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\begin{document}
\begin{center}
\textbf{On Some New Inequalities of Hadamard-Type for }$h-$\textbf{Convex
Functions} \\[0pt]
\hrule
\textbf{A.O. Akdemir}$^{1}$, \textbf{E. Set}$^{2}$, \textbf{M.E. \"{O}zdemir}%
$^{3}$ and \textbf{\c{C}. Y\i ld\i z}$^{3}$
$^{1}$A\u{g}r\i\ \.{I}brahim \c{C}e\c{c}en University, Faculty of Science
and Letters, Department of Mathematics, A\u{g}r\i , Turkey $^{2}$Department
of Mathematics, Faculty of Science and Arts, D\"{u}zce University, D\"{u}%
zce, Turkey $^{3}$Atat\"{u}rk University, K.K. Education Faculty, Department
of Mathematics, 25240, Erzurum, Turkey
\end{center}
\textbf{Abstract}
A function $f:I\rightarrow\mathbb{R}$, $I\subseteq\mathbb{R}$ is an
interval, is said to be a convex function on $I$ if
\begin{equation}
f\left( tx+\left( 1-t\right) y\right) \leq tf\left( x\right) +\left(
1-t\right) f\left( y\right) \label{convex}
\end{equation}
holds for all $x,y\in I$ and $t\in\left[ 0,1\right] $. If the reversed
inequality in (\ref{convex}) holds, then $f$ is concave.
If $[a,b]\subseteq I$, then the following double inequality holds for any
convex function $f$:
\begin{equation}
f\left( \frac{a+b}{2}\right) \leq\frac{1}{b-a}\int_{a}^{b}f\left( x\right)
dx\leq\frac{f\left( a\right) +f\left( b\right) }{2}. \label{1.1}
\end{equation}
This result is known as the Hermite-Hadamard inequality for convex function.
A number of papers have been written on this inequality providing new
proofs, noteworthy extensions, generalizations, refinements, counterparts
and new Hadamard-type inequalities and numerous applications, see \cite{WW}-%
\cite{A} and the references therein.
In \cite{SA}, Varo\v{s}anec gave the following definition:
\begin{definition}
( See \cite{SA}) Let $I,J$ be intervals $\mathbb{R}$, $(0,1)\subseteq J$ and
let $h:J\rightarrow\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$ is
called an $h-$convex function, or that $f$ belongs to the class $SX(h,I)$,
if for all $x,y\in I$\ and $t\in(0,1)$\ we have
\begin{equation}
f\left( tx+\left( 1-t\right) y\right) \leq h\left( t\right) f\left( x\right)
+h\left( 1-t\right) f\left( y\right) . \label{1.5}
\end{equation}
If inequality in (\ref{1.5}) is reversed, then $f$ is said to be $h-$concave.
\end{definition}
Obviously, if $h\left( t\right) =t$, for all $t\in \lbrack 0,1]\subseteq J$,
then all convex functions belong to $SX\left( h,I\right) $\ and all concave
functions are $h-$concave; if $h(t)=\frac{1}{t},$ for all $t\in (0,1)$, then
$SX(h,I)=Q(I);$ if $h(t)=1,$ $SX(h,I)\supseteq P(I);$ and if $h(t)=t^{s},$
where $s\in \left( 0,1\right) ,$ then $SX(h,I)\supseteq K_{s}^{2}.$ For some
recent results about $h-$convex functions we refer the reader to papers \cite%
{SA2}, \cite{BU}, \cite{SAR} and \cite{OZ1}.
\begin{lemma}
Let $f:I\subseteq
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\rightarrow
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$ be a twice differentiable function on $I^{\circ }$ , the interior of $I$,
where $a,b\in I$ with $a