1. Introduction
A function \(f:\mathcal{K}\rightarrow \mathbb{R}\), where \(\mathcal{K}\) is an interval of real
numbers, is called convex if the following inequality holds:
\begin{equation}
\label{e1e}
f(ru_1+(1-r)u_2)\leq rf(u_1)+(1-r)f(u_2),
\end{equation}
(1)
for all \(u_1,u_2\in \mathcal{K}\) and \(r\in[0,1]\). Function \(f\) is called concave
if \(-f\) is convex.
The Hermite-Hadamard inequality [1] for convex functions
\(f:\mathcal{K}\rightarrow \mathbb{R}\) on an interval of real line is:
\begin{equation}
\label{p1}
f\left( \frac{u_1+u_2}{2}\right) \leq
\frac{1}{u_2-u_1}\int^{u_2}_{u_1}f(x)dx\leq\frac{f(u_1)+f(u_2)}{2},
\end{equation}
(2)
where \(u_1,u_2\in \mathcal{K}\) with \(u_1< u_2\). Then Fejér [
2] introduced the weighted
generalization of (2) as follows
\begin{equation}
\label{p2}
f\left( \frac{u_1+u_2}{2}\right) \int_{u_1}^{u_2}g(x)dx\leq
\frac{1}{u_2-u_1}\int^{u_2}_{u_1}f(x)g(x)dx\leq\frac{f(u_1)+f(u_2)}{2}\int_{u_1}^{u_2}g(x)dx,
\end{equation}
(3)
where \(g:[u_1,u_2]\rightarrow \mathbb{R}\) is nonnegative, integrable and
symmetric to \((u_1+u_2)/2\). For more results and details see [
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14].
Definition 1 ([15]).
Let \(\mathcal{K}\subset\mathbb{R}\setminus\{0\}\) be a real interval. A
function \(f:\mathcal{K}\rightarrow \mathbb{R}\) is said to be harmonically
convex, if
\begin{equation}
\label{1ee}
f\left(\frac{u_1u_2}{ru_1+(1-r)u_2}\right)\leq rf(u_2)+(1-r)f(u_1),
\end{equation}
(4)
for all \(u_1,u_2\in \mathcal{K}\) and \(r\in[0,1]\). If the inequality in
(4) is reversed, then \(f\) is said to be harmonically
concave.
Dragomir [
16] gave the Hadamard’s inequality for convex functions on the co-ordinate which is defined as:
Definition 2 ([16]).
A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq \mathbb{R}^{2}
\rightarrow \mathbb{R}\) is called convex on the co-ordinate with
\(u_1< u_2\) and \(v_1< v_2\) if the partial functions
\(f_{y}:[u_1,u_2] \rightarrow \mathbb{R}\), \(f_{y}(a)=f(a,y)\) and
\(f_{x}:[v_1,v_2] \rightarrow \mathbb{R}\), \(f_{x}(c)=f(x,c)\)
are convex for all \(x\in[u_1,u_2]\) and \(y\in [v_1,v_2]\).
Definition 3 ([17]).
A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq \mathbb{R}^{2}
\rightarrow \mathbb{R}\) is called co-ordinate convex on \(\Delta\) with
\(u_1< u_2\) and \(v_1< v_2\), if
\begin{align*}
&f(rx+(1-r)z,\tau y+(1-\tau)w)\leq r\tau
f(x,y)+r(1-\tau)f(x,w)+(1-r)\tau
f(z,y)+(1-r)(1-\tau)f(z,w),
\end{align*}
for all \(r,\tau\in [0,1]\) and \((x,y),(z,w)\in\Delta\). For more results and details see [16,17,18,19].
Definition 4 ([20]).
A function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq (0,\infty)\times (0,\infty)
\rightarrow \mathbb{R}\) is called co-ordinated harmonically convex on \(\Delta\) with
\(u_1< u_2\) and \(v_1< v_2\), if
\begin{align*}
&f\left(\frac{xz}{rx+(1-r)z},\frac{yw}{\tau y+(1-\tau)w}\right)\leq r\tau
f(x,y)+r(1-\tau)f(x,w)+(1-r)\tau
f(z,y)+(1-r)(1-\tau)f(z,w),
\end{align*}
for all \(r,\tau\in [0,1]\) and \((x,y),(z,w)\in\Delta\).
