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\begin{document}
\title[On generalization of midpoint and trapezoid type inequalities...]{On
generalization of midpoint and trapezoid type inequalities involving
fractional integrals}
\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}
\author{S\"{u}meyye S\"{O}NMEZO\u{G}LU}
\email{smyy.mat@gmail.com}
\keywords{\textbf{\thanks{\textbf{2010 Mathematics Subject Classification.}
26D15, 26B25, 26D10.} }Hermite-Hadamard inequality, midpoint inequality,
fractional integral operators, convex function.}

\begin{abstract}
In this paper, we first prove a lemma for twice differentiable functions .
Then we establish some inequalities for mapping whose second derivatives in
absolute value are convex via Riemann-Liouville fractional integrals. These
results generalize the midpoint and trapezoid inequalities involving
Riemann-Liouville fractional integrals given in earlier studies.
\end{abstract}

\maketitle

\section{{\protect\large {Introduction}}}

The inequalities discovered by C. Hermite and J. Hadamard for convex
functions are considerable significant in the literature (see, e.g.,\cite%
{Dragomir1}, \cite{had}, \cite[p.137]{Pecaric}). These inequalities state
that if $f:I\rightarrow \mathbb{R}$ is a convex function on the interval $I$
of real numbers and $a,b\in I$ with $a<b$, then 
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int_{a}^{b}f(x)dx\leq \frac{%
f\left( a\right) +f\left( b\right) }{2}.  \label{E1}
\end{equation}%
Both inequalities hold in the reversed direction if $f$ is concave. We note
that Hadamard's inequality may be regarded as a refinement of the concept of
convexity and it follows easily from Jensen's inequality. Hadamard's
inequality for convex functions has received renewed attention in recent
years and a remarkable variety of refinements and generalizations have been
found (see, for example, \cite{alomari}-\cite{cal}, \cite{dragomir}, \cite%
{kavurmaci}, \cite{kirmaci}, \cite{ozdemir2}, \cite{ozdemir3}, \cite{saglam}%
, \cite{sarikaya2}, \cite{sarikaya6}, \cite{set}, \cite{xi}, \cite{xi1}) and
the references cited therein.

In the following we will give some necessary definitions and mathematical
preliminaries of fractional calculus theory which are used further in this
paper. More details, one can consult (\cite{Gorenflo}, \cite{kilbas}, \cite%
{Miller}, \cite{Podlubni})

\begin{definition}
Let $f\in L_{1}[a,b].$ The Riemann-Liouville integrals $J_{a+}^{\alpha }f$
and $J_{b-}^{\alpha }f$ of order $\alpha >0$ with $a\geq 0$ are defined by 
\begin{equation*}
J_{a+}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}\left( x-t\right)
^{\alpha -1}f(t)dt,\ \ x>a
\end{equation*}%
and%
\begin{equation*}
J_{b-}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int_{x}^{b}\left( t-x\right)
^{\alpha -1}f(t)dt,\ \ x<b
\end{equation*}%
respectively. Here, $\Gamma (\alpha )$ is the Gamma function and $%
J_{a+}^{0}f(x)=J_{b-}^{0}f(x)=f(x).$
\end{definition}

It is remarkable that Sarikaya et al.\cite{sarikaya4} first give the
following interesting integral inequalities of Hermite-Hadamard type
involving Riemann-Liouville fractional integrals.

\begin{theorem}
Let $f:\left[ a,b\right] \rightarrow \mathbb{R}$ be a positive function with 
$0\leq a<b$ and $f\in L_{1}\left[ a,b\right] .$ If $f$ is a convex function
on $[a,b]$, then the following inequalities for fractional integrals hold:%
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{\Gamma (\alpha +1)}{2\left(
b-a\right) ^{\alpha }}\left[ J_{a+}^{\alpha }f(b)+J_{b-}^{\alpha }f(a)\right]
\leq \frac{f\left( a\right) +f\left( b\right) }{2}  \label{S1}
\end{equation}%
with $\alpha >0.$
\end{theorem}

Sar\i kaya and Y\i ld\i r\i m also give the following Hermite-Hadamard type
inequality for the Riemann-Lioville fractional integrals in \cite{sarikaya1}.

\begin{theorem}
Let $f:\left[ a,b\right] \rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a positive function with $a<b$ and $f\in L_{1}\left[ a,b\right] .$ If $%
f $ is a convex function on $\left[ a,b\right] ,$ then the following
inequalities for fractional integrals hold:%
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq \frac{2^{\alpha -1}\Gamma (\alpha +1)}{%
\left( b-a\right) ^{\alpha }}\left[ J_{\left( \frac{a+b}{2}\right)
^{+}}^{\alpha }f(b)+J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f(a)\right]
\leq \frac{f(a)+f(b)}{2}.  \label{hh0}
\end{equation}
\end{theorem}

For the more information fractional calculus and related inequalities please
refer to (\cite{chen}, \cite{hussain}, \cite{iqbal}, \cite{noor}, \cite%
{ozdemir}, \cite{sarikaya3}, \cite{sarikaya5}, \cite{wang}, \cite{whang}, 
\cite{zhang})

\section{{\protect\large {Generalized Midpoint and Trapezoid Type
Inequalities}}}

In this section, we will first present a lemma for twice differentiable
functions . Then we establish some inequalities which generalize the
midpoint and trapezoid inequalities involving Riemann-Liouville fractional
integrals obtained in previous works.

\begin{lemma}
\label{l1} Let $I\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be an open interval, $a,b\in I$ with $a<b.$ If $f:I\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is a twice differentiable mapping such that $f^{\prime \prime }$ is
integrable and $0\leq \lambda \leq 1,\ \alpha \geq 1,$ then we have

\begin{eqnarray}
&&\left[ \left( \lambda -\frac{\alpha +1}{2^{\alpha }}\right) f\left( \frac{%
a+b}{2}\right) -\lambda \left( \frac{f\left( a\right) +f\left( b\right) }{2}%
\right) +\frac{\Gamma \left( \alpha +2\right) }{2\left( b-a\right) ^{\alpha }%
}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha }f\left( b\right)
+J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left( a\right) \right) %
\right]   \label{2.1} \\
&&  \notag \\
&=&\frac{\left( b-a\right) ^{2}}{2}\int_{0}^{1}k\left( t\right) f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) dt  \notag
\end{eqnarray}%
where%
\begin{equation*}
k\left( t\right) =\left\{ 
\begin{array}{ll}
t\left( t^{\alpha }-\lambda \right)  & 0\leq t\leq \frac{1}{2} \\ 
&  \\ 
\left( 1-t\right) \left( \left( 1-t\right) ^{\alpha }-\lambda \right)  & 
\frac{1}{2}\leq t\leq 1.%
\end{array}%
\right. 
\end{equation*}
\end{lemma}

