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\begin{center}{\footnotesize Khayyam J. Math. 1 (2015), no. 1, 115--124}\\\end{center}
\noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=0.24]{KJM.jpg}}
\vspace{0.5cm}

\title[Inequalities for $\alpha-,m-,\left( \alpha ,m\right)-$logarithmically convex functions]{Some integral inequalities for  $\alpha-,m-,\left( \alpha ,m\right)-$logarithmically convex
functions}

\author[M. TUN\c{C}, E. Y\"{U}KSEL]{MEVL\"{U}T TUN\c{C}$^1$$^{*}$, EBRU Y\"{U}KSEL$^2$}

\address{$^{1}$ Department of Mathematics, Faculty of Science and
Arts, Mustafa Kemal University, Hatay, 31000, Turkey.}
\email{mevluttttunc@gmail.com}

\address{$^{2}$ Department of Mathematics, Faculty of Science
and Arts, A\u{g}r\i\ \.{I}brahim \c{C}e\c{c}en University,
A\u{g}r\i , 04000, Turkey.}
\email{yuksel.ebru90@hotmail.com}

\dedicatory{\rm Communicated by S. Hejazian}

\subjclass[2010]{Primary 26A15; Secondary 26A51, 26D10.}

\keywords{ $\alpha-,m-,\left( \alpha ,m\right)-$logarithmically convex, Hadamard's inequality, H\"{o}lder's
inequality, power mean inequality, Cauchy's inequality.}

\date{Received: 12 November 2014; Revised: 15 December 2014; Accepted: 23 December 2014.
\newline \indent $^{*}$ Corresponding author}

\begin{abstract}
In this paper, the authors establish some Hermite-Hadamard type
inequalities by using elementary inequalities for functions whose
first derivative
absolute values are $\alpha $-, $m$-$,$ $\left( \alpha ,m\right) $%
-logarithmically convex.
\end{abstract} \maketitle

\section{Introduction and preliminaries}

In this section, we will present definitions and some results used
in this paper.

Let \ $f:I\subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a convex mapping defined on the interval $I$\ of real numbers and $%
a,b\in I$, with $a<b$. The following double inequalities:%
\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}
\end{equation*}%
hold. This double inequality is known in the literature as the
Hermite-Hadamard inequality for convex functions (see
\cite{bai}-\cite{tnc}).

\begin{definition}
Let $I$ be an interval in $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
.$ Then $f:I\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ $\emptyset \neq I\subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$\ is said to be convex 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{7}
\end{equation}%
for all $x,y\in I$ and $t\in \left[ 0,1\right] $.
\end{definition}

The concepts of $\alpha $-, $m$- and $\left( \alpha ,m\right) $%
-logarithmically convex functions were introduced as follows.

\begin{definition}
\cite{bai}\label{d} A function $f:[0,b]\rightarrow (0,\infty )$ is
said to
be $m$-logarithmically convex if the inequality%
\begin{equation}
f\left( tx+m\left( 1-t\right) y\right) \leq \left[ f\left(
x\right) \right] ^{t}\left[ f\left( y\right) \right] ^{m\left(
1-t\right) }  \label{d1}
\end{equation}%
holds for all $x,y\in \lbrack 0,b]$, $m\in (0,1]$, and $t\in
\lbrack 0,1]$.
\end{definition}

Obviously, if putting $m=1$ in Definition \ref{d}, then $f$ is
just the ordinary logarithmically convex on $\left[ 0,b\right] $.

\begin{definition}
\cite{tnc}\label{ddd} A function $f:[0,b]\rightarrow (0,\infty )$
is said to
be $\alpha $-logarithmically convex if%
\begin{equation}
f\left( tx+\left( 1-t\right) y\right) \leq \left[ f\left( x\right)
\right] ^{t^{\alpha }}\left[ f\left( y\right) \right] ^{\left(
1-t^{\alpha }\right) }
\end{equation}%
holds for all $x,y\in \lbrack 0,b]$, $\alpha \in \left( 0,1\right] $ and $%
t\in \lbrack 0,1]$.
\end{definition}

Clearly, when taking $\alpha =1$ in Definition \ref{ddd}, then $f$
becomes the ordinary logarithmically convex on $\left[ 0,b\right]
$.

\begin{definition}
\cite{bai}\label{dd} A function $f:[0,b]\rightarrow (0,\infty )$
is said to
be $\left( \alpha ,m\right) $-logarithmically convex if%
\begin{equation}
f\left( tx+m\left( 1-t\right) y\right) \leq \left[ f\left(
x\right) \right] ^{t^{\alpha }}\left[ f\left( y\right) \right]
^{m\left( 1-t^{\alpha }\right) }  \label{d2}
\end{equation}%
holds for all $x,y\in \lbrack 0,b]$, $\left( \alpha ,m\right) \in \left( 0,1%
\right] \times \left( 0,1\right] ,$ and $t\in \lbrack 0,1]$.
\end{definition}

Clearly, when taking $\alpha =1$ in Definition \ref{dd}, then $f$
becomes the standard $m$-logarithmically convex function on
$\left[ 0,b\right] $,
and, when taking $m=1$ in Definition \ref{dd}, then $f$ becomes the $\alpha $%
-logarithmically convex function on $\left[ 0,b\right] $.