Clearly, a function \(f:\Delta=[u_1,u_2]\times [v_1,v_2]\subseteq (0,\infty)\times (0,\infty)
\rightarrow \mathbb{R}\) is called harmonically convex on the co-ordinate with
\(u_1< u_2\) and \(v_1< v_2\) if the partial functions
\(f_{y}:[u_1,u_2] \rightarrow \mathbb{R}\), \(f_{y}(a)=f(a,y)\) and
\(f_{x}:[v_1,v_2] \rightarrow \mathbb{R}\), \(f_{x}(b)=f(x,b)\)
are harmonically convex for all \(x\in[u_1,u_2]\) and \(y\in [v_1,v_2]\), see [
21] for more details.
Definition 5 ([22]).
Let \([u_1,u_2]\subset\mathbb{R}\) be a finite interval. The left-
and right-side Katugampola fractional integrals of order
\(\alpha(>0)\) of \(f\in X^{p}_{c}(u_1,u_2)\) are defined by,
\begin{equation*}
^{\rho}I^{\alpha}_{u_1+}f(x)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{u_1}^{x}(x^{\rho}-t^{\rho})^{\alpha-1}t^{\rho-1}f(t)dt,
\end{equation*}
and
\begin{equation*}
^{\rho}I^{\alpha}_{u_2-}f(x)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{x}^{u_2}(t^{\rho}-x^{\rho})^{\alpha-1}t^{\rho-1}f(t)dt,
\end{equation*}
with \(u_1< x
0\), where \(X^{p}_{c}(u_1,u_2)\) \((c \in \mathbb{R}, 1\leq p\leq \infty)\) is the space of those complex valued Lebesgue measurable functions \(f\) on \([u_1,u_2]\) for which \(\|f\|_{X^{p}_{c}}< \infty\), where the norm is defined by
\begin{equation*}
\|f\|_{X^{p}_{c}}=\left( \int_{u_1}^{u_2}|t^{c}f(t)|^{p}\frac{dt}{t}\right)^{1/p}< \infty,
\end{equation*}
for \(1\leq p< \infty\), \(c\in \mathbb{R}\) and for the case \(p=\infty\),
\begin{equation*}
\|f\|_{X^{\infty}_{c}}= ess\ sup_{u_1\leq t\leq u_2}[t^{c}|f(t)|].
\end{equation*}
Definition 6 ([23]).
Let \(f\in L_{1}([u_1,u_2]\times [v_1,v_2])\). The Katugampola fractional integrals \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_1+}\), \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_2-}\), \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_1+}\) and \(^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_2-}\) of order \(\alpha,\beta>0\) with \(a,c\geq 0\) are defined by
\begin{equation*}
^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_1+}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u_1}^{x}\int_{v_1}^{y}(x^{\rho_{1}}-t^{\rho_{1}})^{\alpha-1}(y^{\rho_{2}}-s^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt,
\end{equation*}
with \(x>u_1, \ y>v_1\),
\begin{equation*}
^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_1+,v_2-}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{u_1}^{x}\int_{y}^{v_2}(x^{\rho_{1}}-t^{\rho_{1}})^{\alpha-1}(s^{\rho_{2}}-y^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt,
\end{equation*}
with \(x>u_1, \ y< v_2\),
\begin{equation*}
^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_1+}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{u_2}\int_{v_1}^{y}(t^{\rho_{1}}-x^{\rho_{1}})^{\alpha-1}(y^{\rho_{2}}-s^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt,
\end{equation*}
with \(x
v_1\), and
\begin{equation*}
^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{u_2-,v_2-}f(x,y)=\frac{\rho^{1-\alpha}_{1}\rho^{1-\beta}_{2}}{\Gamma(\alpha)\Gamma(\beta)}\int_{x}^{u_2}\int_{y}^{v_2}(t^{\rho_{1}}-x^{\rho_{1}})^{\alpha-1}(s^{\rho_{2}}-y^{\rho_{2}})^{\beta-1}t^{\rho_{1}-1}s^{\rho_{2}-1}f(t,s)dsdt,
\end{equation*}
with \(x< u_2, \ y< v_2\), respectively, where the Gamma function \(\Gamma\) is defined as \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt\).
In the next section, we give result for harmonically convex functions in one dimension.
2. Hermite-Hadamard-Fejér type inequalities
In this section, we give Hermite-Hadamard-Fejér type inequalities for harmonically convex functions via Katugampola fractional integral in one dimension which will play a key role for the results in the next section.
Latif
et al., [
24] defined following useful definition:
Definition 7 ([24]).
A function \(h:[u_1,u_2]\subseteq\mathbb{R}\backslash \{0\}\rightarrow \mathbb{R}
\) is said to be harmonically symmetric with respect to \(2u_1u_2/(u_1+u_2)\) if
\begin{equation*}
h(x)=h\left(\frac{1}{\frac{1}{u_1}+\frac{1}{u_2}-\frac{1}{x}}\right)
\end{equation*}
holds for all \(x\in[u_1,u_2]\).