\begin{proof}
It suffices to note that

\begin{eqnarray}
I &=&\int_{0}^{1}k\left( t\right) f^{\prime \prime }\left( ta+\left(
1-t\right) b\right) dt  \label{2.2} \\
&&  \notag \\
&=&\int_{0}^{\frac{1}{2}}t\left( t^{\alpha }-\lambda \right) f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) dt+\int_{\frac{1}{2}}^{1}\left(
1-t\right) \left( \left( 1-t\right) ^{\alpha }-\lambda \right) f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) dt  \notag \\
&&  \notag \\
&=&I_{1}+I_{2}.  \notag
\end{eqnarray}%
Integrating by parts twice, we can state:%
\begin{eqnarray}
I_{1} &=&\int_{0}^{\frac{1}{2}}t\left( t^{\alpha }-\lambda \right) f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) dt  \label{2.3} \\
&&  \notag \\
&=&\left. t\left( t^{\alpha }-\lambda \right) \frac{f^{\prime }\left(
ta+\left( 1-t\right) b\right) }{a-b}\right\vert _{0}^{\frac{1}{2}}-\int_{0}^{%
\frac{1}{2}}\frac{f^{\prime }\left( ta+\left( 1-t\right) b\right) }{a-b}%
\left( \left( \alpha +1\right) t^{\alpha }-\lambda \right) dt  \notag \\
&&  \notag \\
&=&\frac{-1}{2\left( b-a\right) }\left( \frac{1}{2^{\alpha }}-\lambda
\right) f^{\prime }\left( \frac{a+b}{2}\right) -\frac{1}{\left( b-a\right)
^{2}}\left( \frac{\alpha +1}{2^{\alpha }}-\lambda \right) f\left( \frac{a+b}{%
2}\right)   \notag \\
&&  \notag \\
&&-\frac{\lambda }{\left( b-a\right) ^{2}}f\left( b\right) +\frac{\alpha
\left( \alpha +1\right) }{\left( b-a\right) ^{2}}\int_{0}^{\frac{1}{2}%
}f\left( ta+\left( 1-t\right) b\right) t^{\alpha -1}dt  \notag \\
&&  \notag \\
&&+\frac{\alpha \left( \alpha +1\right) }{\left( b-a\right) ^{2}}\frac{1}{%
\left( b-a\right) ^{\alpha }}\Gamma \left( \alpha \right) J_{\left( \frac{a+b%
}{2}\right) ^{+}}^{\alpha }f\left( b\right)   \notag
\end{eqnarray}%
and similarly, we get%
\begin{eqnarray}
I_{2} &=&\int_{\frac{1}{2}}^{1}\left( 1-t\right) \left( \left( 1-t\right)
^{\alpha }-\lambda \right) f^{\prime \prime }\left( ta+\left( 1-t\right)
b\right) dt  \label{2.4} \\
&&  \notag \\
&=&\left. \left( 1-t\right) \left( \left( 1-t\right) ^{\alpha }-\lambda
\right) \frac{f^{\prime }\left( ta+\left( 1-t\right) b\right) }{a-b}%
\right\vert _{\frac{1}{2}}^{1}  \notag \\
&&  \notag \\
&&+\int_{\frac{1}{2}}^{1}\frac{f^{\prime }\left( ta+\left( 1-t\right)
b\right) }{a-b}\left( \left( \alpha +1\right) \left( 1-t\right) ^{\alpha
}-\lambda \right)   \notag \\
&&  \notag \\
&=&\frac{1}{2\left( b-a\right) }\left( \frac{1}{2^{\alpha }}-\lambda \right)
f^{\prime }\left( \frac{a+b}{2}\right) -\lambda \frac{f\left( a\right) }{%
\left( b-a\right) ^{2}}  \notag \\
&&  \notag \\
&&-\frac{1}{\left( b-a\right) ^{2}}\left( \frac{\alpha +1}{2^{\alpha }}%
-\lambda \right) f\left( \frac{a+b}{2}\right) +\frac{\alpha \left( \alpha
+1\right) }{\left( b-a\right) ^{\alpha +2}}\Gamma \left( \alpha \right)
J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left( a\right) .  \notag
\end{eqnarray}%
Using (\ref{2.3}) and (\ref{2.4}) in (\ref{2.2}), it follows that%
\begin{eqnarray*}
I &=&I_{1}+I_{2}=\frac{-2}{\left( b-a\right) ^{2}}\left( \frac{\alpha +1}{%
2^{\alpha }}-\lambda \right) f\left( \frac{a+b}{2}\right) -\frac{2\lambda }{%
\left( b-a\right) ^{2}}\left( \frac{f\left( a\right) +f\left( b\right) }{2}%
\right)  \\
&& \\
&&+\frac{\alpha \left( \alpha +1\right) }{\left( b-a\right) ^{\alpha +2}}%
\Gamma \left( \alpha \right) \left( J_{\left( \frac{a+b}{2}\right)
^{+}}^{\alpha }f\left( b\right) +J_{\left( \frac{a+b}{2}\right)
^{-}}^{\alpha }f\left( a\right) \right)  \\
&& \\
&=&\frac{2}{\left( b-a\right) ^{2}}\left[ \left( \lambda -\frac{\alpha +1}{%
2^{\alpha }}\right) f\left( \frac{a+b}{2}\right) -\lambda \left( \frac{%
f\left( a\right) +f\left( b\right) }{2}\right) \right]  \\
&& \\
&&+\frac{\Gamma \left( \alpha +2\right) }{\left( b-a\right) ^{\alpha +2}}%
\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha }f\left( b\right)
+J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left( a\right) \right) .
\end{eqnarray*}%
Then by multipling the above equality with $\frac{\left( b-a\right) ^{2}}{2},
$ this completes the proof.
\end{proof}

\begin{theorem}
\label{t1} Let $I\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be an open intervial, $a,b\in I$ with $a<b$ and $f:I\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a twice differentiable mapping such that $f^{\prime \prime }$ is
integrable and \ $0\leq \lambda \leq 1,\ \alpha \geq 1.$ If $\left\vert
f^{\prime \prime }\right\vert $ is a convex on $\left[ a,b\right] ,$ then
the following inequalities hold:
\end{theorem}

