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\begin{document}
\title[\textbf{The (}$k,s,h$)-\textbf{Riemann-Liouville and the (}$k,s$)-%
\textbf{Hadamard Operators}]{\textbf{The (}$k,s,h$)\textbf{$-$%
Riemann-Liouville and the (}$k,s$)-\textbf{Hadamard Operators: New
Applications}}
\author{Mohamed BEZZIOU}
\address{Laboratory FIMA, UDBKM, University of Khemis Miliana, Algeria}
\email{ m.bezziou@yahoo.fr}
\author{Zoubir DAHMANI}
\address{LPAM, Faculty SEI, UMAB University, Algeria}
\email{zzdahmani@yahoo.fr}
\author{Mehmet Zeki SARIKAYA}
\address{Department of Mathematics, \ Faculty of Science and Arts, D\"{u}zce
University, D\"{u}zce-TURKEY}
\email{sarikayamz@gmail.com}
\keywords{\textbf{\thanks{\textbf{2010 Mathematics Subject Classification.}
26A33, 26D10, 24D15.} }$\left( k,s,h\right) -$Riemann-Liouville fractional
integral, $\left( k,h\right) -$Hadamard fractional operator, Chebyshev's
functional, Gruss inequality.\\
}

\begin{abstract}
This paper deals with new results on Gruss inequality by using recent
fractional integral operators. In fact, based on the $\left( k,s,h\right) -$%
Riemann-Liouville and the $\left( k,h\right) -$Hadamard fractional
operators, we establish several integral results. For our results, some very
recent results on the paper: [A Gr\"{u}ss type inequality for two weighted
functions. J. Math. Computer Sci., 2018.] can be deduced as some special
cases.
\end{abstract}

\maketitle

\section{Introduction}

The integral inequalities are very important in many areas of science,
especially in mathematics, physics, chemistry, biology. Many researchers
have given a lot of attention to the generalization of fractional integral
inequalities related to weighted Chebyshev functional. In fact, they
established many results to Gr\"{u}ss and Chebyshev inequalities. For more
details, we refer the reader to \cite{2,4,5,6,7,8,9,11,13,14,15} and the
references therein.\newline
Let us now cite some recent work that have motivated the present paper. We
begin by the paper \cite{1}, where the authors introduced two new fractional
integral operators: the first one is the $(k,s,h)$-Riemann-Liouville
fractional integral (for a function $f$\ $\in L^{1}\left( \left[ a,b\right]
\right) $ with respect to another measurable, increasing, positive function%
\emph{\ }$h$ with\emph{\ }$h^{\prime }\in C^{1}\left( \left[ a,b\right]
\right) $). It is given by%
\begin{equation}
\text{ }_{k}^{s}J_{a,h}^{\alpha }\left( f\left( t\right) \right) =\frac{%
\left( s+1\right) ^{1-\frac{\alpha }{k}}}{k\Gamma _{k}\left( \alpha \right) }%
\int_{a}^{t}\left( h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right)
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) f\left( \tau \right) d\tau ,  \label{8}
\end{equation}%
where $\Gamma _{k}\left( \alpha \right) =\int_{0}^{\infty }$ $t^{\alpha
-1}e^{-\frac{t^{k}}{k}}dt,\  \alpha >0,\ k>0,\ s\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
-\left \{ -1\right \} .$

The second introduced operator of the paper \cite{1} is the $(k,h)$-Hadamard
fractional integral (of $f\in L^{1}\left( \left[ a,b\right] \right) $ with
respect to $h$). It is defined for $k>0$ by%
\begin{equation}
\text{ }_{k}I_{a,h}^{\alpha }\left( f\left( t\right) \right) =\frac{1}{%
k\Gamma _{k}\left( \alpha \right) }\int_{a}^{t}\left( \log \frac{h\left(
t\right) }{h\left( \tau \right) }\right) ^{\frac{\alpha }{k}-1}\frac{%
h^{\prime }\left( \tau \right) }{h\left( \tau \right) }f\left( \tau \right)
d\tau .  \label{99}
\end{equation}%
It is important to note that based on these two operators, we can state that:

\begin{proposition}
\begin{equation}
\underset{s\longrightarrow -1^{+}}{\lim }\text{ }_{k}^{s}J_{a,h}^{\alpha
}\left( f\left( t\right) \right) =_{k}I_{a,h}^{\alpha }\left( f\left(
t\right) \right)   \label{303}
\end{equation}
\end{proposition}

Now, by considering the weighted functional (see \cite{12}): 
\begin{eqnarray}
&&T\left( f,g,p,q\right)
=\int_{a}^{b}p(x)\int_{a}^{b}q(x)f(x)g(x)dx+\int_{a}^{b}q(x)%
\int_{a}^{b}p(x)f(x)g(x)dx  \label{33} \\
&&-\int_{a}^{b}p(x)f(x)dx\int_{a}^{b}q(x)g(x)dx-\int_{a}^{b}q(x)f(x)dx%
\int_{a}^{b}p(x)g(x)dx  \notag
\end{eqnarray}%
where $f$ and $g$ are two real-valued integrable functions which are
synchronous on $[a,b]$, i.e.: 
\begin{equation}
\left( f(x)-f(y)\right) \left( g(x)-g(y)\right) \geq 0,\text{ for any }%
x,y\in \left[ a,b\right] ,  \label{1}
\end{equation}%
and $p,q$ are two positive integrable functions on a finite interval $[a,b].$
By considering the above functional, we can observe that 
\begin{equation}
T\left( f,g,p,q\right) =\int_{a}^{b}\int_{a}^{b}\left( f(\tau )-f(\rho
)\right) \left( g(\tau )-g(\rho )\right) p\left( \tau \right) q\left( \rho
\right) d\tau d\rho .  \label{001}
\end{equation}%
We continue by citing the work that has motivated this paper. We observe
that in \cite{3}, J. Choi proved the following interesting result:

\begin{theorem}
Let $f,g:\left[ a,b\right] \rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be integrable functions, such that 
\begin{equation}
m\leq f(x)\leq M,n\leq g(x)\leq N;m,M,n,N\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,\text{ }x\in \left[ a,b\right] .  \label{77}
\end{equation}%
If $p,q:\left[ a,b\right] \rightarrow \left[ 0,\infty \right[ $ are two
integrable functions on $[a,b]$ that satisfy 
\begin{equation*}
\min \left \{ \int_{a}^{b}p(t)dt,\int_{a}^{b}q(t)dt\right \} >0,
\end{equation*}%
then,%
\begin{eqnarray}
&&\left \vert T\left( f,g,p,q\right) \right \vert \leq \left[ \left( \frac{%
(M-m)^{2}}{2}+2\left \Vert f\right \Vert _{\infty }\right) \left( \frac{%
(N-n)^{2}}{2}+2\left \Vert g\right \Vert _{\infty }\right) \right] ^{\frac{1}{2%
}}  \label{112} \\
&&\times \left( \int_{a}^{b}p(t)dt\right) \left( \int_{a}^{b}q(t)dt\right) ,
\notag
\end{eqnarray}%
where 
\begin{equation*}
\left \Vert f\right \Vert _{\infty }=\underset{x\in \left[ a,b\right] }{\sup }%
\left \vert f\left( x\right) \right \vert \text{ and }\left \Vert g\right \Vert
_{\infty }=\underset{x\in \left[ a,b\right] }{\sup }\left \vert g\left(
x\right) \right \vert .
\end{equation*}
\end{theorem}

\noindent Based on the two fractional operators of \cite{1}, we establish
several integral results related to Gr\"{u}ss inequality. We begin by
proving an $\alpha $-theorem. Then, using two $\alpha $-$\beta $-auxiliary
results ( lemmas), we establish two more general theorems. For our results,
the above theorem of \cite{3} is deduced as a special case.