Lemma 1.
Let \(\rho>0\). If \(h:[u_1^{\rho},u_2^{\rho}]\subseteq(0,\infty)\rightarrow
\mathbb{R}\) is integrable and harmonically symmetric with respect to
\(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then
\begin{equation}
\label{e32}
^{\rho}I^{\alpha}_{1/u_2+}(h\circ
g)(1/u_1^{\rho})=^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho})
=\frac{1}{2}\left[^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})+\
^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho})\right],
\end{equation}
(5)
with \(\alpha>0\) and \(g(x^{\rho})=1/x^{\rho}\).
Proof.
Since \(h\) is harmonically symmetric with respect to
\(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then by definition we have
\(h(\frac{1}{x^{\rho}})=h\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)\),
for all \(x^{\rho}\in\left[\frac{1}{u_2^{\rho}},\frac{1}{u_1^{\rho}}\right] \). In the
following integral, by setting
\(t^{\rho}=\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}\), we get
\begin{align*}
^{\rho}I^{\alpha}_{1/u_2+}(h\circ
g)(1/u_1^{\rho})&=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}
\int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(\frac{1}{u_1^{\rho}}-t^{\rho}\right)^{\alpha-1}t^{\rho-1}h\left( \frac{1}{t^{\rho}}\right) dt
\\
&=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}
\int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(x^{\rho}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}x^{\rho-1}h\left( \frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right) dx
\\
&=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}
\int^{\frac{1}{u_1}}_{\frac{1}{u_2}}\left(x^{\rho}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}x^{\rho-1}h\left( \frac{1}{x^{\rho}}\right) dx\\
&=\
^{\rho}I^{\alpha}_{1/u_1-}(h\circ g)(1/u_2^{\rho}).
\end{align*}
This completes the proof.
Remark 1.
In Lemma 1, if we take \(\rho\longmapsto 0\), we get Lemma 2 in [25].
Theorem 8.
Let \(\rho>0\). Let \(f:[u_1^{\rho},u_2^{\rho}]\subseteq (0,\infty)\rightarrow\mathbb{R}\) be a
harmonically convex with \(u_1< u_2\) and \(f\in L_1[u_1,u_2]\). If
\(h:[u_1^{\rho},u_2^{\rho}]\subseteq(0,\infty)\rightarrow
\mathbb{R}\) is nonnegative and harmonically symmetric with respect
to \(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\), then the following
inequalities hold:
\begin{align}
f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)\left[^{\rho}I^{\alpha}_{1/u_1-}(h\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})\right]
&\leq\left[^{\rho}I^{\alpha}_{1/u_1-}(fh\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(fh\circ
g)(1/u_1^{\rho})\right]\notag
\end{align}
\begin{align}
\label{e33}
&\leq\frac{f(u_1^{\rho})+f(u_2^{\rho})}{2}\left[^{\rho}I^{\alpha}_{1/u_1-}(h\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})\right],
\end{align}
(6)
with \(\alpha>0\) and \(g(x^{\rho})=1/x^{\rho}\).
Proof.
Since \(f\) is harmonically convex on \([u_1^{\rho},u_2^{\rho}]\), we have
for all \(r\in[0,1]\)
\begin{align}
\label{e34}
f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)&=
f\left(\frac{2u_1^{\rho}u_2^{\rho}}{\left(r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}\right)+\left(r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}\right)}\right)\notag
\\
&\leq
\frac{f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right)
+f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)}{2}.
\end{align}
(7)
Multiplying (7) by
\(r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\)
on both sides and integrate with respect to \([0,1]\), we get
\begin{align*}
\begin{split}
2f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)&\int_{0}^{1}
r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr
\\
&\leq
\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right)
h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr\\
&\;\;\;\;+\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)
h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr.
\end{split}
\end{align*}
Since \(h\) is harmonically symmetric with respect to
\(2u_1^{\rho}u_2^{\rho}/(u_1^{\rho}+u_2^{\rho})\). By setting
\(x^{\rho}=\frac{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}{u_1^{\rho}u_2^{\rho}}\),
we get
\begin{align*}
\begin{split}
2\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)
\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}h\left( \frac{1}{x^{\rho}}\right) dx
\\
\leq&\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\left[\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}
f\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)h\left( \frac{1}{x^{\rho}}\right) dx\right.\\&\left.+\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}
f\left( \frac{1}{x^{\rho}}\right) h\left( \frac{1}{x^{\rho}}\right) dx \right]
\\
=&\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\left[\int_{1/u_2}^{1/u_1}\left(\frac{1}{u_1^{\rho}}-x^{p}\right)^{\alpha-1}
f\left( \frac{1}{x^{\rho}}\right) h\left(\frac{1}{\frac{1}{u_1^{\rho}}+\frac{1}{u_2^{\rho}}-x^{\rho}}\right)dx\right.