\begin{eqnarray}
&&\left\vert \left( \lambda -\frac{\alpha +1}{2^{\alpha }}\right) f\left( 
\frac{a+b}{2}\right) -\lambda \left( \frac{f\left( a\right) +f\left(
b\right) }{2}\right) +\frac{\Gamma \left( \alpha +2\right) }{2\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) \right\vert   \label{2.5} \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}}{2}\left\{ 
\begin{array}{ll}
\left( \dfrac{1}{2^{\alpha +2}\left( \alpha +2\right) }-\dfrac{\lambda }{8}+%
\dfrac{\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}\right) \left[
\left\vert f^{\prime \prime }\left( a\right) \right\vert +\left\vert
f^{\prime \prime }\left( b\right) \right\vert \right] , & 0\leq \lambda \leq 
\frac{1}{2} \\ 
&  \\ 
\left( \dfrac{\lambda }{8}-\dfrac{1}{2^{\alpha +2}\left( \alpha +2\right) }%
\right) \left[ \left\vert f^{\prime \prime }\left( a\right) \right\vert
+\left\vert f^{\prime \prime }\left( b\right) \right\vert \right] , & \frac{1%
}{2}\leq \lambda \leq 1.%
\end{array}%
\right.   \notag
\end{eqnarray}

\begin{proof}
From Lemma \ref{l1} and by defination of $k\left( t\right) $, we get

\begin{eqnarray}
&&\left\vert \left( \lambda -\frac{\alpha +1}{2^{\alpha }}\right) f\left( 
\frac{a+b}{2}\right) -\lambda \left( \frac{f\left( a\right) +f\left(
b\right) }{2}\right) +\frac{\Gamma \left( \alpha +2\right) }{2\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) \right\vert   \label{2.6} \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}}{2}\int_{0}^{1}\left\vert k\left(
t\right) \right\vert \left\vert f^{\prime \prime }\left( ta+\left(
1-t\right) b\right) \right\vert dt  \notag \\
&&  \notag \\
&=&\frac{\left( b-a\right) ^{2}}{2}\left\{ \int_{0}^{\frac{1}{2}}\left\vert
t\left( t^{\alpha }-\lambda \right) \right\vert \left\vert f^{\prime \prime
}\left( ta+\left( 1-t\right) b\right) \right\vert dt\right.   \notag \\
&&  \notag \\
&&\left. +\int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right) \left( \left(
1-t\right) ^{\alpha }-\lambda \right) \right\vert \left\vert f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) \right\vert dt\right\}   \notag
\\
&&  \notag \\
&=&\frac{\left( b-a\right) ^{2}}{2}\left\{ J_{1}+J_{2}\right\} .  \notag
\end{eqnarray}%
We assume that $0\leq \lambda \leq \frac{1}{2},$ then using the convexity of 
$\left\vert f^{\prime \prime }\right\vert ,$we get%
\begin{eqnarray}
J_{1} &\leq &\int_{0}^{\frac{1}{2}}\left\vert t\left( t^{\alpha }-\lambda
\right) \right\vert \left[ t\left\vert f^{\prime \prime }\left( a\right)
\right\vert +\left( 1-t\right) \left\vert f^{\prime \prime }\left( b\right)
\right\vert \right] dt  \label{2.7} \\
&&  \notag \\
&=&\int_{0}^{\lambda ^{\frac{1}{\alpha }}}t\left( \lambda -t^{\alpha
}\right) \left[ t\left\vert f^{\prime \prime }\left( a\right) \right\vert
+\left( 1-t\right) \left\vert f^{\prime \prime }\left( b\right) \right\vert %
\right] dt  \notag \\
&&  \notag \\
&&+\int_{\lambda ^{\frac{1}{\alpha }}}^{\frac{1}{2}}t\left( t^{\alpha
}-\lambda \right) \left[ t\left\vert f^{\prime \prime }\left( a\right)
\right\vert +\left( 1-t\right) \left\vert f^{\prime \prime }\left( b\right)
\right\vert \right] dt  \notag \\
&&  \notag \\
&=&\left\vert f^{\prime \prime }\left( a\right) \right\vert \left[ \frac{%
2\alpha \lambda ^{1+\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{1}{%
2^{\alpha +3}\left( \alpha +3\right) }-\frac{\lambda }{24}\right]   \notag \\
&&  \notag \\
&&+\left\vert f^{\prime \prime }\left( b\right) \right\vert \left[ \frac{%
\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}-\frac{2\alpha \lambda ^{1+%
\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{\alpha +4}{2^{\alpha
+3}\left( \alpha +2\right) \left( \alpha +3\right) }-\frac{\lambda }{12}%
\right]   \notag
\end{eqnarray}%
and similarly, we have%
\begin{eqnarray}
J_{2} &\leq &\int_{\frac{1}{2}}^{1-\lambda ^{\frac{1}{\alpha }}}\left(
1-t\right) \left( \left( 1-t\right) ^{\alpha }-\lambda \right) \left[
t\left\vert f^{\prime \prime }\left( a\right) \right\vert +\left( 1-t\right)
\left\vert f^{\prime \prime }\left( b\right) \right\vert \right] dt
\label{2.8} \\
&&  \notag \\
&&+\int_{1-\lambda ^{\frac{1}{\alpha }}}^{1}\left( 1-t\right) \left( \lambda
-\left( 1-t\right) ^{\alpha }\right) \left[ t\left\vert f^{\prime \prime
}\left( a\right) \right\vert +\left( 1-t\right) \left\vert f^{\prime \prime
}\left( b\right) \right\vert \right] dt  \notag \\
&&  \notag \\
&=&\left\vert f^{\prime \prime }\left( a\right) \right\vert \left[ \frac{%
\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}-\frac{2\alpha \lambda ^{1+%
\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{\alpha +4}{2^{\alpha
+3}\left( \alpha +2\right) \left( \alpha +3\right) }-\frac{\lambda }{12}%
\right]   \notag \\
&&  \notag \\
&&+\left\vert f^{\prime \prime }\left( b\right) \right\vert \left[ \frac{%
2\alpha \lambda ^{1+\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{1}{%
2^{\alpha +3}\left( \alpha +3\right) }-\frac{\lambda }{24}\right] .  \notag
\end{eqnarray}%
Using (\ref{2.7}) and (\ref{2.8}) in (\ref{2.6}),we see thatthe first
inequality of (\ref{2.5}) holds. On the other hand, let $\frac{1}{2}\leq
\lambda \leq 1,$ then, using the convexity of $\left\vert f^{\prime \prime
}\right\vert $ and by simple computation we have%
\begin{eqnarray}
J_{1}^{\prime } &\leq &\int_{0}^{\frac{1}{2}}\left\vert t\left( t^{\alpha
}-\lambda \right) \right\vert \left[ t\left\vert f^{\prime \prime }\left(
a\right) \right\vert +\left( 1-t\right) \left\vert f^{\prime \prime }\left(
b\right) \right\vert \right] dt  \label{2.9} \\
&&  \notag \\
&=&\int_{0}^{\frac{1}{2}}t\left( \lambda -t^{\alpha }\right) \left[
t\left\vert f^{\prime \prime }\left( a\right) \right\vert +\left( 1-t\right)
\left\vert f^{\prime \prime }\left( b\right) \right\vert \right] dt  \notag
\\
&&  \notag \\
&=&\left( \frac{\lambda }{24}-\frac{1}{2^{\alpha +3}\left( \alpha +3\right) }%
\right) \left\vert f^{\prime \prime }\left( a\right) \right\vert +\left( 
\frac{\lambda }{12}-\frac{\alpha +4}{2^{\alpha +3}\left( \alpha +2\right)
\left( \alpha +3\right) }\right) \left\vert f^{\prime \prime }\left(
b\right) \right\vert   \notag
\end{eqnarray}%
and similarly%
\begin{eqnarray}
J_{2}^{\prime } &\leq &\int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right)
\left( \left( 1-t\right) ^{\alpha }-\lambda \right) \right\vert \left\vert
f^{\prime \prime }\left( ta+\left( 1-t\right) b\right) \right\vert dt
\label{2.10} \\
&&  \notag \\
&=&\int_{\frac{1}{2}}^{1}\left( 1-t\right) \left( \lambda -\left( 1-t\right)
^{\alpha }\right) \left[ t\left\vert f^{\prime \prime }\left( a\right)
\right\vert +\left( 1-t\right) \left\vert f^{\prime \prime }\left( b\right)
\right\vert \right] dt  \notag \\
&&  \notag \\
&=&\left( \frac{\lambda }{12}-\frac{\alpha +4}{2^{\alpha +3}\left( \alpha
+2\right) \left( \alpha +3\right) }\right) \left\vert f^{\prime \prime
}\left( a\right) \right\vert +\left( \frac{\lambda }{24}-\frac{1}{2^{\alpha
+3}\left( \alpha +3\right) }\right) \left\vert f^{\prime \prime }\left(
b\right) \right\vert .  \notag
\end{eqnarray}%
Thus if we (\ref{2.9}) and (\ref{2.10}) in (\ref{2.6}), we obtain the second
inequality of (\ref{2.5}). This completes the proof.
\end{proof}

