---
abstract: 'In this paper, we establish some new inequalities for functions whose third derivatives in the absolute value are $m-$convex.'
address: '$^{\bigstar }$Atatürk University, K.K. Education Faculty, Department of Mathematics, 25240 Campus, Erzurum, Turkey'
author:
- 'M.E. Özdemir$^{\bigstar }$'
- 'Merve Avci$^{\bigstar \blacklozenge }$'
- 'Havva Kavurmaci$^{\bigstar }$'
title: 'Simpson Type Inequalities for $m-$convex Functions'
---

[^1]

introduction
============

The following inequality is well known in the literature as Simpson’s inequality:$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{3}\left[ \frac{f(a)+f(b)}{2}%
+2f\left( \frac{a+b}{2}\right) \right] \right\vert  \label{1.1} \\
&\leq &\frac{1}{2880}\left\Vert f^{(4)}\right\Vert _{\infty }\left(
b-a\right) ^{5},  \notag\end{aligned}$$where the mapping $f:[a,b]\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is assumed to be four times continuously differentiable on the interval and $f^{(4)}$ to be bounded on $(a,b)$ , that is,$$\left\Vert f^{(4)}\right\Vert _{\infty }=\sup_{t\in (a,b)}\left\vert
f^{(4)}(t)\right\vert <\infty .$$In [@T], G.Toader defined the concept of $m-$convexity as the following: The function $f:[0,b]\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is said to be $m-$convex, where $m\in \lbrack 0,1],$ if for every $x,y\in
\lbrack 0,b]$ and $t\in \lbrack 0,1]$ we have:$$f(tx+m(1-t)y)\leq tf(x)+m(1-t)f(y).$$Denote by $K_{m}(b)$ the set of the $m-$convex functions on $[a,b]$ for which $f(0)\leq 0.$

Some important inequalities for $m-$convex functions can be found in [OAS]{}-[@BOP].

In [@AH], Alomari and Hussain used the following lemma in order to establish some inequalities for $P-$convex functions.

\[lem 1.1\] Let $f:I\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a function such that $f^{\prime \prime \prime \text{ }}$be absolutely continuous on $I^{\circ }$, the interior of I. Assume that $a,b\in I^{\circ
},$ with $a<b$ and $f^{\prime \prime \prime \text{ }}\in L[a,b].$ Then, the following equality holds,$$\begin{aligned}
&&\int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{2}\right)
+f(b)\right] \\
&=&\left( b-a\right) ^{4}\int_{0}^{1}p(t)f^{\prime \prime \prime
}(ta+(1-t)b)dt,\end{aligned}$$where$$p(t)=\left\{ 
\begin{array}{c}
\frac{1}{6}t^{2}\left( t-\frac{1}{2}\right) ,\text{ \ \ \ }t\in \lbrack 0,%
\frac{1}{2}] \\ 
\\ 
\frac{1}{6}(t-1)^{2}\left( t-\frac{1}{2}\right) ,\text{ \ \ }t\in (\frac{1}{2%
},]\text{\ \ \ .\ \ \ \ }%
\end{array}%
\right.$$

In [@AHO], Avci et. al obtained following results using the above lemma.

\[teo 1.1\] Let $f:I\subset \lbrack 0,\infty )\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a differentiable function on $I^{\circ }$ such that $f^{\prime \prime
\prime \text{ }}\in L[a,b],$ where $a,b\in I^{\circ }$ with $a<b.$ If $%
\left\vert f^{\prime \prime \prime \text{ }}\right\vert ^{q}$ is $s-$convex in the second sense on $[a,b]$ and for some fixed $s\in (0,1]$ and $q>1$ with $\frac{1}{p}+\frac{1}{q}=1,$ then the following inequality holds:$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{%
2}\right) +f(b)\right] \right\vert \\
&\leq &\frac{\left( b-a\right) ^{4}}{48}\left( \frac{1}{2}\right) ^{\frac{1}{%
p}}\left( \frac{\Gamma (2p+1)\Gamma (p+1)}{\Gamma (3p+2)}\right) ^{\frac{1}{p%
}} \\
&&\times \left\{ \left[ \frac{1}{2^{s+1}(s+1)}\left\vert f^{\prime \prime
\prime \text{ }}(a)\right\vert ^{q}+\frac{2^{s+1}-1}{2^{s+1}(s+1)}\left\vert
f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}\right] ^{\frac{1}{q}%
}\right. \\
&&\left. +\left[ \frac{2^{s+1}-1}{2^{s+1}(s+1)}\left\vert f^{\prime \prime
\prime \text{ }}(a)\right\vert ^{q}+\frac{1}{2^{s+1}(s+1)}\left\vert
f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}\right] ^{\frac{1}{q}%
}\right\} .\end{aligned}$$