\\
&\left.+\int_{1/u_2}^{1/u_1}\left(x^{p}-\frac{1}{u_2^{\rho}}\right)^{\alpha-1}
f\left( \frac{1}{x^{\rho}}\right) h\left( \frac{1}{x^{\rho}}\right) dx \right].
\end{split}
\end{align*}
Then by Lemma 1, we have
\begin{align}
\label{e35}
\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&\rho^{\alpha-1}\Gamma(\alpha)f\left(\frac{2u_1^{\rho}u_2^{\rho}}{u_1^{\rho}+u_2^{\rho}}\right)
\left[^{\rho}I^{\alpha}_{1/u_1-}(h\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(h\circ g)(1/u_1^{\rho})\right]\notag
\\
&\leq
\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\rho^{\alpha-1}\Gamma(\alpha)
\left[^{\rho}I^{\alpha}_{1/u_1-}(fh\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(fh\circ
g)(1/u_1^{\rho})\right].
\end{align}
(8)
This completes the first inequality. For second inequality, we first note that if \(f\) is harmonically
convex function, then we have
\begin{equation}
\label{e36}
f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right)
+f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\leq
f(u_1^{\rho})+f(u_2^{\rho}).
\end{equation}
(9)
Multiplying (8) by
\(r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)\)
on both sides and integrate with respect to \(r\in[0,1]\), we get
\begin{align*}
\begin{split}
&\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_1^{\rho}+(1-r^{\rho})u_2^{\rho}}\right)
h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr
\\
&\;\;\;+\int_{0}^{1}r^{\alpha\rho-1}f\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)
h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr
\\
&\leq
(f(u_1^{\rho})+f(u_2^{\rho}))\int_{0}^{1}r^{\alpha\rho-1}h\left(\frac{u_1^{\rho}u_2^{\rho}}{r^{\rho}u_2^{\rho}+(1-r^{\rho})u_1^{\rho}}\right)dr,
\end{split}
\end{align*}
i.e.,
\begin{align*}
\begin{split}
\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}&\rho^{\alpha-1}\Gamma(\alpha)
\left[^{\rho}I^{\alpha}_{1/u_1-}(fh\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(fh\circ
g)(1/u_1^{\rho})\right]
\\
&\leq\left(\frac{u_1^{\rho}u_2^{\rho}}{u_2^{\rho}-u_1^{\rho}}\right)^{\alpha}\rho^{\alpha-1}\Gamma(\alpha)\frac{f(u_1^{\rho})+f(u_2^{\rho})}{2}
\left[^{\rho}I^{\alpha}_{1/u_1-}(h\circ
g)(1/u_2^{\rho})+^{\rho}I^{\alpha}_{1/u_2+}(h\circ
g)(1/u_1^{\rho})\right].
\end{split}
\end{align*}
This completes the proof.
Remark 2.
- 1)   In Theorem 8, if we take \(\rho\rightarrow 1\), we get Theorem 5 in [25].
- 2)   In Theorem 8, if we take \(\rho\rightarrow 1\) and \(\alpha=1\), we get Theorem 8 in [26].
3. Hermite-Hadamard-Fejér type inequalities on co-ordinates
In this section, we established some new results by using Katugampola fractional integrals on co-ordinates. First we give the following result:
Theorem 9.
Let \(\alpha,\beta>0\) and \(\rho_{1},\rho_{2}>0\). Let \(f:\Delta=[u_1^{\rho_{1}},u_2^{\rho_{1}}]\times [v_1^{\rho_{2}},v_2^{\rho_{2}}]\subseteq (0,\infty)\times(0,\infty) \rightarrow
\mathbb{R}\) be a co-ordinated harmonically convex on \(\Delta\), with \(0< u_1< u_2\), \(0< v_1< v_2\). If \(h:\Delta\rightarrow
\mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}\), \(\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\) on \(\Delta\). Then
\begin{align}
\label{t1e1}
f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.\notag
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag
\\
\leq& \frac{1}{4}
\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}(fh\circ g) \left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}(fh\circ g)\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.\notag
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag
\\
\leq&\frac{f(u_1^{\rho_{1}},v_1^{\rho_{2}})+f(u_1^{\rho_{1}},v_2^{\rho_{2}})+f(u_2^{\rho_{1}},v_1^{\rho_{2}})+f(u_2^{\rho_{1}},v_2^{\rho_{2}})}{4}\notag
\\
&\times\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.\notag
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right],
\end{align}
(10)
holds, where \(g(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right) \).