\begin{corollary}
\label{c2} Under the assumptions of Theorem \ref{t1} with $\lambda =0$, then
we get the following inequality
\end{corollary}

\begin{eqnarray*}
&&\left\vert \frac{2^{\alpha -1}\Gamma \left( \alpha +1\right) }{\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) -f\left( \frac{a+b}{2}\right) \right\vert  \\
&& \\
&\leq &\frac{\left( b-a\right) ^{2}}{\left( \alpha +1\right) \left( \alpha
+2\right) }\left[ \frac{\left\vert f^{\prime \prime }\left( a\right)
\right\vert +\left\vert f^{\prime \prime }\left( b\right) \right\vert }{8}%
\right] 
\end{eqnarray*}%
which is proved by Noor and Awan in \cite[Theorem 2 (for $s$=1)]{noor}.

\begin{remark}
If we take $\alpha =1$ in Corollary \ref{c2}, then we get the following
inequality%
\begin{equation*}
\left\vert \frac{1}{b-a}\dint\limits_{a}^{b}f(t)dt-f\left( \frac{a+b}{2}%
\right) \right\vert \leq \frac{\left( b-a\right) ^{2}}{24}\left[ \frac{%
\left\vert f^{\prime \prime }\left( a\right) \right\vert +\left\vert
f^{\prime \prime }\left( b\right) \right\vert }{2}\right]
\end{equation*}%
which is given by Sarikaya et al. in \cite{sarikaya6}.
\end{remark}

\begin{corollary}
\label{c1} Under the assumptions of Theorem \ref{t1} with $\lambda =\frac{%
\alpha +1}{2^{\alpha }}$, then we get the following inequality%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{2^{\alpha
-1}\Gamma \left( \alpha +1\right) }{\left( b-a\right) ^{\alpha }}\left(
J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha }f\left( b\right) +J_{\left( 
\frac{a+b}{2}\right) ^{-}}^{\alpha }f\left( a\right) \right) \right\vert  \\
&& \\
&\leq &\frac{\left( b-a\right) ^{2}}{8\left( \alpha +1\right) \left( \alpha
+2\right) }\left( \alpha \left( \alpha +1\right) ^{1+\frac{2}{\alpha }}+1-%
\frac{\left( \alpha +1\right) \left( \alpha +2\right) }{2}\right) \left[
\left\vert f^{\prime \prime }\left( a\right) \right\vert +\left\vert
f^{\prime \prime }\left( b\right) \right\vert \right] 
\end{eqnarray*}%
for $\alpha \geq 3$ and 
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{2^{\alpha
-1}\Gamma \left( \alpha +1\right) }{\left( b-a\right) ^{\alpha }}\left(
J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha }f\left( b\right) +J_{\left( 
\frac{a+b}{2}\right) ^{-}}^{\alpha }f\left( a\right) \right) \right\vert  \\
&& \\
&\leq &\frac{\left( b-a\right) ^{2}}{8\left( \alpha +1\right) \left( \alpha
+2\right) }\left( \frac{\left( \alpha +1\right) \left( \alpha +2\right) }{2}%
-1\right) \left[ \left\vert f^{\prime \prime }\left( a\right) \right\vert
+\left\vert f^{\prime \prime }\left( b\right) \right\vert \right] 
\end{eqnarray*}%
for $1\leq \alpha \leq 3.$
\end{corollary}

\begin{remark}
If we take $\alpha =1$ in Corollary \ref{c1}, then we get the following
inequality%
\begin{equation*}
\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\dint\limits_{a}^{b}f(t)dt\right\vert \leq \frac{\left( b-a\right) ^{2}}{12}%
\left[ \frac{\left\vert f^{\prime \prime }\left( a\right) \right\vert
+\left\vert f^{\prime \prime }\left( b\right) \right\vert }{2}\right] 
\end{equation*}%
which is given by Sarikaya and Aktan in \cite{sarikaya2}.
\end{remark}