\section{Main Results}

The first main result is given by the following $\alpha $-theorem. It
depends on one fractional integral parameter.

\begin{theorem}
Let $f$ and $g$ be two integrable functions on $\left[ a,b\right] $
satisfying the condition (\ref{77}), let $p$ and $q$ be two positive
functions on $\left[ a,b\right] $ and let $h$ be a measurable, increasing,
positive function on $\left( a,b\right] $ and $h\in C^{1}\left( \left[ a,b%
\right] \right) .$ Then, the following inequality holds%
\begin{eqnarray}
&&\left \vert _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}I_{a,h}^{\alpha }\left[ qfg(t)\right] +_{k}I_{a,h}^{\alpha }\left[ q(t)%
\right] \text{ }_{k}I_{a,h}^{\alpha }\left[ pfg(t)\right] \right.   \notag \\
&&-\text{ }\left. _{k}I_{a,h}^{\alpha }\left[ pf(t)\right]
_{k}I_{a,h}^{\alpha }\left[ qg(t)\right] -\text{ }_{k}I_{a,h}^{\alpha }\left[
qf(t)\right] _{k}I_{a,h}^{\alpha }\left[ pg(t)\right] \right \vert 
\label{67} \\
&\leq &\left[ \left( \frac{\left( M-m\right) ^{2}}{2}+2\left \Vert
f\right \Vert _{\infty }\right) \left( \frac{\left( N-n\right) ^{2}}{2}%
+2\left \Vert g\right \Vert _{\infty }\right) \right] ^{\frac{1}{2}}  \notag \\
&&\times \left( _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \right) \left(
_{k}I_{a,h}^{\alpha }\left[ q(t)\right] \right) ,  \notag
\end{eqnarray}%
where $f,g\in L_{\infty }\left[ a,b\right] ,$ $\alpha >0$ and $k>0.$
\end{theorem}

We need the following lemma to prove Theorem 3.

\begin{lemma}
Let $\Phi $ be an integrable function on $\left[ a,b\right] $ satisfying the
condition $m\leq \Phi \left( x\right) \leq M$ on $\left[ a,b\right] $ and
let $p,q$ be two positive functions on $\left[ a,b\right] $ and let $h$ be a
measurable, increasing, positive function on $\left( a,b\right] $ with $h\in
C^{1}\left( \left[ a,b\right] \right) .$ Then for all $t>0,$ we have 
\begin{eqnarray}
&&_{k}I_{a,h}^{\alpha }\left[ p(t)\right] \text{ }_{k}I_{a,h}^{\alpha }\left[
q\Phi ^{2}(t)\right] +_{k}I_{a,h}^{\alpha }\left[ q(t)\right] \text{ }%
_{k}I_{a,h}^{\alpha }\left[ p\Phi ^{2}(t)\right]   \notag \\
&&-2_{k}I_{a,h}^{\alpha }\left[ q\Phi (t)\right] \text{ }_{k}I_{a,h}^{\alpha
}\left[ p\Phi (t)\right]   \notag \\
&=&\left( \ _{k}I_{a,h}^{\alpha }\left[ \left( M-\Phi (t)\right) q(t)\right]
\right) \left( \ _{k}I_{a,h}^{\alpha }\left[ \left( \Phi (t)-m\right) p(t)%
\right] \right)   \label{010} \\
&&+\left( \ _{k}I_{a,h}^{\alpha }\left[ \left( M-\Phi (t)\right) p(t)\right]
\right) \left( \ _{k}I_{a,h}^{\alpha }\left[ \left( \Phi (t)-m\right) q(t)%
\right] \right)   \notag \\
&&-\left( \ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \right) \left( \
_{k}I_{a,h}^{\alpha }\left[ \left( M-\Phi (t)\right) \left( \Phi
(t)-m\right) p(t)\right] \right)   \notag \\
&&-\left( \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \right) \left( \
_{k}I_{a,h}^{\alpha }\left[ \left( M-\Phi (t)\right) \left( \Phi
(t)-m\right) q(t)\right] \right) ,  \notag
\end{eqnarray}%
where $k>0,$ $s\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
-\left \{ -1\right \} ,$ $\alpha >0.$
\end{lemma}