Proof.
Since \(f\) is co-ordinated harmonically convex on \(\Delta\), we have
\begin{align}
\label{t1e3}
f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)
\leq& \frac{1}{4}\left[f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right)\right.\notag
\\
&+f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\notag
\\
&+
f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right)\notag
\\
& \left.+f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right) \right].
\end{align}
(11)
Multiplying (11) by \(r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\) on both sides and then integrating with respect to \((r,\tau)\) over \([0,1]\times [0,1]\), we get
\begin{align*}
f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)& \int_{0}^{1}\int_{0}^{1}r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1} h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)drd\tau
\\
\leq& \frac{1}{4}\bigg[\int_{0}^{1}\int_{0}^{1} f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right)
\\
&\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau
\\
& +\int_{0}^{1}\int_{0}^{1}f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)
\\
&\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau
\\
&+
\int_{0}^{1}\int_{0}^{1} f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right)
\\
&\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau
\\
& +\int_{0}^{1}\int_{0}^{1}f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)
\\
&\times h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}drd\tau \bigg].
\end{align*}
By change of variables \(x^{\rho_{1}}=\frac{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}}{u_1^{\rho_{1}}u_2^{\rho_{1}}}\) and \(y^{\rho_{2}}=\frac{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}{v_1^{\rho_{2}}v_2^{\rho_{2}}}\) and using the symmetric property of \(h\), we find
\begin{align*}
&\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)
\\
&\;\;\;\;\times \int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1} h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy
\\
& \leq \frac{1}{4}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}\left[\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}\right.
\end{align*}
\begin{align*}
&\;\;\;\times x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy\\
&\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1} x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy
\\
&\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}
x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_1^{\rho_{2}}}-y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy
\\
&\;\;\;\left.+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}
x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy \right]
\\
&= \frac{1}{4}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}\left[\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(\frac{1}{u_1^{\rho_{1}}}-x^{\rho_{1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_{2}}}-y^{\rho_{2}} \right)^{\beta-1}\right.
\\
&\;\;\;\times x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right)dxdy
\\
&\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(\frac{1}{u_1^{\rho_{1}}}-x^{\rho_{1}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}
x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{\frac{1}{u_1^{\rho_{1}}}+\frac{1}{u_2^{\rho_{1}}}-x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy
\\
&\;\;\;+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_{2}}}-y^{\rho_{2}} \right)^{\beta-1}
x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{\frac{1}{v_1^{\rho_{2}}}+\frac{2}{v_2^{\rho_{2}}}-y^{\rho_{2}}}\right)dxdy
\\
&\;\;\;\left.+\int_{1/v_2}^{1/v_1}\int_{1/u_2}^{1/u_1} \left(x^{\rho_{1}}-\frac{1}{u_2^{\rho_{1}}} \right)^{\alpha-1}\left(y^{\rho_{2}}-\frac{1}{v_2^{\rho_{2}}} \right)^{\beta-1}
x^{\rho_{1}-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right) h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dxdy\right].
\end{align*}
Thus, we get
\begin{align*}
\begin{split}
\frac{\Gamma(\alpha)\Gamma(\beta)}{\rho_{1}^{1-\alpha}\rho_2^{1-\beta}}&\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)
\\
&\times\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
\leq& \frac{\Gamma(\alpha)\Gamma(\beta)}{4\rho_{1}^{1-\alpha}\rho_2^{1-\beta}}\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_2^{\rho_{1}}-u_1^{\rho_{1}}}\right)^{\alpha}\left( \frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_2^{\rho_{2}}-v_1^{\rho_{2}}}\right)^{\beta}
\\
&\times \left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}(fh\circ g) \left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}(fh\circ g)\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}(fh\circ g)\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right].
\end{split}
\end{align*}
This completes the first inequality of (10). For the second inequality of (10) we use the co-ordinated harmonically convexity of \(f\) as:
\begin{align*}
&f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right)+f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_1^{\rho_{1}}+(1-r^{\rho_{1}})u_2^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\end{align*}
\begin{align}
\label{t1e5}
&+
f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_1^{\rho_{2}}+(1-\tau^{\rho_{2}})v_2^{\rho_{2}}}\right)+f\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\notag
\\
&\leq f(u_1^{\rho_{1}},v_1^{\rho_{2}})+f(u_2^{\rho_{1}},v_1^{\rho_{2}})+f(u_1^{\rho_{1}},v_2^{\rho_{2}})+f(u_2^{\rho_{1}},v_2^{\rho_{2}}).