\begin{remark}
Under the assumptions of Theorem \ref{t1} with $\lambda =\frac{1}{3}$ and $%
\alpha =1,$ then we get the following inequality%
\begin{equation*}
\left\vert \frac{1}{6}\left[ f\left( a\right) +4f\left( \frac{a+b}{2}\right)
+f\left( b\right) \right] -\frac{1}{b-a}\dint\limits_{a}^{b}f(t)dt\right%
\vert \leq \frac{\left( b-a\right) ^{2}}{81}\left[ \frac{\left\vert
f^{\prime \prime }\left( a\right) \right\vert +\left\vert f^{\prime \prime
}\left( b\right) \right\vert }{2}\right] 
\end{equation*}%
which is given by Sarikaya and Aktan in \cite{sarikaya2}.
\end{remark}

\begin{remark}
Under the assumptions of Theorem \ref{t1} with $\lambda =\frac{1}{2}$ and $%
\alpha =1,$ then we get the following inequality%
\begin{equation*}
\left\vert \frac{1}{b-a}\dint\limits_{a}^{b}f(t)dt-\frac{1}{2}\left[ \frac{%
f\left( a\right) +f\left( b\right) }{2}+f\left( \frac{a+b}{2}\right) \right]
\right\vert \leq \frac{\left( b-a\right) ^{2}}{48}\left[ \frac{\left\vert
f^{\prime \prime }\left( a\right) \right\vert +\left\vert f^{\prime \prime
}\left( b\right) \right\vert }{2}\right] 
\end{equation*}%
which is given by Sarikaya and Aktan in \cite{sarikaya2}.
\end{remark}

\begin{theorem}
\label{t2} Let $I\subset 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be an open intervial, $a,b\in I$ with $a<b$ and $f:I\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a twice differentiable mapping such that $f^{\prime \prime }$ is
integrable and $0\leq \lambda \leq 1,\ \alpha \geq 1.$ If $\left\vert
f^{\prime \prime }\right\vert ^{q}$ is a convex on $\left[ a,b\right] ,$ $%
q\geq 1$ then the following inequalities hold:%
\begin{eqnarray}
&&\left\vert \left[ \left( \lambda -\frac{\alpha +1}{2^{\alpha }}\right)
f\left( \frac{a+b}{2}\right) -\lambda \left( \frac{f\left( a\right) +f\left(
b\right) }{2}\right) +\frac{\Gamma \left( \alpha +2\right) }{2\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) \right] \right\vert   \label{h1} \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}}{2}\left( \frac{\alpha \lambda ^{1+\frac{%
2}{\alpha }}}{\alpha +2}+\frac{1}{2^{\alpha +2}\left( \alpha +2\right) }-%
\frac{\lambda }{8}\right) ^{1-\frac{1}{q}}  \notag \\
&&  \notag \\
&&\times \left\{ \left[ C_{1}\left\vert f^{\prime \prime }\left( a\right)
\right\vert ^{q}+C_{2}\left\vert f^{\prime \prime }\left( b\right)
\right\vert ^{q}\right] ^{\frac{1}{q}}+\left[ C_{2}\left\vert f^{\prime
\prime }\left( a\right) \right\vert ^{q}+C_{1}\left\vert f^{\prime \prime
}\left( b\right) \right\vert ^{q}\right] ^{\frac{1}{q}}\right\}   \notag
\end{eqnarray}%
for $0\leq \lambda \leq \frac{1}{2}$ and%
\begin{eqnarray}
&&\left\vert \left( \lambda -\frac{\alpha +1}{2^{\alpha }}\right) f\left( 
\frac{a+b}{2}\right) -\lambda \left( \frac{f\left( a\right) +f\left(
b\right) }{2}\right) +\frac{\Gamma \left( \alpha +2\right) }{2\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) \right\vert   \label{h2} \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}}{2}\left( \frac{\lambda }{8}-\frac{1}{%
2^{\alpha +2}\left( \alpha +2\right) }\right) ^{^{1-\frac{1}{q}}}  \notag \\
&&  \notag \\
&&\times \left\{ \left[ C_{3}\left\vert f^{\prime \prime }\left( a\right)
\right\vert ^{q}+C_{4}\left\vert f^{\prime \prime }\left( b\right)
\right\vert ^{q}\right] ^{^{\frac{1}{q}}}+\left[ C_{4}\left\vert f^{\prime
\prime }\left( a\right) \right\vert ^{q}+C_{3}\left\vert f^{\prime \prime
}\left( b\right) \right\vert ^{q}\right] ^{^{^{\frac{1}{q}}}}\right\} , 
\notag
\end{eqnarray}%
for $\frac{1}{2}\leq \lambda \leq 1$ where $\frac{1}{p}+\frac{1}{q}=1,$%
\begin{eqnarray*}
C_{1} &=&\left( \frac{2\alpha \lambda ^{1+\frac{3}{\alpha }}}{3\left( \alpha
+3\right) }+\frac{1}{2^{\alpha +3}\left( \alpha +3\right) }-\frac{\lambda }{%
24}\right)  \\
C_{2} &=&\left( \frac{\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}-%
\frac{2\alpha \lambda ^{1+\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+%
\frac{\alpha +4}{2^{\alpha +3}\left( \alpha +2\right) \left( \alpha
+3\right) }-\frac{\lambda }{12}\right)  \\
C_{3} &=&\left( \frac{\lambda }{24}-\frac{1}{2^{\alpha +3}\left( \alpha
+3\right) }\right)  \\
C_{4} &=&\left( \frac{\lambda }{12}-\frac{\alpha +4}{2^{\alpha +3}\left(
\alpha +2\right) \left( \alpha +3\right) }\right) .
\end{eqnarray*}
\end{theorem}