In \cite{dr3}, the following theorem which was obtained by
Dragomir and Agarwal contains the Hermite-Hadamard type integral
inequality.

\begin{theorem}
\cite[Theorem 2.2]{dr3} Let $f:I\subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a differentiable mapping on $I^{\circ }$, the interior of
$I$, $a,b\in
I^{\circ }$ with $a<b$. If $|f^{\prime }\left( x\right) |$ is convex on $%
[a,b]$, then%
\begin{equation}
\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \leq \frac{\left(
b-a\right) \left( \left\vert f^{\prime }\left( a\right)
\right\vert +\left\vert f^{\prime }\left( b\right) \right\vert
\right) }{8}.  \label{109}
\end{equation}
\end{theorem}

\begin{theorem}
\cite[Theorem 2.3]{dr3} Let $f:I\subseteq
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a differentiable mapping on $I^{\circ }$, $a,b\in I^{\circ }$
with $a<b, $ and $p>1$. If the new mapping $|f^{\prime }\left(
x\right) |^{p/p-1}$ is
convex on $[a,b]$, then%
\begin{eqnarray}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert  \\
&\leq &\frac{b-a}{2\left( p+1\right) ^{1/p}}\left[
\frac{\left\vert f^{\prime }\left( a\right) \right\vert ^{p/\left(
p-1\right) }+\left\vert f^{\prime }\left( b\right) \right\vert
^{p/\left( p-1\right) }}{2}\right] ^{\left( p-1\right) /p}.
\notag
\end{eqnarray}
\end{theorem}

The aim of this paper is to establish some integral inequalities
of
Hermite-Hadamard type for $\alpha $-, $m$-$,$ $\left( \alpha ,m\right) $%
-logarithmically convex functions.



\section{Hadamard Type Inequalities}

In order to prove our main theorems, we need the following lemma
\cite{lmmm}.

\begin{lemma}
\label{l1}\cite{lmmm} Let $f:\ I\subset
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a differentiable mapping on $I^{\circ }$\textit{, }$a,b\in $
$I^{\circ } $ with $a$ $<$ $b$. If $f^{\prime }\in $ $L\left[
a,b\right] ,$ then the following equality holds:
\begin{eqnarray}
&&\frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx  \label{a} \\
&=&\frac{b-a}{2}\int_{0}^{1}\int_{0}^{1}\left[ f^{\prime }\left(
ta+\left( 1-t\right) b\right) -f^{\prime }\left( sa+\left(
1-s\right) b\right) \right] \left( s-t\right) dtds.  \notag
\end{eqnarray}
\end{lemma}

A simple proof of this equality can be also done integrating by
parts in the right hand side (see \cite{lmmm}).

The next theorems gives a new result of the upper Hermite-Hadamard
inequality for $\alpha $-, $m$-, $\left( \alpha ,m\right)
$-logarithmically convex functions.

\begin{theorem}
\label{t1}Let$\ I\supset \left[ 0,\infty \right) $ be an open
interval and let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$ such that $f^{\prime }\in L\left(
a,b\right) $ for $0\leq a<b<\infty . $ If $\left\vert f^{\prime
}\left( x\right) \right\vert $ is $\left( \alpha ,m\right)
$-logarithmically convex on $\left[ 0,\frac{b}{m}\right] $ for
$\left( \alpha ,m\right) \in \left( 0,1\right] ^{2},$ then
\begin{eqnarray}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\left\{
\begin{array}{cc}
\frac{\left( b-a\right) }{3}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m},\text{ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ }
& \eta =1 \\
\frac{\left( b-a\right) }{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\frac{-\alpha ^{2}\ln ^{2}\eta
-2\alpha \ln \eta +2\eta
^{\alpha }-2}{\alpha ^{3}\ln ^{3}\eta }, & \eta <1%
\end{array}%
\right.  \notag
\end{eqnarray}%
where $\eta =\left\vert f^{\prime }\left( a\right) \right\vert
/\left\vert f^{\prime }\left( \frac{b}{m}\right) \right\vert
^{m}.$
\end{theorem}