\[co 1.1\] If we choose $s=1$ in Theorem \[teo 1.1\], we have$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{%
2}\right) +f(b)\right] \right\vert  \label{1.2} \\
&\leq &\frac{\left( b-a\right) ^{4}}{96}\left( \frac{1}{4}\right) ^{\frac{1}{%
q}}\left( \frac{\Gamma (2p+1)\Gamma (p+1)}{\Gamma (3p+2)}\right) ^{\frac{1}{p%
}}  \notag \\
&&\times \left\{ \left( \left\vert f^{\prime \prime \prime \text{ }%
}(a)\right\vert ^{q}+3\left\vert f^{\prime \prime \prime \text{ }%
}(b)\right\vert ^{q}\right) ^{\frac{1}{q}}+\left( 3\left\vert f^{\prime
\prime \prime \text{ }}(a)\right\vert ^{q}+\left\vert f^{\prime \prime
\prime \text{ }}(b)\right\vert ^{q}\right) ^{\frac{1}{q}}\right\} .  \notag\end{aligned}$$

\[teo 1.2\] Suppose that all the assumptions of Theorem \[teo 2.1\] are satisfied. Then$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{%
2}\right) +f(b)\right] \right\vert \\
&\leq &\frac{\left( b-a\right) ^{4}}{6}\left( \frac{1}{192}\right) ^{1-\frac{%
1}{q}} \\
&&\times \left\{ \left( \frac{2^{-4-s}}{(3+s)(4+s)}\left\vert f^{\prime
\prime \prime \text{ }}(a)\right\vert ^{q}+\frac{2^{-4-s}\left(
34+2^{4+s}(-2+s)+11s+s^{2}\right) }{(1+s)(2+s)(3+s)(4+s)}\left\vert
f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}\right) ^{\frac{1}{q}%
}\right. \\
&&\left. +\left( \frac{2^{-4-s}\left( 34+2^{4+s}(-2+s)+11s+s^{2}\right) }{%
(1+s)(2+s)(3+s)(4+s)}\left\vert f^{\prime \prime \prime \text{ }%
}(a)\right\vert ^{q}+\frac{2^{-4-s}}{(3+s)(4+s)}\left\vert f^{\prime \prime
\prime \text{ }}(b)\right\vert ^{q}\right) ^{\frac{1}{q}}\right\} .\end{aligned}$$

\[co 1.2\] If we choose $s=1$ in Theorem \[teo 1.2\], we have$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{%
2}\right) +f(b)\right] \right\vert  \label{1.3} \\
&\leq &\frac{\left( b-a\right) ^{4}}{1152}\left\{ \left( \frac{3\left\vert
f^{\prime \prime \prime \text{ }}(a)\right\vert ^{q}+7\left\vert f^{\prime
\prime \prime \text{ }}(b)\right\vert ^{q}}{10}\right) ^{\frac{1}{q}}+\left( 
\frac{7\left\vert f^{\prime \prime \prime \text{ }}(a)\right\vert
^{q}+3\left\vert f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}}{10}%
\right) ^{\frac{1}{q}}\right\} .  \notag\end{aligned}$$

In [@BBS], Barani et. al proved the following results.

\[teo 1.3\] Let $f:I\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a function such that $f^{\prime \prime \prime \text{ }}$be absolutely continuous on $I^{\circ }.$ Assume that $a,b\in I^{\circ },$ with $a<b$ and $%
f^{\prime \prime \prime \text{ }}\in L[a,b].$ If $\left\vert f^{\prime
\prime \prime \text{ }}\right\vert $ is a $P-$convex function on $[a,b]$ then, the following inequality holds:$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{%
2}\right) +f(b)\right] \right\vert \\
&\leq &\frac{\left( b-a\right) ^{4}}{1152}\left\{ \left\vert f^{\prime
\prime \prime \text{ }}(a)\right\vert +\left\vert f^{\prime \prime \prime 
\text{ }}\left( \frac{a+b}{2}\right) \right\vert +\left\vert f^{\prime
\prime \prime \text{ }}(b)\right\vert \right\} .\end{aligned}$$

\[co 1.3\] Let $f$ as in Theorem \[teo 1.3\]. If $f^{\prime \prime
\prime \text{ }}\left( \frac{a+b}{2}\right) =0,$ then we have$$\begin{aligned}
&&\left\vert \int_{a}^{b}f(x)dx-\frac{b-a}{6}\left[ f(a)+4f\left( \frac{a+b}{%
2}\right) +f(b)\right] \right\vert  \label{1.4} \\
&\leq &\frac{\left( b-a\right) ^{4}}{1152}\left\{ \left\vert f^{\prime
\prime \prime \text{ }}(a)\right\vert +\left\vert f^{\prime \prime \prime 
\text{ }}(b)\right\vert \right\} .  \notag\end{aligned}$$