\begin{proof}
Let $\Phi $ be an integrable function on $\left[ a,b\right] $ satisfying the
condition (\ref{77}) on $\left[ a,b\right] .$ For any $\tau ,\rho $ $\in %
\left[ a,b\right] $, we have the following identity%
\begin{eqnarray}
&&\Phi ^{2}\left( \tau \right) +\Phi ^{2}(\rho )-2\Phi (\tau )\Phi (\rho ) 
\notag \\
&=&\left[ M-\Phi (\rho )\right] \left[ \Phi (\tau )-m\right] +\left[ M-\Phi
(\tau )\right] \left[ \Phi (\rho )-m\right]   \label{015} \\
&&-\left[ M-\Phi (\tau )\right] \left[ \Phi (\tau )-m\right] -\left[ M-\Phi
(\rho )\right] \left[ \Phi (\rho )-m\right] .  \notag
\end{eqnarray}%
We also consider the quantities: 
\begin{equation}
\left \{ 
\begin{array}{l}
\ _{k}^{s}H_{\alpha ,h}\left( t,\tau \right) =\frac{\left( s+1\right) ^{1-%
\frac{\alpha }{k}}}{k\Gamma _{k}\left( \alpha \right) }\left( h^{s+1}\left(
t\right) -h^{s+1}\left( \tau \right) \right) ^{\frac{\alpha }{k}%
-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right) p\left( \tau
\right)  \\ 
\ _{k}^{s}H_{\alpha ,h}\left( t,\rho \right) =\frac{\left( s+1\right) ^{1-%
\frac{\alpha }{k}}}{k\Gamma _{k}\left( \alpha \right) }\left( h^{s+1}\left(
t\right) -h^{s+1}\left( \rho \right) \right) ^{\frac{\alpha }{k}%
-1}h^{s}\left( \rho \right) h^{\prime }\left( \rho \right) q\left( \rho
\right) .%
\end{array}%
\right.   \label{222}
\end{equation}%
Multiplying (\ref{015}) by $\ _{k}^{s}H_{\alpha ,h}\left( t,\tau \right)
\times \ _{k}^{s}H_{\alpha ,h}\left( t,\rho \right) ,\  \left( \tau ,\rho
\right) \in \left( a,t\right) ^{2},\ s\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
-\left \{ -1\right \} $ and integrating with respect to $\tau $ and $\rho $
over $\left( a,t\right) ^{2},$ we get%
\begin{eqnarray*}
&&\frac{1}{k^{2}\Gamma _{k}^{2}\left( \alpha \right) }\left \{
\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \Phi ^{2}(\tau )\right. \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \Phi ^{2}(\rho )\right]
d\tau d\rho  \\
&&-2\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \Phi (\tau )\right.  \\
&&\times \left. \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left(
\rho \right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \Phi \left( \rho \right) %
\right] d\tau d\rho \right \} 
\end{eqnarray*}%
\begin{eqnarray*}
&=&\frac{1}{k^{2}\Gamma _{k}^{2}\left( \alpha \right) }\left \{
\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) \left( \Phi (\tau )-m\right) p\left( \tau \right) \right.
\right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( M-\Phi (\rho )\right) q\left( \rho
\right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) \left( M-\Phi (\tau )\right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( \Phi (\rho )-m\right) q\left( \rho
\right) \right] d\tau d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) \left( M-\Phi (\tau )\right) \left( \Phi (\tau )-m\right)
p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \right.  \\
&&\times \left. \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left(
\rho \right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( M-\Phi (\rho )\right) \left( \Phi
(\rho )-m\right) q\left( \rho \right) \right] d\tau d\rho \right \} .
\end{eqnarray*}%
Therefore, it yields that 
\begin{eqnarray*}
&&\frac{1}{k^{2}\Gamma _{k}^{2}\left( \alpha \right) }\left \{
\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right)
p\left( \tau \right) \Phi ^{2}(\tau )\right. \right.  \\
&&\times \left. \left( \log \frac{h\left( t\right) }{h\left( \rho \right) }%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime }\left(
\rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right)
p\left( \tau \right) \right.  \\
&&\times \left. \left( \log \frac{h\left( t\right) }{h\left( \rho \right) }%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime }\left(
\rho \right) q\left( \rho \right) \Phi ^{2}(\rho )\right] d\tau d\rho  \\
&&-2\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right)
p\left( \tau \right) \Phi (\tau )\right.  \\
&&\times \left. \left. \left( \log \frac{h\left( t\right) }{h\left( \rho
\right) }\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime
}\left( \rho \right) q\left( \rho \right) \Phi \left( \rho \right) \right]
d\tau d\rho \right \} 
\end{eqnarray*}%
\begin{eqnarray*}
&=&\frac{1}{k^{2}\Gamma _{k}^{2}\left( \alpha \right) }\left \{
\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right) \left(
\Phi (\tau )-m\right) p\left( \tau \right) \right. \right.  \\
&&\times \left. \left( \log \frac{h\left( t\right) }{h\left( \rho \right) }%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime }\left(
\rho \right) \left( M-\Phi (\rho )\right) q\left( \rho \right) \right] d\tau
d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right) \left(
M-\Phi (\tau )\right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \log \frac{h\left( t\right) }{h\left( \rho \right) }%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime }\left(
\rho \right) \left( \Phi (\rho )-m\right) q\left( \rho \right) \right] d\tau
d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right) \left(
M-\Phi (\tau )\right) \left( \Phi (\tau )-m\right) p\left( \tau \right)
\right.  \\
&&\times \left. \left( \log \frac{h\left( t\right) }{h\left( \rho \right) }%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime }\left(
\rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \log \frac{h\left( t\right) }{h\left( \tau \right) }\right) ^{\frac{%
\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau \right)
p\left( \tau \right) \right.  \\
&&\times \left. \left. \left( \log \frac{h\left( t\right) }{h\left( \rho
\right) }\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right) h^{\prime
}\left( \rho \right) \left( M-\Phi (\rho )\right) \left( \Phi (\rho
)-m\right) q\left( \rho \right) \right] d\tau d\rho \right \} .
\end{eqnarray*}%
This completes the proof of the above lemma.
\end{proof}

Let us now prove Theorem 3.