\begin{proof}
By Lemma \ref{l1} and since $\left\vert f^{\prime }\right\vert $ is an $%
\left( \alpha ,m\right) $-logarithmically convex on $\left[ 0,\frac{b}{m}%
\right] $, then we have%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\int_{0}^{1}\int_{0}^{1}\left\vert \left(
f^{\prime }\left( ta+\left( 1-t\right) b\right) \right) -\left(
f^{\prime }\left( sa+\left( 1-s\right) b\right) \right)
\right\vert \left\vert s-t\right\vert
dtds \\
&\leq &\frac{b-a}{2}\left[ \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \left\vert f^{\prime }\left( a\right) \right\vert
^{t^{\alpha }}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m\left( 1-t^{\alpha
}\right) }dtds\right] \\
&&+\frac{b-a}{2}\left[ \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \left\vert f^{\prime }\left( a\right) \right\vert
^{s^{\alpha }}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m\left( 1-s^{\alpha }\right) }dtds\right]
\end{eqnarray*}%
Let $0<k\leq 1,$ $0\leq m\leq 1,$ and $0<n\leq 1$. Then%
\begin{equation}
k^{m^{n}}\leq k^{nm}.  \label{1}
\end{equation}
When $\eta =1,$ by (\ref{1}), we get%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left[
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
dtds+\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert dtds\right] \\
&=&\frac{b-a}{3}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}
\end{eqnarray*}%
When $0<\eta <1,$ by (\ref{1}), we get
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left[
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \eta ^{\alpha
t}dtds+\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \eta
^{\alpha s}dtds\right] \\
&=&\frac{b-a}{2}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left[ \frac{-\alpha ^{2}\ln ^{2}\eta -2\alpha \ln
\eta +4\eta ^{\alpha
}+\alpha ^{2}\eta ^{\alpha }\ln ^{2}\eta -2\alpha \eta ^{\alpha }\ln \eta -4%
}{2\alpha ^{3}\ln ^{3}\eta }\right. \\
&&+\left. \frac{-\alpha \ln \eta +2\eta ^{\alpha }-\alpha \eta
^{\alpha }\ln \eta -2}{2\alpha ^{2}\ln ^{2}\eta }\right]
\end{eqnarray*}%
which completes the proof.
\end{proof}

\begin{corollary}
Let$\ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert $ is
$m$-logarithmically
convex on $\left[ 0,\frac{b}{m}\right] $ for $m\in \left( 0,1\right] $, then
\begin{equation*}
\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \leq \left\{
\begin{array}{cc}
\frac{\left( b-a\right) }{3}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m},\text{ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ } & \eta =1
\\
\frac{\left( b-a\right) }{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\frac{-\ln ^{2}\eta -2\ln \eta
+2\eta -2}{\ln ^{3}\eta }, &
\eta <1%
\end{array}%
\right.
\end{equation*}%
where $\eta $ is same as Theorem \ref{t1}.
\end{corollary}

\begin{corollary}
Let$\ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert $is $\alpha $%
-logarithmically convex on $\left[ 0,b\right] $ for $\alpha \in \left( 0,1%
\right] $ , then%
\begin{equation}
\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \leq \left\{
\begin{array}{cc}
\frac{\left( b-a\right) }{3}\left\vert f^{\prime }\left( b\right)
\right\vert ,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ }
& \eta =1 \\
\frac{\left( b-a\right) }{2}\left\vert f^{\prime }\left( b\right)
\right\vert \frac{4\eta ^{\alpha }-4\alpha \ln \eta -2\alpha
^{2}\ln
^{2}\eta -4}{2\alpha ^{3}\ln ^{3}\eta }, & \eta <1%
\end{array}%
\right.  \notag
\end{equation}%
where $\eta =\left\vert f^{\prime }\left( a\right) \right\vert
/\left\vert f^{\prime }\left( b\right) \right\vert .$
\end{corollary}

\begin{theorem}
\label{t2}Let$\ I\supset \left[ 0,\infty \right) $ be an open
interval and let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$ such that $f^{\prime }\in L\left(
a,b\right) $ for $0\leq a<b<\infty
. $ If $\left\vert f^{\prime }\left( x\right) \right\vert ^{q}$ is an $%
\left( \alpha ,m\right) $-logarithmically convex on $\left[ 0,\frac{b}{m}%
\right] $ for $\left( \alpha ,m\right) \in \left( 0,1\right] ^{2}$ and $%
p,q>1 $ with $\frac{1}{p}+\frac{1}{q}=1,$ then%
\begin{eqnarray}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\left\{
\begin{array}{cc}
\left( b-a\right) \left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left( \frac{2}{\left( p+1\right) \left(
p+2\right) }\right) ^{\frac{1}{p}},\text{ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ } &
\eta =1 \\
\left( b-a\right) \left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left( \frac{2}{\left( p+1\right) \left(
p+2\right) }\right)
^{\frac{1}{p}}\times \left( \frac{\eta \left( \alpha q,\alpha q\right) -1}{%
\ln \eta \left( \alpha q,\alpha q\right) }\right) ^{\frac{1}{q}}, & \eta <1%
\end{array}%
\right.  \notag
\end{eqnarray}%
where $\eta \left( \alpha ,\alpha \right) $ is same as Theorem
\ref{t1}.
\end{theorem}