The aim of this study is to establish some inequalities for $m-$convex functions. In order to obtain our results, we modified Lemma \[lem 1.1\] given in the [@AH].

main results
============

\[lem 2.1\] Let $f:I\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ be a function such that $f^{\prime \prime \prime \text{ }}$be absolutely continuous on $I^{\circ }$, the interior of I. Assume that $a,b\in I^{\circ
},$ with $a<b,$ $m\in (0,1]$ and $f^{\prime \prime \prime \text{ }}\in
L[a,b].$ Then, the following equality holds,$$\begin{aligned}
&&\int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b}{2}%
\right) +f(mb)\right] \\
&=&\left( mb-a\right) ^{4}\int_{0}^{1}p(t)f^{\prime \prime \prime
}(ta+m(1-t)b)dt,\end{aligned}$$where$$p(t)=\left\{ 
\begin{array}{c}
\frac{1}{6}t^{2}\left( t-\frac{1}{2}\right) ,\text{ \ \ \ }t\in \lbrack 0,%
\frac{1}{2}] \\ 
\\ 
\frac{1}{6}(t-1)^{2}\left( t-\frac{1}{2}\right) ,\text{ \ \ }t\in (\frac{1}{2%
},]\text{\ \ \ .\ \ \ \ }%
\end{array}%
\right.$$

A simple proof of the equality can be done by performing an integration by parts in the integrals from the right side and changing the variable. The details are left to the interested reader.

\[teo 2.1\] Let $f:I\subset \lbrack 0,b^{\ast }]\rightarrow 
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
,$ be a differentiable function on $I^{\circ }$ such that $f^{\prime \prime
\prime \text{ }}\in L[a,b]$ where $a,b\in I$ with $a<b,$ $b^{\ast }>0.$ If $%
\left\vert f^{\prime \prime \prime \text{ }}\right\vert ^{q}$ is $m-$convex on $[a,b]$ for $m\in (0,1],$ $q>1,$ then the following inequality holds:$$\begin{aligned}
&&\left\vert \int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b%
}{2}\right) +f(mb)\right] \right\vert \\
&\leq &\frac{\left( mb-a\right) ^{4}}{96}\left( \frac{\Gamma (2p+1)\Gamma
(p+1)}{\Gamma (3p+1)}\right) ^{\frac{1}{p}} \\
&&\times \left\{ \left( \frac{\left\vert f^{\prime \prime \prime \text{ }%
}(a)\right\vert ^{q}+3m\left\vert f^{\prime \prime \prime \text{ }%
}(b)\right\vert ^{q}}{4}\right) ^{\frac{1}{q}}+\left( \frac{3\left\vert
f^{\prime \prime \prime \text{ }}(a)\right\vert ^{q}+m\left\vert f^{\prime
\prime \prime \text{ }}(b)\right\vert ^{q}}{4}\right) ^{\frac{1}{q}}\right\}
.\end{aligned}$$