\begin{proof}
We consider the functional 
\begin{equation}
G\left( \tau ,\rho \right) =\left( f\left( \tau \right) -f\left( \rho
\right) \right) \left( g\left( \tau \right) -g\left( \rho \right) \right)
,\tau ,\rho \in \left( a,t\right) ,t\in \left( a,b\right] ,  \label{100}
\end{equation}%
where $f$ and $g$ be two integrable functions on $\left[ a,b\right] $
satisfying the condition (\ref{77}).\newline
Multiplying (\ref{100}) by $\ _{k}^{s}H_{\alpha ,h}\left( t,\tau \right)
\times \ _{k}^{s}H_{\alpha ,h}\left( t,\rho \right) ,\  \tau ,\rho \in \left(
a,t\right) ,$ and integrating the resulting identity with respect to $\tau $
and $\rho $ over $\left( a,t\right) ^{2},$ we have%
\begin{eqnarray}
&&\int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau \right) 
\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\rho \right) \left( f\left( \tau
\right) -f\left( \rho \right) \right) \left( g\left( \tau \right) -g\left(
\rho \right) \right) d\tau d\rho   \notag \\
&=&\text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \
_{k}^{s}J_{a,h}^{\alpha }\left[ qfg(t)\right] +\text{ }_{k}^{s}J_{a,h}^{%
\alpha }\left[ q(t)\right] \ _{k}^{s}J_{a,h}^{\alpha }\left[ pfg(t)\right] 
\label{101} \\
&&-\text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ pf(t)\right] \
_{k}^{s}J_{a,h}^{\alpha }\left[ qg(t)\right] -_{k}^{s}J_{a,h}^{\alpha }\left[
qf(t)\right] \ _{k}^{s}J_{a,h}^{\alpha }\left[ pg(t)\right] .  \notag
\end{eqnarray}%
Thanks to the Cauchy Schwarz integral inequality for double integrals, we get%
\begin{eqnarray}
&&\left[ \int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau
\right) \text{ }_{k}^{s}H_{\alpha ,h}\left( t,\rho \right) \left( f\left(
\tau \right) -f\left( \rho \right) \right) \left( g\left( \tau \right)
-g\left( \rho \right) \right) d\tau d\rho \right] ^{2}  \notag \\
&\leq &\int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau
\right) \text{ }_{k}^{s}H_{\alpha ,h}\left( t,\rho \right) \left( f\left(
\tau \right) -f\left( \rho \right) \right) ^{2}d\tau d\rho   \label{102} \\
&&\times \int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau
\right) \text{ }_{k}^{s}H_{\alpha ,h}\left( t,\rho \right) \left( g\left(
\tau \right) -g\left( \rho \right) \right) ^{2}d\tau d\rho .  \notag
\end{eqnarray}%
Then, we obtain 
\begin{eqnarray}
&&\int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau \right) 
\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\rho \right) \left[ f\left( \tau
\right) -f\left( \rho \right) \right] ^{2}d\tau d\rho   \label{88} \\
&=&\ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \ _{k}^{s}J_{a,h}^{\alpha }%
\left[ pf^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \
_{k}^{s}J_{a,h}^{\alpha }\left[ qf^{2}(t)\right]   \notag \\
&&-2\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ pf(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ qf(t)\right] \right)   \notag
\end{eqnarray}%
\begin{eqnarray}
&=&\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-f(t)\right) q(t)\right]
\right) \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( f(t)-m\right) p(t)%
\right] \right)   \notag \\
&&+\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-f(t)\right) p(t)\right]
\right) \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( f(t)-m\right) q(t)%
\right] \right)   \notag \\
&&-\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-f(t)\right) \left( f(t)-m\right)
p(t)\right] \right)   \notag \\
&&-\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-f(t)\right) \left( f(t)-m\right)
q(t)\right] \right) ,\text{ }  \notag
\end{eqnarray}%
where 
\begin{eqnarray}
&&\left( _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \right) \left(
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-f(t)\right) \left( f(t)-m\right)
p(t)\right] \right)   \label{017} \\
&&+\left( _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \right) \left(
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-f(t)\right) \left( f(t)-m\right)
q(t)\right] \right) \geq 0,\text{ }  \notag
\end{eqnarray}%
and%
\begin{eqnarray}
&&\int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau \right) 
\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\rho \right) \left[ g\left( \tau
\right) -g\left( \rho \right) \right] ^{2}d\tau d\rho   \label{00} \\
&=&\ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \ _{k}^{s}J_{a,h}^{\alpha }%
\left[ pg^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \
_{k}^{s}J_{a,h}^{\alpha }\left[ qg^{2}(t)\right]   \notag \\
&&-2\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ pg(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ qg(t)\right] \right)   \notag
\end{eqnarray}%
\begin{eqnarray*}
&=&\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-g(t)\right) q(t)\right]
\right) \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( g(t)-m\right) p(t)%
\right] \right)  \\
&&+\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-g(t)\right) p(t)\right]
\right) \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ \left( g(t)-m\right) q(t)%
\right] \right)  \\
&&-\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-g(t)\right) \left( g(t)-m\right)
p(t)\right] \right)  \\
&&-\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-g(t)\right) \left( g(t)-m\right)
q(t)\right] \right) ,\text{ }
\end{eqnarray*}%
such that%
\begin{eqnarray}
&&\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-g(t)\right) \left( g(t)-m\right)
p(t)\right] \right)   \label{016} \\
&&+\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-g(t)\right) \left( g(t)-m\right)
q(t)\right] \right) \geq 0.\text{ }  \notag
\end{eqnarray}%
Now, by Lemma 4 and thanks to (\ref{77}) and (\ref{017}), the following
inequality holds: 
\begin{eqnarray}
&&\frac{\ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)}{2}%
\left \{ \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ qf^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\alpha }%
\left[ q(t)\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ pf^{2}(t)\right]
\right.   \notag \\
&&\left. -2\ _{k}^{s}J_{a,h}^{\alpha }\left[ qf(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ pf(t)\right] \right \}   \label{018} \\
&\leq &-Mm\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha
}q(t)\right] ^{2}+\frac{M\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t)\right] }{2}\left[ \ _{k}^{s}J_{a,h}^{\alpha
}p(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t)\right.   \notag \\
&&+\left. \ _{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)
\right] +\frac{m\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t)\right] }{2}\left[ \ _{k}^{s}J_{a,h}^{\alpha
}p(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t)\right.   \notag \\
&&+\left. \ _{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)
\right] -\ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)\
_{k}^{s}J_{a,h}^{\alpha }pf(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t).  \notag
\end{eqnarray}%
Therefore, we obtain%
\begin{eqnarray*}
&&\frac{\ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)}{2}%
\left \{ \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ qf^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\alpha }%
\left[ q(t)\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ pf^{2}(t)\right]
\right.  \\
&&\left. -2\ _{k}^{s}J_{a,h}^{\alpha }\left[ qf(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ pf(t)\right] \right \}  \\
&\leq &\left( M\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t)\right] -\frac{1}{2}\left[ \
_{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t)+\
_{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)\right] \right) 
\\
&&\times \left( \frac{1}{2}\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }qf(t)+\ _{k}^{s}J_{a,h}^{\alpha }q(t)\
_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] -m\left[ \ _{k}^{s}J_{a,h}^{\alpha
}p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)\right] \right)  \\
&&+\frac{1}{4}\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }qf(t)+\ _{k}^{s}J_{a,h}^{\alpha }q(t)\
_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] ^{2} \\
&&-\ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t) \\
&\leq &\left( M\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t)\right] -\frac{1}{2}\left[ \
_{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t)+\
_{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)\right] \right) 
\\
&&\times \left( \frac{1}{2}\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }qf(t)+\ _{k}^{s}J_{a,h}^{\alpha }q(t)\
_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] -m\left[ \ _{k}^{s}J_{a,h}^{\alpha
}p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)\right] \right)  \\
&&+\frac{1}{2}\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }qf(t)+\ _{k}^{s}J_{a,h}^{\alpha }q(t)\
_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] ^{2} \\
&&-\ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t).
\end{eqnarray*}%
Then, we get 
\begin{eqnarray}
&&\frac{_{k}^{s}J_{a,h}^{\alpha }p(t)_{k}^{s}J_{a,h}^{\alpha }q(t)}{2}%
\left \{ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ qf^{2}(t)\right] +_{k}^{s}J_{a,h}^{\alpha }%
\left[ q(t)\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ pf^{2}(t)\right]
\right.   \notag \\
&&\left. -2_{k}^{s}J_{a,h}^{\alpha }\left[ qf(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ pf(t)\right] \right \}   \label{020} \\
&\leq &\left( M\left[ _{k}^{s}J_{a,h}^{\alpha }p(t)_{k}^{s}J_{a,h}^{\alpha
}q(t)\right] -\frac{1}{2}\left[ _{k}^{s}J_{a,h}^{\alpha
}p(t)_{k}^{s}J_{a,h}^{\alpha }qf(t)+_{k}^{s}J_{a,h}^{\alpha
}q(t)_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] \right)   \notag \\
&&\times \left( \frac{1}{2}\left[ _{k}^{s}J_{a,h}^{\alpha
}p(t)_{k}^{s}J_{a,h}^{\alpha }qf(t)+_{k}^{s}J_{a,h}^{\alpha
}q(t)_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] -m\left[ _{k}^{s}J_{a,h}^{\alpha
}p(t)_{k}^{s}J_{a,h}^{\alpha }q(t)\right] \right)   \notag \\
&&+\frac{1}{2}\left[ _{k}^{s}J_{a,h}^{\alpha }p(t)_{k}^{s}J_{a,h}^{\alpha
}qf(t)\right] ^{2}+\frac{1}{2}\left[ _{k}^{s}J_{a,h}^{\alpha
}q(t)_{k}^{s}J_{a,h}^{\alpha }pf(t)\right] ^{2}.  \notag
\end{eqnarray}%
Now, using the fact that $4xy\leq (x+y)^{2}$ for all $x,y\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ and using also the two following inequalities 
\begin{eqnarray}
\left[ \ _{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }qf(t)\right]
^{2} &\leq &\left \Vert f\right \Vert _{\infty }^{2}\left[ \
_{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)\right] ^{2},
\label{021} \\
\left[ \ _{k}^{s}J_{a,h}^{\alpha }q(t)\ _{k}^{s}J_{a,h}^{\alpha }pf(t)\right]
^{2} &\leq &\left \Vert f\right \Vert _{\infty }^{2}\left[ \
_{k}^{s}J_{a,h}^{\alpha }p(t)\ _{k}^{s}J_{a,h}^{\alpha }q(t)\right] ^{2}, 
\notag
\end{eqnarray}%
we can write 
\begin{eqnarray}
&&\ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }_{k}^{s}J_{a,h}^{%
\alpha }\left[ qf^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)%
\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ pf^{2}(t)\right]   \notag \\
&&-2\ _{k}^{s}J_{a,h}^{\alpha }\left[ qf(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ pf(t)\right]   \label{022} \\
&\leq &\left( \frac{\left( M-m\right) ^{2}}{2}+2\left \Vert f\right \Vert
_{\infty }^{2}\right) \text{ }_{k}^{s}J_{a,h}^{\alpha }p(t)\
_{k}^{s}J_{a,h}^{\alpha }q(t).  \notag
\end{eqnarray}%
Similarly, we have%
\begin{eqnarray}
&&\ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }_{k}^{s}J_{a,h}^{%
\alpha }\left[ qg^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)%
\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ pg^{2}(t)\right]   \notag \\
&&-2\ _{k}^{s}J_{a,h}^{\alpha }\left[ qg(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left[ pg(t)\right]   \label{96} \\
&\leq &\left( \frac{\left( N-n\right) ^{2}}{2}+2\left \Vert g\right \Vert
_{\infty }^{2}\right) \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right]
\right) \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \right) . 
\notag
\end{eqnarray}%
Consequently, by (\ref{022}), (\ref{96}) and (\ref{303}), we end the proof
of Theorem 3.
\end{proof}