\begin{proof}
Since $\left\vert f^{\prime }\right\vert ^{q}$ is an $\left(
\alpha ,m\right) $-logarithmically convex on $\left[
0,\frac{b}{m}\right] $, from
Lemma \ref{l1} and the well known H\"{o}lder inequality, we have%
\begin{eqnarray}
&&  \label{y} \\
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert  \notag \\
&\leq &\frac{b-a}{2}\int_{0}^{1}\int_{0}^{1}\left\vert \left(
f^{\prime }\left( ta+\left( 1-t\right) b\right) \right) -\left(
f^{\prime }\left( sa+\left( 1-s\right) b\right) \right)
\right\vert \left\vert s-t\right\vert
dtds  \notag \\
&\leq &\frac{b-a}{2}\int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \left\vert f^{\prime }\left( a\right) \right\vert
^{t^{\alpha }}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m\left( 1-t^{\alpha
}\right) }dtds  \notag \\
&&+\frac{b-a}{2}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
\left\vert f^{\prime }\left( a\right) \right\vert ^{s^{\alpha
}}\left\vert f^{\prime }\left( \frac{b}{m}\right) \right\vert
^{m\left( 1-s^{\alpha }\right) }dtds
\notag \\
&\leq &\frac{b-a}{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left(
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
^{p}dtds\right) ^{\frac{1}{p}}  \notag \\
&&\times \left[ \left( \int_{0}^{1}\int_{0}^{1}\eta ^{qt^{\alpha
}}dtds\right) ^{\frac{1}{q}}+\left( \int_{0}^{1}\int_{0}^{1}\eta
^{qs^{\alpha }}dtds\right) ^{\frac{1}{q}}\right]  \notag
\end{eqnarray}%
If $\ \eta =1,$ by (\ref{1}), we obtain%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\left( b-a\right) \left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left(
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
^{p}dtds\right) ^{\frac{1}{p}} \\
&=&\left( b-a\right) \left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left( \frac{2}{\left(
p+1\right) \left( p+2\right) }\right) ^{\frac{1}{p}}
\end{eqnarray*}%
If $\eta <1,$ by (\ref{1}), we obtain%
\begin{eqnarray}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left(
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
^{p}dtds\right) ^{\frac{1}{p}}  \notag \\
&&\times \left[ \left( \int_{0}^{1}\int_{0}^{1}\eta ^{qt^{\alpha
}}dtds\right) ^{\frac{1}{q}}+\left( \int_{0}^{1}\int_{0}^{1}\eta
^{qs^{\alpha }}dtds\right) ^{\frac{1}{q}}\right]  \notag \\
&=&\left( b-a\right) \left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left( \frac{2}{\left(
p+1\right) \left( p+2\right) }\right)
^{\frac{1}{p}}\times \left( \frac{\eta \left( \alpha q,\alpha q\right) -1}{%
\ln \eta \left( \alpha q,\alpha q\right) }\right) ^{\frac{1}{q}}
\notag
\end{eqnarray}

\textit{which completes the proof.}
\end{proof}

\begin{corollary}
Let$\ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert ^{q}$is an $m$%
-logarithmically convex on $\left[ 0,\frac{b}{m}\right] $ for
$m\in \left(0,1\right] $ and $p=q=2,$ then
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert  \\
&\leq &\left( b-a\right) \left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\sqrt{\frac{1}{6}}\times
\left\{
\begin{array}{cc}
1,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } & \eta =1 \\
\left( \frac{\eta \left( 2,2\right) -1}{\ln \eta \left( 2,2\right)
}\right)
^{\frac{1}{2}}, & \eta <1%
\end{array}%
\right.
\end{eqnarray*}
\end{corollary}

\begin{corollary}
Let$\ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert $is $\alpha $%
-logarithmically convex on $\left[ 0,b\right] $ for $\alpha \in \left( 0,1%
\right] $ , then%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert  \\
&\leq &\left( b-a\right) \left\vert f^{\prime }\left( b\right)
\right\vert
\left( \frac{2}{\left( p+1\right) \left( p+2\right) }\right) ^{\frac{1}{p}%
}\times \left\{
\begin{array}{cc}
1,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } & \eta =1 \\
\left( \frac{\eta \left( \alpha q,\alpha q\right) -1}{\ln \eta
\left( \alpha
q,\alpha q\right) }\right) ^{\frac{1}{q}}, & \eta <1%
\end{array}%
\right.
\end{eqnarray*}

where $\eta =\left\vert f^{\prime }\left( a\right) \right\vert
/\left\vert f^{\prime }\left( b\right) \right\vert .$
\end{corollary}