\end{align}
(12)
Thus multiplying (12) by \(r^{\rho_{1}\alpha-1}\tau^{\rho_{2}\beta-1}h\left( \frac{u_1^{\rho_{1}}u_2^{\rho_{1}}}{r^{\rho_{1}}u_2^{\rho_{1}}+(1-r^{\rho_{1}})u_1^{\rho_{1}}},\frac{v_1^{\rho_{2}}v_2^{\rho_{2}}}{\tau^{\rho_{2}} v_2^{\rho_{2}}+(1-\tau^{\rho_{2}})v_1^{\rho_{2}}}\right)\) and integrating with respect to \((r,\tau)\) over \([0,1]\times [0,1]\), we get the second inequality of (10).
Hence the proof is completed.
Theorem 10.
Let \(\alpha,\beta>0\) and \(\rho_{1},\rho_{2}>0\). Let \(f:\Delta=[u_1^{\rho_{1}},u_2^{\rho_{1}}]\times [v_1^{\rho_{2}},v_2^{\rho_{2}}]\subseteq (0,\infty)\times(0,\infty) \rightarrow
\mathbb{R}\) be a co-ordinated harmonically convex on \(\Delta\), with \(0< u_1< u_2\), \(0< v_1< v_2\) and \(f\in L_1[\Delta]\). If \(h:\Delta\rightarrow
\mathbb{R}\) is nonnegative and harmonically symmetric with respect to \(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}\),\(\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\) on \(\Delta\). Then the following inequalities hold:
\begin{align*}
f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
\leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\bigg[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)
\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \bigg]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\bigg[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \bigg]
\\
&+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\bigg[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \bigg]
\\
\leq& 2\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&+\left.\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
\leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\bigg[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_2^{\rho_{2}}\right)
\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \bigg]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\end{align*}
\begin{align}
\label{tt1e1}
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_2^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag
\\
&+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\notag
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]\notag
\\
\end{align}
\begin{align}
\leq& \frac{f(u_1^{\rho_1},v_1^{\rho_2})+f(u_1^{\rho_1},v_2^{\rho_2})+f(u_2^{\rho_1},v_1^{\rho_2})+f(u_2^{\rho_1},v_2^{\rho_2})}{4}\notag
\\
&\times \left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.\notag
\\
&\left.+^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right],
\end{align}
(13)
where \(g(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right) \), \(g_{1}(x^{\rho_{1}},y^{\rho_{2}})=\left(\frac{1}{x^{\rho_{1}}},y^{\rho_{2}} \right)\) and \(g_{2}(x^{\rho_{1}},y^{\rho_{2}})=\left(x^{\rho_{1}},\frac{1}{y^{\rho_{2}}} \right) \), respectively.
Proof.
Since \(f\) is co-ordinated harmonically convex on \(\Delta\), then the function \(f_{1/x^{\rho_{1}}}:[v_1^{\rho_{2}},v_2^{\rho_{2}}]\rightarrow \mathbb{R}\), defined by \(f_{1/x^{\rho_{1}}}(y^{\rho_{2}})=f(\frac{1}{x^{\rho_{1}}},y^{\rho_{2}})\) is harmonically convex on \([v_1^{\rho_{2}},v_2^{\rho_{2}}]\) for all \(x^{\rho_{1}}\in \left[ \frac{1}{u_2^{\rho_{1}}},\frac{1}{u_1^{\rho_{1}}}\right] \). Then from (6), we have
\begin{align}
\label{tte2}
\frac{\rho_2^{1-\beta}}{\Gamma(\beta)}f&\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right) \left[\int_{1/v_2}^{1/v_1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dy\right.\notag
\\
&\;\;\;\;\left.+\int_{1/v_2}^{1/v_1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dy\right]\notag
\\
&\leq \frac{\rho_2^{1-\beta}}{\Gamma(\beta)}\left[\int_{1/v_2}^{1/v_1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} y^{\rho_{2}-1}fh\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dy\right.\notag\\
&\;\;\;\;\left.+\int_{1/v_2}^{1/v_1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} y^{\rho_{2}-1}fh\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dy\right]\notag
\\
&\leq\frac{f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right) +f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right) }{2}\left[\int_{1/v_2}^{1/v_1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dy\right.\notag
\\
&\;\;\;\;\left.+\int_{1/v_2}^{1/v_1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dy\right].