\begin{corollary}
Let $f$ be an integrable functions on $\left[ a,b\right] $ satisfying the
condition (\ref{77}), $p$ be a positive functions on $\left[ a,b\right] $, $h
$ be a measurable, increasing and positive function on $\left( a,b\right] $
and $h\in C^{1}\left( \left[ a,b\right] \right) .$ Then for all $t>0,$ the
following inequality is valid: 
\begin{eqnarray}
&&\left \vert \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}I_{a,h}^{\alpha }\left[ pf^{2}(t)\right] -\left( \ _{k}I_{a,h}^{\alpha }%
\left[ pf(t)\right] \right) ^{2}\right \vert   \label{0213} \\
&\leq &\left( \frac{\left( M-m\right) ^{2}}{4}+\left \Vert f\right \Vert
_{\infty }\right) \left[ \ _{k}I_{a,h}^{\alpha }p(t)\right] ^{2},  \notag
\end{eqnarray}%
where $f\in L_{\infty }\left[ a,b\right] ,\  \alpha >0$ and $k>0.$
\end{corollary}

\begin{proof}
Applying Theorem 3 for $f\left( x\right) =g\left( x\right) $ and $p\left(
x\right) =q\left( x\right) ,$ we obtain (\ref{0213}).
\end{proof}

\begin{remark}
Taking $\alpha =k=1$ and $h\left( x\right) =e^{x},\ t=b$ in Theorem 3, we
obtain Theorem 2.
\end{remark}

Now we use two real positive parameters to prove the following $\alpha $-$%
\beta $-theorem.

\begin{theorem}
Let $f$ and $g$ be two integrable functions on $\left[ a,b\right] $
satisfying the condition (\ref{77}), let $p$ and $q$ be two positive
functions on $\left[ a,b\right] $ and let $h$ be a measurable, increasing,
positive function on $\left( a,b\right] $ with $h\in C^{1}\left( \left[ a,b%
\right] \right) .$ Then, we have 
\begin{eqnarray}
&&\left \vert \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}I_{a,h}^{\beta }\left[ qfg(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ p(t)%
\right] \text{ }_{k}I_{a,h}^{\alpha }\left[ qfg(t)\right] \right.   \notag \\
&&+\ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \text{ }_{k}I_{a,h}^{\beta }%
\left[ pfg(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ q(t)\right] \text{ }%
_{k}I_{a,h}^{\alpha }\left[ pfg(t)\right]   \notag \\
&&-\text{ }_{k}I_{a,h}^{\alpha }\left[ pf(t)\right] \ _{k}I_{a,h}^{\beta }%
\left[ qg(t)\right] -\text{ }_{k}I_{a,h}^{\beta }\left[ pf(t)\right] \
_{k}I_{a,h}^{\alpha }\left[ qg(t)\right]   \label{206} \\
&&-\left. \text{ }_{k}I_{a,h}^{\alpha }\left[ qf(t)\right]
_{k}I_{a,h}^{\beta }\left[ pg(t)\right] -\text{ }_{k}I_{a,h}^{\beta }\left[
qf(t)\right] _{k}I_{a,h}^{\alpha }\left[ pg(t)\right] \right \vert   \notag \\
&\leq &\left[ \left( \frac{\left( M-m\right) ^{2}}{2}+2\left \Vert
f\right \Vert _{\infty }\right) \left( \frac{\left( N-n\right) ^{2}}{2}%
+2\left \Vert g\right \Vert _{\infty }\right) \right] ^{\frac{1}{2}}  \notag \\
&&\times \left[ \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \
_{k}I_{a,h}^{\beta }\left[ q(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ p(t)%
\right] \ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \right] ,  \notag
\end{eqnarray}%
where $\alpha ,\beta >0,\ k>0$ and $f,g\in L_{\infty }\left[ a,b\right] .$
\end{theorem}