\begin{proof}
Suppose that $q\geq 1.$ From Lemma \ref{l1} and using the well known power
mean inequality, we have%
\begin{eqnarray}
&&\left\vert \left( \lambda -\frac{\alpha +1}{2^{\alpha }}\right) f\left( 
\frac{a+b}{2}\right) -\lambda \left( \frac{f\left( a\right) +f\left(
b\right) }{2}\right) +\frac{\Gamma \left( \alpha +2\right) }{2\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) \right\vert   \label{2.12} \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}}{2}\int_{0}^{1}\left\vert k\left(
t\right) \right\vert \left\vert f^{\prime \prime }\left( ta+\left(
1-t\right) b\right) \right\vert dt  \notag \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}}{2}\left\{ \int_{0}^{\frac{1}{2}%
}\left\vert t\left( t^{\alpha }-\lambda \right) \right\vert \left\vert
f^{\prime \prime }\left( ta+\left( 1-t\right) b\right) \right\vert dt\right. 
\notag \\
&&  \notag \\
&&\left. +\int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right) \left( \left(
1-t\right) ^{\alpha }-\lambda \right) \right\vert \left\vert f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) \right\vert dt\right\}   \notag
\\
&&  \notag \\
&=&\frac{\left( b-a\right) ^{2}}{2}\left\{ \left( \int_{0}^{\frac{1}{2}%
}\left\vert t\left( t^{\alpha }-\lambda \right) \right\vert dt\right) ^{1-%
\frac{1}{q}}\left( \int_{0}^{\frac{1}{2}}\left\vert t\left( t^{\alpha
}-\lambda \right) \right\vert \left\vert f^{\prime \prime }\left( ta+\left(
1-t\right) b\right) \right\vert ^{q}dt\right) ^{\frac{1}{q}}\right.   \notag
\\
&&  \notag \\
&&\left. +\left( \int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right) \left(
\left( 1-t\right) ^{\alpha }-\lambda \right) \right\vert dt\right) ^{^{1-%
\frac{1}{q}}}\left( \int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right)
\left( \left( 1-t\right) ^{\alpha }-\lambda \right) \right\vert \left\vert
f^{\prime \prime }\left( ta+\left( 1-t\right) b\right) \right\vert
^{q}dt\right) ^{^{\frac{1}{q}}}\right\}   \notag
\end{eqnarray}%
Let $0\leq \lambda \leq \frac{1}{2}.$ Then since $\left\vert f^{\prime
}\right\vert ^{q}$ is convex on $\left[ a,b\right] ,$we know that for $t\in %
\left[ 0,1\right] $%
\begin{equation*}
\left\vert f^{\prime }\left( ta+\left( 1-t\right) b\right) \right\vert
^{q}\leq t\left\vert f^{\prime }\left( a\right) \right\vert ^{q}+\left(
1-t\right) \left\vert f^{\prime }\left( b\right) \right\vert ^{q}
\end{equation*}%
hence, by simple computation%
\begin{eqnarray}
&&\int_{0}^{\frac{1}{2}}\left\vert t\left( t^{\alpha }-\lambda \right)
\right\vert \left\vert f^{\prime \prime }\left( ta+\left( 1-t\right)
b\right) \right\vert ^{q}dt  \label{2.13} \\
&&  \notag \\
&\leq &\int_{0}^{\lambda ^{\frac{1}{\alpha }}}t\left( \lambda -t^{\alpha
}\right) \left[ t\left\vert f^{\prime \prime }\left( a\right) \right\vert
^{q}+\left( 1-t\right) \left\vert f^{\prime \prime }\left( b\right)
\right\vert ^{q}\right] dt  \notag \\
&&  \notag \\
&&+\int_{\lambda ^{\frac{1}{\alpha }}}^{\frac{1}{2}}t\left( t^{\alpha
}-\lambda \right) \left[ t\left\vert f^{\prime \prime }\left( a\right)
\right\vert ^{q}+\left( 1-t\right) \left\vert f^{\prime \prime }\left(
b\right) \right\vert ^{q}\right] dt  \notag \\
&&  \notag \\
&=&\left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}\left[ \frac{%
2\alpha \lambda ^{1+\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{1}{%
2^{\alpha +3}\left( \alpha +3\right) }-\frac{\lambda }{24}\right]   \notag \\
&&  \notag \\
&&+\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}\left[ \frac{%
\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}-\frac{2\alpha \lambda ^{1+%
\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{\alpha +4}{2^{\alpha
+3}\left( \alpha +2\right) \left( \alpha +3\right) }-\frac{\lambda }{12}%
\right] ,  \notag
\end{eqnarray}%
\begin{eqnarray}
&&\int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right) \left( \left(
1-t\right) ^{\alpha }-\lambda \right) \right\vert \left\vert f^{\prime
\prime }\left( ta+\left( 1-t\right) b\right) \right\vert ^{q}dt  \label{2.14}
\\
&&  \notag \\
&\leq &\int_{\frac{1}{2}}^{1-\lambda ^{\frac{1}{\alpha }}}\left( 1-t\right)
\left( \left( 1-t\right) ^{\alpha }-\lambda \right) \left[ t\left\vert
f^{\prime \prime }\left( a\right) \right\vert ^{q}+\left( 1-t\right)
\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}\right] dt 
\notag \\
&&  \notag \\
&&+\int_{1-\lambda ^{\frac{1}{\alpha }}}^{1}\left( 1-t\right) \left( \lambda
-\left( 1-t\right) ^{\alpha }\right) \left[ t\left\vert f^{\prime \prime
}\left( a\right) \right\vert ^{q}+\left( 1-t\right) \left\vert f^{\prime
\prime }\left( b\right) \right\vert ^{q}\right] dt  \notag \\
&&  \notag \\
&=&\left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}\left[ \frac{%
\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}-\frac{2\alpha \lambda ^{1+%
\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{\alpha +4}{2^{\alpha
+3}\left( \alpha +2\right) \left( \alpha +3\right) }-\frac{\lambda }{12}%
\right]   \notag \\
&&  \notag \\
&&+\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}\left[ \frac{%
2\alpha \lambda ^{1+\frac{3}{\alpha }}}{3\left( \alpha +3\right) }+\frac{1}{%
2^{\alpha +3}\left( \alpha +3\right) }-\frac{\lambda }{24}\right] ,  \notag
\end{eqnarray}%
\begin{equation}
\int_{0}^{\frac{1}{2}}\left\vert t\left( t^{\alpha }-\lambda \right)
\right\vert dt=\int_{0}^{\lambda ^{\frac{1}{\alpha }}}t\left( \lambda
-t^{\alpha }\right) dt+\int_{\lambda ^{\frac{1}{\alpha }}}^{\frac{1}{2}%
}t\left( t^{\alpha }-\lambda \right) dt=\frac{\alpha \lambda ^{1+\frac{2}{%
\alpha }}}{\alpha +2}+\frac{1}{2^{\alpha +2}\left( \alpha +2\right) }-\frac{%
\lambda }{8}  \label{2.15}
\end{equation}%
and%
\begin{eqnarray}
&&\int_{\frac{1}{2}}^{1}\left\vert \left( 1-t\right) \left( \left(
1-t\right) ^{\alpha }-\lambda \right) \right\vert dt  \label{2.16} \\
&&  \notag \\
&=&\int_{\frac{1}{2}}^{1-\lambda ^{\frac{1}{\alpha }}}\left( 1-t\right)
\left( \left( 1-t\right) ^{\alpha }-\lambda \right) dt+\int_{1-\lambda ^{%
\frac{1}{\alpha }}}^{1}\left( 1-t\right) \left( \lambda -\left( 1-t\right)
^{\alpha }\right) dt  \notag \\
&&  \notag \\
&=&\frac{\alpha \lambda ^{1+\frac{2}{\alpha }}}{\alpha +2}+\frac{1}{%
2^{\alpha +2}\left( \alpha +2\right) }-\frac{\lambda }{8}.  \notag
\end{eqnarray}%
Substituting the equalities (\ref{2.13})-(\ref{2.16}) in (\ref{2.12}), the
we obtain the inequality (\ref{h1}). One can prove the inequality (\ref{h2})
similar to (\ref{h1}). It is omited to readers.
\end{proof}