\begin{theorem}
Let$\ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f:\ I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert ^{q}$ is $\left(
\alpha
,m\right) $-logarithmically convex on $\left[ 0,\frac{b}{m}\right] $ for $%
\left( \alpha ,m\right) \in \left( 0,1\right] ^{2}$, and then%
\begin{eqnarray}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \leq
\frac{b-a}{3}\left\vert f^{\prime }\left( \frac{b}{m}\right) \right\vert ^{m}  \nonumber \\
&&\times \left\{
\begin{array}{ll}
1, & \eta =1 \\
\frac{3}{2}\left( \frac{1}{3}\right) ^{1-\frac{1}{q}}\left[ \left( \frac{%
2\varphi -2}{\left[ \ln \varphi \right] ^{3}}-\frac{\varphi
+1}{\left[ \ln
\varphi \right] ^{2}}-\frac{1-\varphi }{2\ln \varphi }\right) ^{\frac{1}{q}%
}+\left( \frac{\varphi -1}{\left[ \ln \varphi \right] ^{2}}-\frac{\varphi +1%
}{2\ln \varphi }\right) ^{\frac{1}{q}}\right] & \eta <1%
\end{array}%
\right.  \notag
\end{eqnarray}%
where $\eta \left( \alpha ,\alpha \right) $ is same as Theorem
\ref{t1}, and $\varphi =\eta \left( \alpha q,\alpha q\right)
$.$\allowbreak $
\end{theorem}

\begin{proof}
Since $\left\vert f^{\prime }\right\vert ^{q}$ is an $\left(
\alpha
,m\right) $-logarithmically convex on $\left[ 0,\frac{b}{m}\right] $, for $%
q\geq 1$, from Lemma \ref{l1} and the well known power mean
integral inequality, we get%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\int_{0}^{1}\int_{0}^{1}\left\vert \left(
f^{\prime }\left( ta+\left( 1-t\right) b\right) \right) -\left(
f^{\prime }\left( sa+\left( 1-s\right) b\right) \right)
\right\vert \left\vert s-t\right\vert
dtds \\
&\leq &\frac{b-a}{2}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert dtds\right) ^{1-\frac{1}{q}}\left(
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \left\vert
f^{\prime }\left( ta+\left( 1-t\right) b\right)
\right\vert ^{q}dtds\right) ^{\frac{1}{q}} \\
&&+\frac{b-a}{2}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert dtds\right) ^{1-\frac{1}{q}}\left(
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \left\vert
f^{\prime }\left( sa+\left( 1-s\right) b\right)
\right\vert ^{q}dtds\right) ^{\frac{1}{q}} \\
&\leq &\frac{b-a}{2}\left\vert f^{\prime }\left(
\frac{b}{m}\right) \right\vert ^{m}\left(
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert dtds\right)
^{1-\frac{1}{q}}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{qt^{\alpha }}dtds\right) ^{\frac{1}{q}} \\
&&+\frac{b-a}{2}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert
^{m}\left( \int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert dtds\right) ^{1-%
\frac{1}{q}}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{qs^{\alpha }}dtds\right) ^{\frac{1}{q}}
\end{eqnarray*}%
When $\eta =1,$ by (\ref{1}), we obtain%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\left( \frac{1}{3}\right)
^{1-\frac{1}{q}}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert dtds\right) ^{\frac{1}{q}}
\\
&&+\frac{b-a}{2}\left( \frac{1}{3}\right)
^{1-\frac{1}{q}}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert dtds\right) ^{\frac{1}{q}}
\\
&=&\frac{b-a}{3}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}
\end{eqnarray*}%
When $\eta <1,$ by (\ref{1}), we obtain%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \\
&\leq &\frac{b-a}{2}\left( \frac{1}{3}\right)
^{1-\frac{1}{q}}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{\alpha
qt}dtds\right) ^{\frac{1}{q}} \\ 
&&+\frac{b-a}{2}\left( \frac{1}{3}\right)
^{1-\frac{1}{q}}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert ^{m}\left( \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{\alpha
qs}dtds\right) ^{\frac{1}{q}} \\
&=&\frac{b-a}{2}\left( \frac{1}{3}\right)
^{1-\frac{1}{q}}\left\vert
f^{\prime }\left( \frac{b}{m}\right) \right\vert ^{m} \\
&&\times \left\{ \left[ \frac{2\eta \left( \alpha q,\alpha q\right) -2}{%
\left[ \ln \left( \eta \left( \alpha q,\alpha q\right) \right) \right] ^{3}}-%
\frac{\eta \left( \alpha q,\alpha q\right) +1}{\left[ \ln \left(
\eta \left( \alpha q,\alpha q\right) \right) \right]
^{2}}-\frac{1-\eta \left( \alpha q,\alpha q\right) }{2\ln \left(
\eta \left( \alpha q,\alpha q\right) \right)
}\right] ^{\frac{1}{q}}\right. \\
&&+\left. \left[ \frac{\eta \left( \alpha q,\alpha q\right)
-1}{\left[ \ln \left( \eta \left( \alpha q,\alpha q\right) \right)
\right] ^{2}}-\frac{\eta \left( \alpha q,\alpha q\right) +1}{2\ln
\left( \eta \left( \alpha q,\alpha q\right) \right) }\right]
^{\frac{1}{q}}\right\} ,
\end{eqnarray*}%
\textit{which completes the proof.}
\end{proof}