\end{align}
(14)
Multiplying both sides of (14) by \(\frac{x^{\rho_1-1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}}{\rho_1^{\alpha-1}\Gamma(\alpha)}\) and \(\frac{x^{\rho_1-1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}}{\rho_1^{\alpha-1}\Gamma(\alpha)}\), and integrating with respect to \(x\) over \(\left[ \frac{1}{u_2},\frac{1}{u_1}\right] \), respectively, we get
\begin{align}
\label{tte3}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)\right.\notag
\\
&\times h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}\notag
\\
&\left.\times f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1} y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1} y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)} \left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\times x^{\rho_1-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\notag
\\
&\times \left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\notag
\\
&\left.+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}
f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right],
\end{align}
(15)
and
\begin{align*}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)\right.
\\
&\times h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}
\\
&\times \left.f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]
\\
\leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1} y^{\rho_{2}-1}\right.
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}
\\
&\left.\times x^{\rho_1-1} y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]
\end{align*}
\begin{align}
\label{tte4}
\leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)} \left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\times x^{\rho_1-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\notag
\\
&\times \left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\notag
\\
&+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\notag
\\
&\left.\times f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right].
\end{align}
(16)
Using similar arguments for the mapping \(f_{\frac{1}{y^{\rho_2}}}:[u_1^{\rho_1},u_2^{\rho_1}]\rightarrow \mathbb{R}\), \(f_{\frac{1}{y^{\rho_{2}}}}(x^{\rho_{1}})=f(x^{\rho_{1}},\frac{1}{y^{\rho_{2}}})\), we have
\begin{align}
\label{tte5}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right)\right.\notag
\\
&\times h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}\notag
\\
&\left.\times f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1} y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1} y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)} \left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left(u_1^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}\notag
\\
&\times x^{\rho_1-1}y^{\rho_{2}-1}f\left( u_1^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\notag
\\
&\times \left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}f\left( u_2^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\notag
\\
&+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\notag
\\
&\left.\times f\left(u_2^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right],
\end{align}
(17)
and
\begin{align}
\label{tte6}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right)\right.\notag
\\
&\times h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}\notag
\\
&\left.\times f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1} y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2}\right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1} y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)} \left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left(u_1^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\times x^{\rho_1-1}y^{\rho_{2}-1}f\left( u_1^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\notag
\\
&\times \left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}f\left( b^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\notag
\\
&\left.+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1} f\left(u_2^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right].
\end{align}
(18)
By adding the inequalities (15)\(\sim\)(18), we get
\begin{align*}
^{\rho_{1}}I^{\alpha}_{1/u_1-}&\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\&+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)
\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\end{align*}
\begin{align*}
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
\leq &2\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}fh\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}fh\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
\leq &\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},v_2^{\rho_{2}}\right)\right.
\\
&\left.\times ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_1^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},v_2^{\rho_{2}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right)\right.
\\
&\left.\times ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_1^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(u_2^{\rho_{1}}, \frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&\left.\times ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right].
\end{align*}
This completes the second and third inequality of (13).
Now, using the first inequality of (6), we find
\begin{align}
\label{tt1e8}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}&f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}
{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\right.\notag
\\
&\times x^{\rho_{1}-1} y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\times \left.x^{\rho_{1}-1}y^{\rho_{2}-1} h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq&\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}\right.\notag
\\
&\times f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\times \left.x^{\rho_{1}-1}y^{\rho_{2}-1} f\left(\frac{1}{x^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right].
\end{align}
(19)
\begin{align*}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}f&\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}
{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\right.
\\
&\times x^{\rho_{1}-1} y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}
\\
&\times\left.x^{\rho_{1}-1}y^{\rho_{2}-1}h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\\
\leq&\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}\right.\notag
\end{align*}
\begin{align}
\label{tt1e9}
&\times f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}\notag
\\
&\times \left.x^{\rho_{1}-1}y^{\rho_{2}-1}f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{1}{y^{\rho_{2}}} \right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right].
\end{align}
(20)
Adding (19) and (20) and using the fact that \(h\) is symmetric, we get
\begin{align*}
\begin{split}
f\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)&\left[\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_1-}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right)
+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_1-,1/v_2+}h\circ g\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right)\right.