\begin{remark}
Under the assumptions Theorem \ref{t2} with $\alpha =1$, then Theorem \ref%
{t2} reduces to Theorem 4 in \cite{sarikaya2}.
\end{remark}

\begin{remark}
Under the assumptions of Theorem \ref{t2}  with $\lambda =\frac{1}{3}$ and $%
\alpha =1,$ then we get the following inequality%
\begin{eqnarray*}
&&\left\vert \frac{1}{6}\left[ f\left( a\right) +4f\left( \frac{a+b}{2}%
\right) +f\left( b\right) \right] -\frac{1}{b-a}\dint\limits_{a}^{b}f(t)dt%
\right\vert  \\
&& \\
&\leq &\frac{\left( b-a\right) ^{2}}{162}\left[ \left( \frac{59\left\vert
f^{\prime \prime }\left( a\right) \right\vert ^{q}+133\left\vert f^{\prime
\prime }\left( b\right) \right\vert ^{q}}{2^{6}\times 3}\right) ^{\frac{1}{q}%
}+\left( \frac{133\left\vert f^{\prime \prime }\left( a\right) \right\vert
^{q}+59\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}}{%
2^{6}\times 3}\right) ^{\frac{1}{q}}\right] 
\end{eqnarray*}%
which is given by Sarikaya and Aktan in \cite{sarikaya2}.
\end{remark}

\begin{corollary}
Under the assumptions Theorem \ref{t2} with $\lambda =0$, then we get the
following inequality%
\begin{eqnarray}
&&\left\vert \frac{2^{\alpha -1}\Gamma \left( \alpha +1\right) }{\left(
b-a\right) ^{\alpha }}\left( J_{\left( \frac{a+b}{2}\right) ^{+}}^{\alpha
}f\left( b\right) +J_{\left( \frac{a+b}{2}\right) ^{-}}^{\alpha }f\left(
a\right) \right) -f\left( \frac{a+b}{2}\right) \right\vert   \label{2.17} \\
&&  \notag \\
&\leq &\frac{\left( b-a\right) ^{2}2^{\alpha -1}}{\alpha +1}\left\{ \left( 
\frac{1}{2^{\alpha +2}\left( \alpha +2\right) }\right) ^{1-\frac{1}{q}%
}\right.   \notag \\
&&  \notag \\
&&\times \left[ \left\vert f^{\prime \prime }\left( a\right) \right\vert ^{q}%
\frac{1}{2^{\alpha +3}\left( \alpha +3\right) }+\left\vert f^{\prime \prime
}\left( b\right) \right\vert ^{q}\frac{\alpha +4}{2^{\alpha +3}\left( \alpha
+2\right) \left( \alpha +3\right) }\right] ^{\frac{1}{q}}  \notag \\
&&  \notag \\
&&+\left. \left[ \left\vert f^{\prime \prime }\left( a\right) \right\vert
^{q}\frac{\alpha +4}{2^{\alpha +3}\left( \alpha +2\right) \left( \alpha
+3\right) }+\left\vert f^{\prime \prime }\left( b\right) \right\vert ^{q}%
\frac{1}{2^{\alpha +3}\left( \alpha +3\right) }\right] ^{\frac{1}{q}%
}\right\} .  \notag
\end{eqnarray}
\end{corollary}

\begin{thebibliography}{99}
\bibitem{alomari} M. Alomari, M. Darus, U. S. Kirmaci, \textit{Refinements
of Hadamard-type inequalities for quasi-convex functions with applications
to trapezoidal formula and to special means}, Comput. Math. Appl., 59
(2010), 225--232.

\bibitem{anas} G. A. Anastassiou, \textit{General Fractional
Hermite--Hadamard Inequalities Using m-Convexity and }$(s,m)$\textit{%
-Convexity}, Frontiers in Time Scales and Inequalities. 2016. 237-255.

\bibitem{AAG} A.G. Azpeitia, \textit{Convex functions and the Hadamard
inequality}, Rev. Colombiana Math., 28 (1994), 7-12.

\bibitem{cal} J. de la Cal, J. Carcamob, L. Escauriaza, \textit{A general
multidimensional Hermite-Hadamard type inequality}, J. Math. Anal. Appl.,
356 (2009), 659--663.

\bibitem{chen} H. Chen and U.N. Katugampola, \textit{Hermite--Hadamard and
Hermite--Hadamard--Fej\'{e}r type inequalities for generalized fractional
integrals}, J. Math. Anal. Appl. 446 (2017) 1274--1291

\bibitem{Dragomir1} S.S. Dragomir and C.E.M. Pearce, Selected Topics on
Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria
University, 2000. Online:[http://rgmia.org/papers/monographs/Master2.pdf].

\bibitem{dragomir} S.S. Dragomir, R.P. Agarwal, \textit{Two inequalities for
differentiable mappings and applications to special means of real numbers
and to trapezoidal formula}, Appl. Math. lett. 11 (5) (1998) 91--95.

\bibitem{Gorenflo} R. Gorenflo, F. Mainardi, \textit{Fractional calculus:
integral and differential equations of fractional order}, Springer Verlag,
Wien (1997), 223-276.

\bibitem{had} J. Hadamard, \textit{Etude sur les proprietes des fonctions
entieres en particulier d'une fonction consideree par Riemann}, J. Math.
Pures Appl. 58 (1893), 171-215.

