\prod}{\partial t_1\partial t_2}\left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right){\text d}t_2 {\text d}t_1 \Bigg].
\end{align}
(35)
Now using co-ordinated harmonically convexity of \(\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2}\right| \), we get
\begin{align}
\label{tt1e3}
& \Bigg|\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}
+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}\notag\\& \hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
& \hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi\Bigg|
\notag\\
& \leq\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\Bigg[\int_{0}^{1}\int_{0}^{1}\Bigg\lbrace \frac{t_1^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}
\notag\\
& \hspace{0.5cm}+\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\Bigg\rbrace
\Bigg\lbrace t_1t_2\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_1)\right|+(1-t_1)t_2\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_1)\right|
\notag\\
& \hspace{0.5cm}+t_1(1-t_2)\left|\frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_2)\right|
+(1-t_1)(1-t_2)\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_2)\right|\Bigg\rbrace {\text d}t_2 {\text d}t_1\Bigg]
\notag\\
& =\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}
\Bigg[\int_{0}^{1}\int_{0}^{1}t_1t_2\left\lbrace \frac{t_1^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\right\rbrace\notag\\
& \hspace{0.5cm}\times \left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_1)\right|{\text d}t_1{\text d}t_2
+\int_{0}^{1}\int_{0}^{1}(1-t_1)t_2\left\lbrace \frac{t_1^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\right\rbrace\notag\\
& \hspace{0.5cm}\times \left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_1)\right|{\text d}t_1{\text d}t_2
+\int_{0}^{1}\int_{0}^{1}t_1(1-t_2)\left\lbrace \frac{t_1^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\right\rbrace \notag\\& \hspace{0.5cm}\times\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_2)\right|{\text d}t_1{\text d}t_2+\int_{0}^{1}\int_{0}^{1}(1-t_1)(1-t_2)
\notag\\
& \hspace{0.5cm}\times\Bigg\lbrace \frac{t_1^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{(1-t_1)^{\alpha}t_2^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}+\frac{t_1^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}
+\frac{(1-t_1)^{\alpha}(1-t_2)^{\beta}}{A_{t_1}^{2}B_{t_2}^{2}}\Bigg\rbrace \left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_2)\right|{\text d}t_1{\text d}t_2 \Bigg].
\end{align}
(36)
After calculating above integrations, we get the required result.
Theorem 14.
Let \(\prod:\Delta=[f_1,f_2]\times [g_1,g_2]\subseteq (0,\infty)\times (0,\infty) \rightarrow
\mathbb{R}\) be a partial differentiable mapping on \(\Delta\) with \(0< f_1< f_2\) and \(0< g_11\), is a harmonically convex on the co-ordinates on \(\Delta\), then following holds:
\begin{align}
\label{tt2e1}
&\Bigg|\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}
+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi\Bigg|
\notag\\
&\leq\frac{f_1g_1(f_2-f_1)(g_2-g_1)}{4f_2g_2[(p\alpha+1)(p\beta+1)]^{1/p}}\left[\psi_{1}^{1/p}+\psi_{2}^{1/p}+\psi_{3}^{1/p}+\psi_{4}^{1/p} \right]
\notag\\
&\hspace{0.5cm}\times\left(\frac{\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_1)\right|^{q}+\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_2)\right|^{q}+\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_1)\right|^{q}+ \left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_2)\right|^{q}}{4} \right)^{1/q},
\end{align}
(37)
where
\begin{equation}
\psi_{1}=\ _{2}F_{1}\left( 2p,p\alpha+1;p\alpha+2;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2p,p\beta+1;p\beta+2;1-\frac{g_1}{g_2}\right),
\end{equation}
(38)
\begin{align}
&\psi_{2}=\ _{2}F_{1}\left( 2p,1;p\alpha+2;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2p,p\beta+1;p\beta+2;1-\frac{g_1}{g_2}\right),
\\
\end{align}
(39)
\begin{align}
&\psi_{3}=\ _{2}F_{1}\left( 2p,p\alpha+1;p\alpha+2;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2p,1;p\beta+2;1-\frac{g_1}{g_2}\right),
\\
\end{align}
(40)
\begin{align}
&\psi_{4}=\ _{2}F_{1}\left( 2p,1;p\alpha+2;1-\frac{f_1}{f_2}\right) \ _{2}F_{1}\left( 2p,1;p\beta+2;1-\frac{g_1}{g_2}\right).
\end{align}
(41)
Proof.