\\
&\left.+\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_1-}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) +\ ^{\rho_{1},\rho_{2}}I^{\alpha,\beta}_{1/u_2+,1/v_2+}h\circ g\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
\leq& \ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}\left[ f\circ g_{1}\left( \frac{1}{u_2^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)
\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_1-}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}\left[ f\circ g_{1}\left( \frac{1}{u_1^{\rho_{1}}},\frac{2v_1^{\rho_{2}}v_2^{\rho_{2}}}{v_1^{\rho_{2}}+v_2^{\rho_{2}}}\right)\ ^{\rho_{2}}I^{\beta}_{1/v_2+}h\circ g_2\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+ \ ^{\rho_{2}}I^{\beta}_{1/v_1-}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_2^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_2^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_1-}h\circ g_1\left( \frac{1}{u_2^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right]
\\
&+\ ^{\rho_{2}}I^{\beta}_{1/v_2+}\left[ f\circ g_{2}\left(\frac{2u_1^{\rho_{1}}u_2^{\rho_{1}}}{u_1^{\rho_{1}}+u_2^{\rho_{1}}}, \frac{1}{v_1^{\rho_{2}}}\right)
\ ^{\rho_{1}}I^{\alpha}_{1/u_2+}h\circ g_1\left( \frac{1}{u_1^{\rho_{1}}},\frac{1}{v_1^{\rho_{2}}}\right) \right].
\end{split}
\end{align*}
This completes the first inequality of (13). Now, to achieve the last inequality of (13), applying the second inequality of (6) as:
\begin{align}
\label{tt1e10}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},v_2^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\frac{f(u_1^{\rho_1},v_2^{\rho_2})+f(u_2^{\rho_1},v_2^{\rho_2})}{2}\notag
\\
&\times\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right.\notag
\\
&\left.+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right],
\end{align}
(21)
\begin{align}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1}y^{\rho_{2}-1}f\left( \frac{1}{x^{\rho_{1}}},v_1^{\rho_2}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq &\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\frac{f(u_1^{\rho_1},v_1^{\rho_2})+f(u_2^{\rho_1},v_1^{\rho_2})}{2}\notag
\\
&\times\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right.\notag
\\
&\left.+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right],
\end{align}
(22)
\begin{align}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( u_2^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1}y^{\rho_{2}-1}f\left(u_2^{\rho_1}, \frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\frac{f(u_2^{\rho_1},v_1^{\rho_2})+f(u_2^{\rho_1},v_2^{\rho_2})}{2}\notag
\\
&\times\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right.\notag
\\
&\left.+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(x^{\rho_1}-\frac{1}{u_2^{\rho_1}} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right],
\end{align}
(23)
and
\begin{align}
\label{tt1e11}
\frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{2\Gamma(\alpha)\Gamma(\beta)}&\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1}x^{\rho_1-1}y^{\rho_{2}-1}\right.\notag
\\
&\times f\left( u_1^{\rho_1},\frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx
+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}\notag
\\
&\left.\times x^{\rho_1-1}y^{\rho_{2}-1}f\left(u_1^{\rho_1}, \frac{1}{y^{\rho_{2}}}\right)h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right]\notag
\\
\leq& \frac{\rho_1^{1-\alpha}\rho_2^{1-\beta}}{\Gamma(\alpha)\Gamma(\beta)}\frac{f(u_1^{\rho_1},v_1^{\rho_2})+f(u_1^{\rho_1},v_2^{\rho_2})}{2}\notag
\\
&\times\left[\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(y^{\rho_2}-\frac{1}{v_2^{\rho_2}} \right)^{\beta-1} x^{\rho_{1}-1} y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right.\notag
\\
&\left.+\int_{1/u_2}^{1/u_1}\int_{1/v_2}^{1/v_1}\left(\frac{1}{u_1^{\rho_1}}-x^{\rho_1} \right)^{\alpha-1}\left(\frac{1}{v_1^{\rho_2}}-y^{\rho_2} \right)^{\beta-1}x^{\rho_{1}-1}y^{\rho_{2}-1}
h\left( \frac{1}{x^{\rho_{1}}},\frac{1}{y^{\rho_{2}}}\right)dydx\right].
\end{align}
(24)
By adding the inequalities (21)\(\sim\)(24), we get the last inequality of (13).
Remark 3.
- 1)   From Theorems 9 and 10, we can get new Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via Riemann-Liouville fractional integral by taking \(\rho_1=\rho_2=1\).
- 2)   From Theorems 9 and 10, we can get new Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via classical integral by taking \(\rho_1=\rho_2=1\) and \(\alpha=\beta=1\).
4. Conclusion
In this paper, firstly we established the Hermite-Hadamard-Fejér type inequalities for harmonically convex function in one dimension which is further used to establish the Hermite-Hadamard-Fejér type inequalities for harmonically convex function via Katugampola fractional integral. The results provided in our paper are the generalizations of some earlier results.
Acknowledgments
This research is supported by National University of Science and Technology(NUST), Islamabad, Pakistan.
Author 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.