\bibitem{hussain} S. Hussain, M. I. Bhatti and M. Iqbal, \textit{%
Hadamard-type inequalities for }$\mathit{s}$\textit{-convex functions} I,
Punjab Univ. Jour. of Math., Vol.41, pp:51-60, (2009).

\bibitem{iqbal} M. Iqbal, S. Qaisar and M. Muddassar, \textit{A short note
on integral inequality of type Hermite-Hadamard through convexity}, J.
Computational analaysis and applications, 21(5), 2016, pp.946-953.

\bibitem{kavurmaci} H. Kavurmaci, M. Avci and M E. Ozdemir, \textit{New
inequalities of hermite-hadamard type for convex functions with applications}%
, Journal of Inequalities and Applications 2011, Art No. 86, Vol 2011.

\bibitem{kilbas} A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, \textit{%
Theory and Applications of Fractional Differential Equations}, North-Holland
Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.

\bibitem{kirmaci} U. S. Kirmaci, \textit{Inequalities for differentiable
mappings and applications to special means of real numbers to midpoint
formula}, Appl. Math. Comput., vol. 147, no. 5, pp. 137--146, 2004, doi:
10.1016/S0096-3003(02)00657-4.

\bibitem{Miller} S. Miller and B. Ross, \textit{An introduction to the
Fractional Calculus and Fractional Differential Equations}, John Wiley \&
Sons, USA, 1993, p.2.

\bibitem{noor} M. A. Noor and M. U. Awan, Some integral inequalities for two
kinds of convexities via fractional integrals, TJMM, 5(2), 2013, pp. 129-136.

\bibitem{ozdemir} M. E. \"{O}zdemir, M. Avc\i -Ard\i \c{c} and H. Kavurmac\i
-\"{O}nalan, \textit{Hermite-Hadamard type inequalities for }$s$\textit{%
-convex and }$s$\textit{-concave functions via fractional integrals, }%
Turkish J.Science,1(1), 28-40, 2016.

\bibitem{ozdemir2} M. E. \"{O}demir, M. Avci, and E. Set, \textit{On some
inequalities of Hermite--Hadamard-type via }$m$\textit{-convexity}, Appl.
Math. Lett. 23 (2010), pp. 1065--1070.

\bibitem{ozdemir3} M. E. \"{O}demir, M. Avci, and H. Kavurmaci, \textit{%
Hermite--Hadamard-type inequalities via }$(\alpha ,m)$\textit{-convexity},
Comput. Math. Appl. 61 (2011), pp. 2614--2620.

\bibitem{Pecaric} J.E. Pe\v{c}ari\'{c}, F. Proschan and Y.L. Tong, \textit{%
Convex Functions, Partial Orderings and Statistical Applications}, Academic
Press, Boston, 1992.

\bibitem{Podlubni} I. Podlubni, \textit{Fractional Differential Equations},
Academic Press, San Diego, 1999.

\bibitem{Raina} R.K. Raina, \textit{On generalized Wright's hypergeometric
functions and fractional calculus operators}, East Asian Math. J., 21(2)
(2005), 191-203.

\bibitem{saglam} A. Saglam, M. Z. Sarikaya ve H. Yildirim, \textit{Some new
inequalities of Hermite-Hadamard's type}, Kyungpook Mathematical Journal,
50(2010), 399-410.

\bibitem{sarikaya1} M.Z. Sarikaya and H. Yildirim, \textit{On
Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals%
}, Miskolc Mathematical Notes, 7(2) (2016), pp. 1049--1059.

\bibitem{sarikaya2} M. Z. Sarikaya and N. Aktan, \textit{On the
generalization some integral inequalities and their applications}
Mathematical and Computer Modelling, 54, Issues 9-10, pp. 2175-2182.

\bibitem{sarikaya3} M.Z. Sarikaya and H. Ogunmez, \textit{On new
inequalities via Riemann-Liouville fractional integration}, Abstract and
Applied Analysis, Volume 2012 (2012), Article ID 428983, 10 pages.

\bibitem{sarikaya4} M.Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, \textit{%
Hermite -Hadamard's inequalities for fractional integrals and related
fractional inequalities}, Mathematical and Computer Modelling,
DOI:10.1016/j.mcm.2011.12.048, 57 (2013) 2403--2407.

\bibitem{sarikaya5} M.Z. Sarikaya and H. Budak, \textit{Generalized
Hermite-Hadamard type integral inequalities for fractional integrals},
Filomat 30:5 (2016), 1315--1326.

\bibitem{sarikaya6} M. Z. Sarikaya, A. Saglam, and H. Yildirim, \textit{New
inequalities of Hermite-Hadamard type for functions whose second derivatives
absolute values are convex and quasi-convex}, International Journal of Open
Problems in Computer Science and Mathematics ( IJOPCM), 5(3), 2012, pp:1-14.

\bibitem{set} E. Set, M. E. Ozdemir and M. Z. Sarikaya, \textit{New
inequalities of Ostrowski's type for }$\mathit{s}$\textit{-convex functions
in the second sense with applications}, Facta Universitatis, Ser. Math.
Inform. Vol. 27, No 1 (2012), 67-82.

\bibitem{wang} J. Wang, X. Li, M. Feckan and Y. Zhou, \textit{%
Hermite-Hadamard-type inequalities for Riemann-Liouville fractional
integrals via two kinds of convexity}, Appl. Anal. (2012).
doi:10.1080/00036811.2012.727986.

\bibitem{whang} J.Wang, X. Lia and Y. Zhou, \textit{Hermite-Hadamard
Inequalities Involving Riemann-Liouville Fractional Integrals via s-convex
functions and applications to special means, }Filomat 30:5 (2016),
1143--1150.

\bibitem{xi} B-Y, Xi and F. Qi, \textit{Some Hermite-Hadamard type
inequalities for differentiable convex functions and applications,} Hacet.
J. Math. Stat.. 42(3), 243--257 (2013).

\bibitem{xi1} B-Y, Xi and F. Qi, \textit{Hermite-Hadamard type inequalities
for functions whose derivatives are of convexities,} Nonlinear Funct. Anal.
Appl.. 18(2), 163--176 (2013)

\bibitem{zhang} Y. Zhang and J-R. Wang, \textit{On some new Hermite-Hadamard
inequalities involving Riemann-Liouville fractional integrals,} Journal of
Inequalities and Applications 2013, 2013:220.
\end{thebibliography}

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