To prove the above result, we prove the following auxiliary result:

\begin{lemma}
Let $f$ and $g$ be two integrable functions on $\left[ a,b\right] $
satisfying the condition (\ref{77}), let $p$ and $q$ be two positive
functions on $\left[ a,b\right] $ and let $h$ be a measurable, increasing,
positive function on $\left( a,b\right] $ with $h\in C^{1}\left( \left[ a,b%
\right] \right) .$ Then for all $\alpha ,\beta >0$, we have%
\begin{eqnarray}
&&\text{ }\left \{ \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \ _{k}I^{\beta }%
\left[ qfg(t)\right] +\text{ }_{k}I_{a,h}^{\beta }\left[ p(t)\right] \
_{k}I^{\alpha }\left[ qfg(t)\right] \right.   \notag \\
&&+\text{ }_{k}I_{a,h}^{\alpha }\left[ q(t)\right] \ _{k}I^{\beta }\left[
pfg(t)\right] +\text{ }_{k}I_{a,h}^{\beta }\left[ q(t)\right] \
_{k}I^{\alpha }\left[ pfg(t)\right]   \notag \\
&&-\text{ }_{k}I_{a,h}^{\alpha }\left[ pf(t)\right] \ _{k}I_{a,h}^{\beta }%
\left[ qg(t)\right] -\text{ }_{k}I_{a,h}^{\beta }\left[ pf(t)\right] \
_{k}I_{a,h}^{\alpha }\left[ qg(t)\right]   \notag \\
&&\left. -\ _{k}I_{a,h}^{\alpha }\left[ qf(t)\right] \ _{k}I_{a,h}^{\beta }%
\left[ pg(t)\right] -\ _{k}I_{a,h}^{\beta }\left[ qf(t)\right] \
_{k}I_{a,h}^{\alpha }\left[ pg(t)\right] \right \} ^{2}  \label{0222} \\
&\leq &\left \{ \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \ _{k}I^{\beta }%
\left[ qf^{2}(t)\right] +\text{ }_{k}I_{a,h}^{\beta }\left[ p(t)\right] \
_{k}I^{\alpha }\left[ qf^{2}(t)\right] \right.   \notag \\
&&+\text{ }_{k}I_{a,h}^{\alpha }\left[ q(t)\right] \ _{k}I^{\beta }\left[
pf^{2}(t)\right] +\text{ }_{k}I_{a,h}^{\beta }\left[ q(t)\right] \
_{k}I^{\alpha }\left[ pf^{2}(t)\right]   \notag \\
&&-\left. 2\text{ }_{k}I_{a,h}^{\alpha }\left[ pf(t)\right] \
_{k}I_{a,h}^{\beta }\left[ qf(t)\right] -2\text{ }_{k}I_{a,h}^{\beta }\left[
pf(t)\right] \ _{k}I_{a,h}^{\alpha }\left[ qf(t)\right] \right \}   \notag
\end{eqnarray}%
\begin{eqnarray*}
&&\times \left \{ \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \ _{k}I^{\beta }%
\left[ qg^{2}(t)\right] +\text{ }_{k}I_{a,h}^{\beta }\left[ p(t)\right] \
_{k}I^{\alpha }\left[ qg^{2}(t)\right] \right.  \\
&&+\text{ }_{k}I_{a,h}^{\alpha }\left[ q(t)\right] \ _{k}I^{\beta }\left[
pg^{2}(t)\right] +\text{ }_{k}I_{a,h}^{\beta }\left[ q(t)\right] \
_{k}I^{\alpha }\left[ pg^{2}(t)\right]  \\
&&-\left. 2\text{ }_{k}I_{a,h}^{\alpha }\left[ pg(t)\right] \
_{k}I_{a,h}^{\beta }\left[ qg(t)\right] -2\text{ }_{k}I_{a,h}^{\beta }\left[
pg(t)\right] \ _{k}I_{a,h}^{\alpha }\left[ qg(t)\right] \right \} .
\end{eqnarray*}
\end{lemma}