\begin{corollary}
Let$\ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f: I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert ^{q}$is $m$%
-logarithmically convex on $\left[ 0,\frac{b}{m}\right] $ for
$m\in \left(
0,1\right] $, then%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert  \\
&\leq &\left\{
\begin{array}{cc}
\frac{b-a}{3}\left\vert f^{\prime }\left( \frac{b}{m}\right)
\right\vert
^{m}, & \eta =1 \\
\begin{array}{c}
\frac{\left( b-a\right) }{2}\left( \frac{1}{3}\right) ^{1-\frac{1}{q}%
}\left\vert f^{\prime }\left( \frac{b}{m}\right) \right\vert ^{m}\left\{ %
\left[ \frac{2\eta \left( q,q\right) -2}{\left[ \ln \eta \left( q,q\right) %
\right] ^{3}}-\frac{\eta \left( q,q\right) +1}{\left[ \ln \eta
\left( q,q\right) \right] ^{2}}-\frac{1-\eta \left( q,q\right)
}{2\ln \eta \left(
q,q\right) }\right] ^{\frac{1}{q}}\right.  \\
\left. +\left[ \frac{\eta \left( q,q\right) -1}{\left[ \ln \eta
\left( q,q\right) \right] ^{2}}-\frac{\eta \left( q,q\right)
+1}{2\ln \eta \left( q,q\right) }\right] ^{\frac{1}{q}}\right\}
\end{array}%
, & \eta <1%
\end{array}%
\right.
\end{eqnarray*}
\end{corollary}

\begin{corollary}
Let$ I\supset \left[ 0,\infty \right) $ be an open interval and
let $f: I\rightarrow \left( 0,\infty \right) $ be a
differentiable function on $I$
such that $f^{\prime }\in L\left( a,b\right) $ for $0\leq a<b<\infty .$ If $%
\left\vert f^{\prime }\left( x\right) \right\vert $ is $\alpha $%
-logarithmically convex on $\left[ 0,b\right] $ for $\alpha \in \left( 0,1%
\right] $ , then%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \leq
\frac{b-a}{3}\left\vert
f^{\prime }\left( b\right) \right\vert  \\
&\leq &\left\{
\begin{array}{cc}
1,\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } & \eta =1 \\
\begin{array}{c}
\frac{3}{2}\left( \frac{1}{3}\right) ^{1-\frac{1}{q}}\left\{ \left[ \frac{%
2\eta \left( \alpha q,\alpha q\right) -2}{\left[ \ln \left( \eta
\left( \alpha q,\alpha q\right) \right) \right] ^{3}}-\frac{\eta
\left( \alpha q,\alpha q\right) +1}{\left[ \ln \left( \eta \left(
\alpha q,\alpha q\right) \right) \right] ^{2}}-\frac{1-\eta \left(
\alpha q,\alpha q\right) }{2\ln
\left( \eta \left( \alpha q,\alpha q\right) \right) }\right] ^{\frac{1}{q}%
}\right.  \\
\left. +\left[ \frac{\eta \left( \alpha q,\alpha q\right)
-1}{\left[ \ln \left( \eta \left( \alpha q,\alpha q\right) \right)
\right] ^{2}}-\frac{\eta \left( \alpha q,\alpha q\right) +1}{2\ln
\left( \eta \left( \alpha q,\alpha q\right) \right) }\right]
^{\frac{1}{q}}\right\}
\end{array}%
, & \eta <1%
\end{array}%
\right.
\end{eqnarray*}

where $\eta =\left\vert f^{\prime }\left( a\right) \right\vert
/\left\vert f^{\prime }\left( b\right) \right\vert .$
\end{corollary}