Applying the Holder’s inequality for double integrals in (35), we get
\begin{align}
\label{tt2e2}
&\Bigg|\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}
+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi\Bigg|
\notag\\
&\leq\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\Bigg[\left( \int_{0}^{1}\int_{0}^{1}\frac{t_1^{p\alpha}t_2^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
+\left( \int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{p\alpha}t_2^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
\notag\\
&\hspace{0.5cm}+\left( \int_{0}^{1}\int_{0}^{1}\frac{t_1^{p\alpha}(1-t_2)^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
+\left( \int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{p\alpha}(1-t_2)^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p} \Bigg]
\notag\\
&\hspace{0.5cm}\times\left(\int_{0}^{1}\int_{0}^{1}\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} \left(\frac{f_1f_2}{A_{t_1}},\frac{g_1g_2}{B_{t_2}}\right)\right|^{q} dt_1dt_2\right)^{1/q}.
\end{align}
(42)
Using co-ordinated harmonically convexity of \(\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2}\right|^{q} \), we get
\begin{align*}\label{tt2e3}
&\Bigg|\frac{\prod(f_1,g_1)+\prod(f_1,g_2)+\prod(f_2,g_1)+\prod(f_2,g_2)}{4}\hspace{0.5cm}+\frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{4}
\left(\frac{f_1f_2}{f_2-f_1} \right)^{\alpha}\left(\frac{g_1g_2}{g_2-g_1} \right)^{\beta}
\notag\\
&\hspace{0.5cm}\times\Bigg[J^{\alpha,\beta}_{1/f_2+,1/g_1+}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_1} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_2+}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_1} \right)
\notag\\
&\hspace{0.5cm}+J^{\alpha,\beta}_{1/f_2+,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_1},\frac{1}{g_2} \right)+J^{\alpha,\beta}_{1/f_1-,1/g_1-}(\prod\circ \Omega)\left(\frac{1}{f_2},\frac{1}{g_2} \right)\Bigg]-\Xi\Bigg|
\notag\\
&\leq\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\Bigg[\left( \int_{0}^{1}\int_{0}^{1}\frac{t_1^{p\alpha}t_2^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
+\left( \int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{p\alpha}t_2^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
\notag\\
&\hspace{0.5cm}+\left( \int_{0}^{1}\int_{0}^{1}\frac{t_1^{p\alpha}(1-t_2)^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
+\left( \int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{p\alpha}(1-t_2)^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p} \Bigg]
\notag\\
&\hspace{0.5cm}\times\Bigg(\int_{0}^{1}\int_{0}^{1} \Bigg\{ t_1t_2\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,f_2)\right|^{q}+(1-t_1)t_2\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_1)\right|^{q}
\notag\\
&\hspace{0.5cm}+t_1(1-t_2)\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_2)\right|^{q}+(1-t_1)(1-t_2)\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_2)\right|^{q} \Bigg\}dt_2 dt_1\Bigg)^{1/q}
\notag\\
&=\frac{f_1f_2g_1g_2(f_2-f_1)(g_2-g_1)}{4}\Bigg[\left( \int_{0}^{1}\int_{0}^{1}\frac{t_1^{p\alpha}t_2^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
+\left( \int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{p\alpha}t_2^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
\notag
\end{align*}
\begin{align}
&\hspace{0.5cm}+\left( \int_{0}^{1}\int_{0}^{1}\frac{t_1^{p\alpha}(1-t_2)^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p}
+\left( \int_{0}^{1}\int_{0}^{1}\frac{(1-t_1)^{p\alpha}(1-t_2)^{p\beta}}{A_{t_1}^{2p}B_{t_2}^{2p}}dt_2 dt_1\right)^{1/p} \Bigg]
\notag\\
&\hspace{0.5cm}\times\left(\frac{\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_1)\right|^{q}+\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_1,g_2)\right|^{q}+\left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_1)\right|^{q}+ \left| \frac{\partial^{2}\prod}{\partial t_1\partial t_2} (f_2,g_2)\right|^{q}}{4} \right)^{1/q} .\end{align}
(43)
By calculating all integrals, we get the required result (37).
3. Conclusion
In Theorem 11 and 12, we have proved some new Hermite-Hadamard type inequalities for co-ordinated harmonically convex on a rectangle via Riemann-Liouville fractional integrals. In Lemma 1, we have proved a fractional integral identity and then with the help of this Lemma 1 we proved some fractional Hermite-Hadamard type inequalities on the co-ordinates.
Acknowledgments
The present investigation is supported by National University of Science and Technology(NUST), Islamabad, Pakistan.
Authors Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Conflicts of interest
The authors declare no conflict of interest.