\begin{proof}
Using the functional $G\left( \tau ,\rho \right) $ which gives in (\ref{100}%
) and multiplying (\ref{015}) by $\ _{k}^{s}H_{\alpha ,h}\left( t,\tau
\right) \times \ _{k}^{s}H_{\beta ,h}\left( t,\rho \right) ,\  \left( \tau
,\rho \right) \in \left( a,t\right) ^{2},\ s\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
-\left \{ -1\right \} $ and integrating with respect to $\tau $ and $\rho $
over $\left( a,t\right) ^{2},$ we get%
\begin{eqnarray}
&&\int_{a}^{t}\int_{a}^{t}\text{ }_{k}^{s}H_{\alpha ,h}\left( t,\tau \right) 
\text{ }_{k}^{s}H_{\beta ,h}\left( t,\rho \right) \left( f\left( \tau
\right) -f\left( \rho \right) \right) \left( g\left( \tau \right) -g\left(
\rho \right) \right) d\tau d\rho   \notag \\
&=&\text{ }_{k}^{s}J_{a,h}^{\alpha }p(t)\text{ }_{k}^{s}J_{a,h}^{\beta
}\left( qfg\right) (t)+\text{ }_{k}^{s}J_{a,h}^{\beta }p(t)\text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left( qfg\right) (t)  \label{111} \\
&&+\text{ }_{k}^{s}J_{a,h}^{\alpha }q(t)\text{ }_{k}^{s}J_{a,h}^{\beta
}\left( pfg\right) (t)+\text{ }_{k}^{s}J_{a,h}^{\beta }q(t)\text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left( pfg\right) (t)  \notag \\
&&-\ _{k}^{s}J_{a,h}^{\alpha }\left( pf\right) (t)\text{ }%
_{k}^{s}J_{a,h}^{\beta }\left( qg\right) (t)-\ _{k}^{s}J_{a,h}^{\beta
}\left( pf\right) (t)\text{ }_{k}^{s}J_{a,h}^{\alpha }\left( qg\right) (t) 
\notag \\
&&-\text{ }_{k}^{s}J_{a,h}^{\alpha }\left( qf\right) (t)\text{ }%
_{k}^{s}J_{a,h}^{\beta }\left( pg\right) (t)-\text{ }_{k}^{s}J_{a,h}^{\beta
}\left( qf\right) (t)\text{ }_{k}^{s}J_{a,h}^{\alpha }\left( pg\right) (t). 
\notag
\end{eqnarray}%
\ Now, by using Cauchy Schwarz integral inequalities, we have%
\begin{eqnarray}
&&\left[ \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \right) \text{ 
}\left( \ _{k}^{s}J_{a,h}^{\beta }\left[ qfg(t)\right] \right) +\left( \
_{k}^{s}J_{a,h}^{\beta }\left[ p(t)\right] \right) \text{ }\left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ qfg(t)\right] \right) \right.   \notag \\
&&+\text{ }_{k}^{s}J_{a,h}^{\alpha }q(t)\text{ }_{k}^{s}J_{a,h}^{\beta
}\left( pfg\right) (t)+\text{ }_{k}^{s}J_{a,h}^{\beta }q(t)\text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left( pfg\right) (t)  \notag \\
&&-\ _{k}^{s}J_{a,h}^{\alpha }\left( pf\right) (t)\text{ }%
_{k}^{s}J_{a,h}^{\beta }\left( qg\right) (t)-\ _{k}^{s}J_{a,h}^{\beta
}\left( pf\right) (t)\text{ }_{k}^{s}J_{a,h}^{\alpha }\left( qg\right) (t) 
\notag \\
&&-\left. \left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ qf(t)\right] \right)
\left( \ _{k}^{s}J_{a,h}^{\beta }\left[ pg(t)\right] \right) -\text{ }\left(
\ _{k}^{s}J_{a,h}^{\beta }\left[ qf(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ pg(t)\right] \right) \right] ^{2}  \notag \\
&\leq &\left \{ \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\beta }\left[ qf^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\beta }%
\left[ p(t)\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ qf^{2}(t)\right]
\right.   \label{155} \\
&&+\text{ }_{k}^{s}J_{a,h}^{\alpha }q(t)\text{ }_{k}^{s}J_{a,h}^{\beta
}\left( pf^{2}\right) (t)+\text{ }_{k}^{s}J_{a,h}^{\beta }q(t)\text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left( pf^{2}\right) (t)  \notag \\
&&-2\ _{k}^{s}J_{a,h}^{\alpha }\left( pf\right) (t)\text{ }%
_{k}^{s}J_{a,h}^{\beta }\left( qf\right) (t)-\left. 2\
_{k}^{s}J_{a,h}^{\beta }\left[ pf(t)\right] \ _{k}^{s}J_{a,h}^{\alpha }\left[
qf(t)\right] \right \}   \notag \\
&&\times \left \{ \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}^{s}J_{a,h}^{\beta }\left[ qg^{2}(t)\right] +\ _{k}^{s}J_{a,h}^{\beta }%
\left[ p(t)\right] \text{ }_{k}^{s}J_{a,h}^{\alpha }\left[ qg^{2}(t)\right]
\right.   \notag \\
&&+\text{ }_{k}^{s}J_{a,h}^{\alpha }q(t)\text{ }_{k}^{s}J_{a,h}^{\beta
}\left( pg^{2}\right) (t)+\text{ }_{k}^{s}J_{a,h}^{\beta }q(t)\text{ }%
_{k}^{s}J_{a,h}^{\alpha }\left( pg^{2}\right) (t)  \notag \\
&&-\left. 2\ _{k}^{s}J_{a,h}^{\alpha }\left[ pg(t)\right] \
_{k}^{s}J_{a,h}^{\beta }\left[ qg(t)\right] -2\ _{k}^{s}J_{a,h}^{\beta }%
\left[ pg(t)\right] \ _{k}^{s}J_{a,h}^{\alpha }\left[ qg(t)\right] \right \} .
\notag
\end{eqnarray}%
At the end, applying (\ref{303}), we get (\ref{0222}).
\end{proof}

\begin{lemma}
Let $\varphi $ be an integrable function on $\left[ a,b\right] $ satisfying
the condition (\ref{77}) on $\left[ a,b\right] ,$ let $p$ and $q$ be two
positive functions on $\left[ a,b\right] $ and let $h$ be a measurable,
increasing, positive function on $\left( a,b\right] $ with $h\in C^{1}\left( %
\left[ a,b\right] \right) .$ Then for all $\alpha ,\beta >0$, we have 
\begin{eqnarray}
&&\ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \text{ }_{k}I_{a,h}^{\beta }%
\left[ q\varphi ^{2}(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ p(t)\right] 
\text{ }_{k}I_{a,h}^{\alpha }\left[ q\varphi ^{2}(t)\right]   \notag \\
&&+\ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \text{ }_{k}I_{a,h}^{\beta }%
\left[ p\varphi ^{2}(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ q(t)\right] 
\text{ }_{k}I_{a,h}^{\alpha }\left[ p\varphi ^{2}(t)\right]   \notag \\
&&-2\left( \ _{k}I_{a,h}^{\alpha }\left[ p\varphi (t)\right] \right) \left(
\ _{k}I_{a,h}^{\beta }\left[ q\varphi (t)\right] \right) -2\left( \
_{k}I_{a,h}^{\beta }\left[ p\varphi (t)\right] \right) \left( \
_{k}I_{a,h}^{\alpha }\left[ q\varphi (t)\right] \right)   \notag \\
&=&\left( \ _{k}I_{a,h}^{\alpha }\left[ \left( M-\varphi (t)\right) q(t)%
\right] \right) \left( \ _{k}I_{a,h}^{\beta }\left[ \left( \varphi
(t)-m\right) p(t)\right] \right)   \notag \\
&&+\left( \ _{k}I_{a,h}^{\beta }\left[ \left( M-\varphi (t)\right) q(t)%
\right] \right) \left( \ _{k}I_{a,h}^{\alpha }\left[ \left( \varphi
(t)-m\right) p(t)\right] \right)   \label{044} \\
&&+\left( \ _{k}I_{a,h}^{\alpha }\left[ \left( M-\varphi (t)\right) p(t)%
\right] \right) \left( \ _{k}I_{a,h}^{\beta }\left[ \left( \varphi
(t)-m\right) q(t)\right] \right)   \notag \\
&&+\left( \ _{k}I_{a,h}^{\beta }\left[ \left( M-\varphi (t)\right) p(t)%
\right] \right) \left( \ _{k}I_{a,h}^{\alpha }\left[ \left( \varphi
(t)-m\right) q(t)\right] \right)   \notag \\
&&-\left( \ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \right) \left( \
_{k}I_{a,h}^{\beta }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) p(t)\right] \right)   \notag \\
&&-\left( \ _{k}I_{a,h}^{\beta }\left[ q(t)\right] \right) \left( \
_{k}I_{a,h}^{\alpha }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) p(t)\right] \right)   \notag \\
&&-\left( \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \right) \left( \
_{k}I_{a,h}^{\beta }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) q(t)\right] \right)   \notag \\
&&-\left( \ _{k}I_{a,h}^{\beta }\left[ p(t)\right] \right) \left( _{\
k}I_{a,h}^{\alpha }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) q(t)\right] \right)   \notag
\end{eqnarray}%
where $k>0.$
\end{lemma}