From Lemma \[lem 2.1\] and using Hölder inequality we have$$\begin{aligned}
&&\left\vert \int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b%
}{2}\right) +f(mb)\right] \right\vert \\
&\leq &\frac{\left( mb-a\right) ^{4}}{6}\left\{ \left( \int_{0}^{\frac{1}{2}%
}\left( t^{2}\left( \frac{1}{2}-t\right) \right) ^{p}dt\right) ^{\frac{1}{p}%
}\left( \int_{0}^{\frac{1}{2}}\left\vert f^{\prime \prime \prime \text{ }%
}(ta+m(1-t)b)\right\vert ^{q}dt\right) ^{\frac{1}{q}}\right. \\
&&\left. +\left( \int_{\frac{1}{2}}^{1}\left( (t-1)^{2}\left( t-\frac{1}{2}%
\right) \right) ^{p}dt\right) ^{\frac{1}{p}}\left( \int_{\frac{1}{2}%
}^{1}\left\vert f^{\prime \prime \prime \text{ }}(ta+m(1-t)b)\right\vert
^{q}dt\right) ^{\frac{1}{q}}\right\} .\end{aligned}$$Since $m-$convexity of $\left\vert f^{\prime \prime \prime \text{ }%
}\right\vert ^{q},$ we have$$\begin{aligned}
\int_{0}^{\frac{1}{2}}\left\vert f^{\prime \prime \prime \text{ }%
}(ta+m(1-t)b)\right\vert ^{q}dt &\leq &\int_{0}^{\frac{1}{2}}\left[
t\left\vert f^{\prime \prime \prime \text{ }}(a)\right\vert
^{q}+m(1-t)\left\vert f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}%
\right] dt \\
&=&\frac{\left\vert f^{\prime \prime \prime \text{ }}(a)\right\vert
^{q}+3m\left\vert f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}}{8},\end{aligned}$$$$\begin{aligned}
\int_{\frac{1}{2}}^{1}\left\vert f^{\prime \prime \prime \text{ }%
}(ta+m(1-t)b)\right\vert ^{q}dt &\leq &\int_{\frac{1}{2}}^{1}\left[
t\left\vert f^{\prime \prime \prime \text{ }}(a)\right\vert
^{q}+m(1-t)\left\vert f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}%
\right] dt \\
&=&\frac{3\left\vert f^{\prime \prime \prime \text{ }}(a)\right\vert
^{q}+m\left\vert f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}}{8}.\end{aligned}$$Using the fact that $$\int_{0}^{\frac{1}{2}}\left( t^{2}\left( \frac{1}{2}-t\right) \right)
^{p}dt=\int_{\frac{1}{2}}^{1}\left( (t-1)^{2}\left( t-\frac{1}{2}\right)
\right) ^{p}dt=\frac{\Gamma \left( 2p+1\right) \Gamma \left( p+1\right) }{%
2^{3p+1}\Gamma \left( 3p+1\right) },$$where $\Gamma $ is the Gamma function, we obtain$$\begin{aligned}
&&\left\vert \int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b%
}{2}\right) +f(mb)\right] \right\vert \\
&\leq &\frac{\left( mb-a\right) ^{4}}{96}\left( \frac{\Gamma (2p+1)\Gamma
(p+1)}{\Gamma (3p+1)}\right) ^{\frac{1}{p}} \\
&&\times \left\{ \left( \frac{\left\vert f^{\prime \prime \prime \text{ }%
}(a)\right\vert ^{q}+3m\left\vert f^{\prime \prime \prime \text{ }%
}(b)\right\vert ^{q}}{4}\right) ^{\frac{1}{q}}+\left( \frac{3\left\vert
f^{\prime \prime \prime \text{ }}(a)\right\vert ^{q}+m\left\vert f^{\prime
\prime \prime \text{ }}(b)\right\vert ^{q}}{4}\right) ^{\frac{1}{q}}\right\}
,\end{aligned}$$which is the required.

\[rem 2.1\] In Theorem \[teo 2.1\], if we choose $m=1,$ we have the inequality in (\[1.2\]).

\[teo 2.2\] Let the assumptions of Theorem \[teo 2.1\] hold with $%
q\geq 1.$ Then $$\begin{aligned}
&&\left\vert \int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b%
}{2}\right) +f(mb)\right] \right\vert \\
&\leq &\frac{\left( mb-a\right) ^{4}}{1152}\left\{ \left( \frac{3\left\vert
f^{\prime \prime \prime \text{ }}(a)\right\vert ^{q}+7m\left\vert f^{\prime
\prime \prime \text{ }}(b)\right\vert ^{q}}{10}\right) ^{\frac{1}{q}}+\left( 
\frac{7\left\vert f^{\prime \prime \prime \text{ }}(a)\right\vert
^{q}+3m\left\vert f^{\prime \prime \prime \text{ }}(b)\right\vert ^{q}}{10}%
\right) ^{\frac{1}{q}}\right\} .\end{aligned}$$