\begin{theorem}
\label{t4}Let $f:I\subset
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
_{+}\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
_{+}$ be differentiable on $I^{\circ },$ $a,b\in I$, with $a<b$ and $%
f^{\prime }\in L\left( \left[ a,b\right] \right) .$ If $\left\vert
f^{\prime
}\right\vert $ is an $\left( \alpha ,m\right) $-logarithmically convex $%
\left[ 0,\frac{b}{m}\right] $ for $\left( \alpha ,m\right) \in \left( 0,1%
\right] ^{2}$ and $\mu _{1},\mu _{2},\tau _{1},\tau _{2}>0$ with
$\mu
_{1}+\tau _{1}=1$ and $\mu _{2}+\tau _{2}=1$, then
\begin{align}
&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert \leq \frac{\left( b-a\right) }{2}%
\left\vert f^{\prime }\left( \frac{b}{m}\right) \right\vert ^{m}   \\
&\times  \left\{
\begin{array}{ll}
\frac{2\mu _{1}^{3}}{\left( 2\mu _{1}+1\right) \left( \mu _{1}+1\right) }+%
\frac{2\mu _{2}^{3}}{\left( 2\mu _{2}+1\right) \left( \mu _{2}+1\right) }%
+\tau _{1}+\tau _{2}, & \eta =1 \\
\frac{2\mu _{1}^{3}}{\left( 2\mu _{1}+1\right) \left( \mu _{1}+1\right) }+%
\frac{2\mu _{2}^{3}}{\left( 2\mu _{2}+1\right) \left( \mu _{2}+1\right) }%
+\tau _{1}\frac{\eta \left( \frac{\alpha }{\tau _{1}},\frac{\alpha
}{\tau_{1}}\right) -1}{\ln \eta \left( \frac{\alpha }{\tau _{1}},\frac{\alpha }{%
\tau _{1}}\right) }+\tau _{2}\frac{\eta \left( \frac{\alpha }{\tau _{2}},%
\frac{\alpha }{\tau _{2}}\right) -1}{\ln \eta \left( \frac{\alpha }{\tau _{2}},\frac{\alpha }{\tau _{2}}\right) }, & \eta <1\nonumber%
\end{array}%
\right. 
\end{align}%
where $\eta \left( \alpha ,\alpha \right) $ is same as Theorem
\ref{t1}.
\end{theorem}
%
\begin{proof}
Since $\left\vert f^{\prime }\right\vert ^{q}$ is an $\left(
\alpha ,m\right) $-logarithmically convex on $\left[
0,\frac{b}{m}\right] $, from
Lemma \ref{l1}, we have%
\begin{eqnarray}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert   \label{f} \\
&\leq &\frac{\left( b-a\right)
}{2}\int_{0}^{1}\int_{0}^{1}\left\vert \left( f^{\prime }\left(
ta+\left( 1-t\right) b\right) \right) -\left( f^{\prime }\left(
sa+\left( 1-s\right) b\right) \right) \right\vert \left\vert
s-t\right\vert dtds  \notag \\
&\leq &\frac{\left( b-a\right) }{2}\left[
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \left\vert
f^{\prime }\left( a\right) \right\vert ^{t^{\alpha }}\left\vert
f^{\prime }\left( \frac{b}{m}\right) \right\vert ^{m\left(
1-t^{\alpha }\right) }dtds\right]   \notag \\
&&+\frac{\left( b-a\right) }{2}\left[
\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \left\vert
f^{\prime }\left( a\right) \right\vert ^{s^{\alpha }}\left\vert
f^{\prime }\left( \frac{b}{m}\right) \right\vert ^{m\left(
1-s^{\alpha }\right) }dtds\right]   \notag \\
&=&\frac{\left( b-a\right) }{2}\left\vert f^{\prime }\left( \frac{b}{m}%
\right) \right\vert ^{m}\left[ \int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{t^{\alpha
}}dtds+\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \eta
^{s^{\alpha }}dtds\right]   \notag
\end{eqnarray}%
for all $t\in \left[ 0,1\right] .$ Using the well known inequality
$rt\leq
\mu r^{\frac{1}{\mu }}+\tau t^{\frac{1}{\tau }},$ on the right side of (\ref%
{f}), we get
\begin{eqnarray}
&&\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \eta
^{t^{\alpha }}dtds+\int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{s^{\alpha
}}dtds  \label{w} \\
&\leq &\mu _{1}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert ^{\frac{1}{%
\mu _{1}}}dtds+\tau _{1}\int_{0}^{1}\int_{0}^{1}\eta ^{\frac{t^{\alpha }}{%
\tau _{1}}}dtds  \notag \\
&&+\mu _{2}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
^{\frac{1}{\mu _{2}}}dtds+\tau _{2}\int_{0}^{1}\int_{0}^{1}\eta
^{\frac{s^{\alpha }}{\tau _{2}}}dtds  \notag
\end{eqnarray}%
When $\eta =1,$ by (\ref{1}), we get
\begin{eqnarray}
&&\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \eta
^{t^{\alpha }}dtds+\int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{s^{\alpha
}}dtds  \label{e} \\
&\leq &\frac{2\mu _{1}^{3}}{\left( 2\mu _{1}+1\right) \left( \mu
_{1}+1\right) }+\frac{2\mu _{2}^{3}}{\left( 2\mu _{2}+1\right)
\left( \mu _{2}+1\right) }+\tau _{1}+\tau _{2}  \notag
\end{eqnarray}%
When $\eta <1,$ by (\ref{1}), we get%
\begin{eqnarray}
&&  \label{r} \\
&&\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert \eta
^{t^{\alpha }}dtds+\int_{0}^{1}\int_{0}^{1}\left\vert
s-t\right\vert \eta ^{s^{\alpha
}}dtds  \notag \\
&\leq &\mu _{1}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert ^{\frac{1}{%
\mu _{1}}}dtds+\tau _{1}\int_{0}^{1}\int_{0}^{1}\eta ^{\frac{t^{\alpha }}{%
\tau _{1}}}dtds  \notag \\
&&+\mu _{2}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert
^{\frac{1}{\mu _{2}}}dtds+\tau _{2}\int_{0}^{1}\int_{0}^{1}\eta
^{\frac{s^{\alpha }}{\tau
_{2}}}dtds  \notag \\
&\leq &\mu _{1}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert ^{\frac{1}{%
\mu _{1}}}dtds+\mu _{2}\int_{0}^{1}\int_{0}^{1}\left\vert s-t\right\vert ^{%
\frac{1}{\mu _{2}}}dtds  \notag \\
&&+\tau _{1}\int_{0}^{1}\int_{0}^{1}\eta ^{\frac{\alpha t}{\tau _{1}}%
}dtds+\tau _{2}\int_{0}^{1}\int_{0}^{1}\eta ^{\frac{\alpha s}{\tau
_{2}}}dtds
\notag \\
&=&\frac{2\mu _{1}^{3}}{\left( 2\mu _{1}+1\right) \left( \mu _{1}+1\right) }+%
\frac{2\mu _{2}^{3}}{\left( 2\mu _{2}+1\right) \left( \mu
_{2}+1\right) }
\notag \\
&&+\tau _{1}\frac{\eta \left( \frac{\alpha }{\tau
_{1}},\frac{\alpha }{\tau
_{1}}\right) -1}{\ln \eta \left( \frac{\alpha }{\tau _{1}},\frac{\alpha }{%
\tau _{1}}\right) }+\tau _{2}\frac{\eta \left( \frac{\alpha }{\tau _{2}},%
\frac{\alpha }{\tau _{2}}\right) -1}{\ln \eta \left( \frac{\alpha }{\tau _{2}%
},\frac{\alpha }{\tau _{2}}\right) }  \notag
\end{eqnarray}