\begin{proof}
Let us multiplying (\ref{015}) by $\ _{k}^{s}H_{\alpha ,h}\left( t,\tau
\right) \times \ _{k}^{s}H_{\beta ,h}\left( t,\rho \right) ,\  \left( \tau
,\rho \right) \in \left( a,t\right) ^{2},\ s\in 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
-\left \{ -1\right \} $ witch gives in (\ref{222}) and integrating with
respect to $\tau $ and $\rho $ over $\left( a,t\right) ^{2},$ we get%
\begin{eqnarray*}
&&\frac{1}{k^{2}\Gamma _{k}\left( \alpha \right) \Gamma _{k}\left( \beta
\right) }\left \{ \int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{%
\lim }\left[ \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) 
}{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime
}\left( \tau \right) p\left( \tau \right) \varphi ^{2}\left( \tau \right)
\right. \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) p\left( \tau \right) \varphi ^{2}\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \varphi ^{2}(\rho )%
\right] d\tau d\rho 
\end{eqnarray*}%
\begin{eqnarray*}
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \varphi ^{2}(\rho )%
\right] d\tau d\rho  \\
&&-2\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \varphi \left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \Phi ^{2}(\rho )\right]
d\tau d\rho  \\
&&-2\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) p\left( \tau \right) \varphi \left( \tau \right) \right.  \\
&&\left. \times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left(
\rho \right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \varphi ^{2}(\rho )%
\right] d\tau d\rho \right \} 
\end{eqnarray*}%
\begin{eqnarray*}
&=&\frac{1}{k^{2}\Gamma _{k}\left( \alpha \right) \Gamma _{k}\left( \beta
\right) }\left \{ \int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{%
\lim }\left[ \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) 
}{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime
}\left( \tau \right) \left( \varphi (\tau )-m\right) p\left( \tau \right)
\right. \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( M-\varphi (\rho )\right) q\left( \rho
\right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) \left( \varphi (\tau )-m\right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( M-\varphi (\rho )\right) q\left( \rho
\right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) \left( M-\varphi (\tau )\right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( \varphi (\rho )-m\right) q\left( \rho
\right) \right] d\tau d\rho  \\
&&+\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) \left( M-\varphi (\tau )\right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) \left( \varphi (\rho )-m\right) q\left( \rho
\right) \right] d\tau d\rho 
\end{eqnarray*}%
\begin{eqnarray*}
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) \left( M-\varphi (\tau )\right) \left( \varphi (\tau )-m\right)
p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) \left( M-\varphi (\tau )\right) \left( \varphi (\tau )-m\right)
p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) q\left( \rho \right) \right] d\tau d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left(
\tau \right) p\left( \tau \right) \right.  \\
&&\times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \rho
\right) }{s+1}\right) ^{\frac{\beta }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) p\left( \rho \right) \left( M-\varphi (\rho
)\right) \left( \varphi (\rho )-m\right) q\left( \rho \right) \right] d\tau
d\rho  \\
&&-\int_{a}^{t}\int_{a}^{t}\underset{s\longrightarrow -1^{+}}{\lim }\left[
\left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left( \tau \right) }{s+1}%
\right) ^{\frac{\beta }{k}-1}h^{s}\left( \tau \right) h^{\prime }\left( \tau
\right) p\left( \tau \right) \right.  \\
&&\left. \times \left. \left( \frac{h^{s+1}\left( t\right) -h^{s+1}\left(
\rho \right) }{s+1}\right) ^{\frac{\alpha }{k}-1}h^{s}\left( \rho \right)
h^{\prime }\left( \rho \right) p\left( \rho \right) \left( M-\varphi (\rho
)\right) \left( \varphi (\rho )-m\right) q\left( \rho \right) \right] d\tau
d\rho \right \} 
\end{eqnarray*}%
Thanks to (\ref{303}), we end the proof.
\end{proof}

Now, we are ready to prove Theorem 7.

\begin{proof}
By (\ref{77}) and (\ref{8}), we have 
\begin{eqnarray}
&&-\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ p(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\beta }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) q(t)\right] \right)   \label{157} \\
&&-\left( \ _{k}^{s}J_{a,h}^{\beta }\left[ p(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) q(t)\right] \right)   \notag \\
&&-\left( \ _{k}^{s}J_{a,h}^{\alpha }\left[ q(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\beta }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) p(t)\right] \right)   \notag \\
&&-\left( \ _{k}^{s}J_{a,h}^{\beta }\left[ q(t)\right] \right) \left( \
_{k}^{s}J_{a,h}^{\alpha }\left[ \left( M-\varphi (t)\right) \left( \varphi
(t)-m\right) p(t)\right] \right) \leq 0.  \notag
\end{eqnarray}%
Now, from Lemma 8, by using (\ref{8}) and Cauchy Schwarz integral inequality
for double integrals with two parameters $\alpha ,\beta >0$, applying also
the Lemma 9, (\ref{157}) and $\ $(\ref{303}), we get (\ref{206}). This
completes the proof of the Theorem 7.
\end{proof}

\begin{remark}
Taking $\alpha =\beta $ in Theorem 7, we obtain Theorem 3.
\end{remark}

\begin{theorem}
Let $f$ be integrable function on $\left[ a,b\right] $ satisfying the
condition (\ref{77}), $p$ and $q$ be two positive functions on $\left[ a,b%
\right] $ and let $h$ be a measurable, increasing and positive function on $%
\left( a,b\right] $ with $h\in C^{1}\left( \left[ a,b\right] \right) .$ Then
for all $t>0,$ the following inequality holds%
\begin{eqnarray}
&&\left \vert \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \text{ }%
_{k}I_{a,h}^{\beta }\left[ qf^{2}(t)\right] +\ _{k}I_{a,h}^{\beta }\left[
p(t)\right] \text{ }_{k}I_{a,h}^{\alpha }\left[ qf^{2}(t)\right] \right.  
\notag \\
&&+\ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \text{ }_{k}I_{a,h}^{\beta }%
\left[ pf^{2}(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ q(t)\right] \text{ }%
_{k}I_{a,h}^{\alpha }\left[ pf^{2}(t)\right]   \label{188} \\
&&-\left. 2\text{ }_{k}I_{a,h}^{\alpha }\left[ pf(t)\right] \
_{k}I_{a,h}^{\beta }\left[ qf(t)\right] -2\text{ }_{k}I_{a,h}^{\beta }\left[
pf(t)\right] \ _{k}I_{a,h}^{\alpha }\left[ qf(t)\right] \right \vert   \notag
\\
&\leq &\left( \frac{\left( M-m\right) ^{2}}{2}+2\left \Vert f\right \Vert
_{\infty }\right)   \notag \\
&&\times \left[ \ _{k}I_{a,h}^{\alpha }\left[ p(t)\right] \
_{k}I_{a,h}^{\beta }\left[ q(t)\right] +\ _{k}I_{a,h}^{\beta }\left[ p(t)%
\right] \ _{k}I_{a,h}^{\alpha }\left[ q(t)\right] \right] ,  \notag
\end{eqnarray}%
where $\alpha ,\beta >0,\ k>0.$
\end{theorem}

\begin{proof}
Applying Theorem 7 for $f\left( x\right) =g\left( x\right) ,$ we obtain (\ref%
{188}).
\end{proof}

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\end{document}