From Lemma \[lem 2.1\] , using the well known power-mean inequality and $%
m- $convexity of $\left\vert f^{\prime \prime \prime \text{ }}\right\vert
^{q}$, we have $$\begin{aligned}
&&\left\vert \int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b%
}{2}\right) +f(mb)\right] \right\vert \\
&\leq &\frac{\left( b-a\right) ^{4}}{6}\left\{ \left( \int_{0}^{\frac{1}{2}%
}t^{2}\left( \frac{1}{2}-t\right) dt\right) ^{1-\frac{1}{q}}\left( \int_{0}^{%
\frac{1}{2}}t^{2}\left( \frac{1}{2}-t\right) \left\vert f^{\prime \prime
\prime \text{ }}(ta+m(1-t)b)\right\vert ^{q}dt\right) ^{\frac{1}{q}}\right.
\\
&&\left. +\left( \int_{\frac{1}{2}}^{1}(t-1)^{2}\left( t-\frac{1}{2}\right)
dt\right) ^{1-\frac{1}{q}}\left( \int_{\frac{1}{2}}^{1}(t-1)^{2}\left( t-%
\frac{1}{2}\right) \left\vert f^{\prime \prime \prime \text{ }%
}(ta+m(1-t)b)\right\vert ^{q}dt\right) ^{\frac{1}{q}}\right\} \\
&\leq &\frac{\left( b-a\right) ^{4}}{6}\left\{ \left( \int_{0}^{\frac{1}{2}%
}t^{2}\left( \frac{1}{2}-t\right) dt\right) ^{1-\frac{1}{q}}\left( \int_{0}^{%
\frac{1}{2}}t^{2}\left( \frac{1}{2}-t\right) \left[ t\left\vert f^{\prime
\prime \prime \text{ }}(a)\right\vert ^{q}+m(1-t)\left\vert f^{\prime \prime
\prime \text{ }}(b)\right\vert ^{q}\right] dt\right) ^{\frac{1}{q}}\right. \\
&&\left. +\left( \int_{\frac{1}{2}}^{1}(t-1)^{2}\left( t-\frac{1}{2}\right)
dt\right) ^{1-\frac{1}{q}}\left( \int_{\frac{1}{2}}^{1}(t-1)^{2}\left( t-%
\frac{1}{2}\right) \left[ t\left\vert f^{\prime \prime \prime \text{ }%
}(a)\right\vert ^{q}+m(1-t)\left\vert f^{\prime \prime \prime \text{ }%
}(b)\right\vert ^{q}\right] dt\right) ^{\frac{1}{q}}\right\} .\end{aligned}$$By simple calculations we obtain$$\begin{aligned}
&&\left\vert \int_{a}^{mb}f(x)dx-\frac{mb-a}{6}\left[ f(a)+4f\left( \frac{a+b%
}{2}\right) +f(mb)\right] \right\vert \\
&\leq &\frac{\left( b-a\right) ^{4}}{6}\left( \frac{1}{192}\right) ^{1-\frac{%
1}{q}}\left\{ \left( \frac{7\left\vert f^{\prime \prime \prime \text{ }%
}(a)\right\vert ^{q}+3m\left\vert f^{\prime \prime \prime \text{ }%
}(b)\right\vert ^{q}}{1920}\right) ^{\frac{1}{q}}+\left( \frac{3\left\vert
f^{\prime \prime \prime \text{ }}(a)\right\vert ^{q}+7m\left\vert f^{\prime
\prime \prime \text{ }}(b)\right\vert ^{q}}{1920}\right) ^{\frac{1}{q}%
}\right\}\end{aligned}$$which is the desired result.

\[rem 2.2\] In Theorem \[teo 2.2\], if we choose $m=1,$ we have the inequality in (\[1.3\]).

\[rem 2.3\] In Theorem \[teo 2.2\], if we choose $m=1$ and $q=1,$ we have the inequality in (\[1.4\]).

[9]{} M. Alomari, S. Hussain, Two inequalities of Simpson type for quasiconvex functions and applications, Appl. Math. E-notes 11 (2001), 110-117.

G.H. Toader, Some generalisations of the convexity, Proc. Colloq. Approx. Optim (1984) 329 338.

A. Barani, S. Barani and S.S. Dragomir, Simpson’s type inequalities for functions whose third derivatives in the absolute values are $P-$convex, RGMIA Res. Rep. Coll., 14(2011), Preprints, Article 95.

M. Avci, H. Kavurmaci and M.E. Özdemir, Simpson type inequalities for functions whose third derivatives in the absolute value are $s-$convex and $s-$concave functions, RGMIA Res. Rep. Coll., Submitted.

M.E. Özdemir, M. Avci, E. Set, On some inequalities of Hermite–Hadamard type via $m-$convexity, Appl. Math. Lett. 23 (9) (2010) 1065–1070.

M.E. Özdemir, E. Set and M.Z. Sarikaya, Some new Hadamard type inequalities for co-ordinated $m-$convex and $(\alpha ,m)-$convex functions, Hacattepe Journal Of Mathematics And Statistics, 40, 219-229, (2011).

M.K. Bakula, M. E Özdemir, J. Pečarić, Hadamard type inequalities for $m-$convex and $(\alpha ,m)-$convex functions, J. Inequal. Pure Appl. Math. 9 (2008). Article 96,                                           online: http:// jipam.vu.edu.au.

[^1]: $^{\blacklozenge }$Corresponding author