from (\ref{f})-(\ref{r}), which completes the proof.
\end{proof}

\begin{corollary}
Under the assumptions of Theorem \ref{t4}, and $\mu =\mu _{1}=\mu _{2}>0,$ $%
\tau =\tau _{1}=\tau _{2}>0$ with $\mu +\tau =1,$ then we have%
\begin{eqnarray*}
&&\left\vert \frac{f\left( a\right) +f\left( b\right) }{2}-\frac{1}{b-a}%
\int_{a}^{b}f\left( x\right) dx\right\vert  \\
&\leq &\frac{\left( b-a\right) }{2}\left\vert f^{\prime }\left( \frac{b}{m}%
\right) \right\vert ^{m}\times \left\{
\begin{array}{cc}
\frac{4\mu ^{3}}{\left( 2\mu +1\right) \left( \mu +1\right)
}+2\tau , & \eta
=1 \\
\frac{4\mu ^{3}}{\left( 2\mu +1\right) \left( \mu +1\right) }+2\tau \frac{%
\eta \left( \frac{\alpha }{\tau },\frac{\alpha }{\tau }\right)
-1}{\ln \eta
\left( \frac{\alpha }{\tau },\frac{\alpha }{\tau }\right) }, & \eta <1%
\end{array}%
\right.
\end{eqnarray*}
\end{corollary}

\bibliographystyle{amsplain}
\begin{thebibliography}{99}

\bibitem{bai} R.F. Bai,  F. Qi, B.Y. Xi, \emph{Hermite-Hadamard type
inequalities for the }$m$\emph{- and }$\left( \alpha ,m\right) $\emph{%
-logarithmically convex functions.} Filomat,\textit{\ }\textbf{27}
(2013), 1--7.

\bibitem{dr2} S.S. Dragomir, C.E.M. Pearce, \emph{Selected Topics on
Hermite-Hadamard Inequalities and Applications,} RGMIA monographs,
Victoria University, 2000. [Online:\
http://www.staff.vu.edu.au/RGMIA/monographs/hermite-hadamard.html].

\bibitem{dr3} S.S. Dragomir, R.P. Agarwal, \emph{Two inequalities for
differantiable mappings and applications to special means of real
numbers and trapezoidal formula,} Appl. Math. Lett.,
\textbf{11}(5) (1998), 91-95.

\bibitem{hdm} J. Hadamard, \emph{\'{E}tude sur les propri\'{e}t\'{e}s des
fonctions enti\`{e}res et en particulier d'une fonction
consider\'{e}e par Riemann,} J. Math Pures Appl., \textbf{58},
(1893) 171--215.

\bibitem{mit} D.S. Mitrinovi\'{c}, J. Pe\v{c}ari\'{c}, A.M.\ Fink, \emph{%
Classical and New Inequalities in Analysis,} KluwerAcademic,
Dordrecht, 1993.

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

\bibitem{lmmm} M.Z. Sar\i kaya, E. Set, M.E. \"{O}zdemir, \emph{New
inequalities of Hermite-Hadamard Type,} Volume \textbf{12}, Issue
4, 2009, Art.11, RGMIA Online:
http://rgmia.org/papers/v12n4/set2.pdf

\bibitem{tnc} M. Tun\c{c}, E. Y\"{u}ksel, \.{I}. Karabay\i r, \emph{On some
inequalities for functions whose second derivetives absolute values are }$%
\alpha $\emph{-}$,$\emph{\ }$m$\emph{-}$,$\emph{\ }$\left( \alpha ,m\right) $%
\emph{-logarithmically convex,} Georgian Mathematical Journal,
Accepted.

\end{thebibliography}

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