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\fancyhead[CE]{G. Tınaztepe, I. Yeşilce Işık, S. Kemali and G. Adilov} 
\fancyhead[CO]{Certain Inequalities for $s$-Convex Functions Via Caputo Fractional
Derivative and Caputo-Fabrizio Integral Operator}



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\begin{document}
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{\noindent Journal of Mathematical Extension \\
Vol. XX, No. XX, (2014), pp-pp (Will be inserted by layout editor)}\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
‎\vspace*{9mm}
‎
\begin{center}

{\Large \bf 
Certain Inequalities for $s$-Convex Functions Via Caputo Fractional
Derivative and Caputo-Fabrizio Integral Operator\\}



\let\thefootnote\relax\footnote{\scriptsize Received: XXXX; Accepted: XXXX (Will be inserted by editor)}

\textbf{Gültekin Tınaztepe}\vspace*{-2mm}\\
 {\small Akdeniz University} \vspace{2mm}

\textbf{Ilknur Ye\c{s}ilce I\c{s}ık}\vspace*{-2mm}\\
 {\small Aksaray University} \vspace{2mm}

\textbf{Serap Kemali $^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
{\small Akdeniz University} \vspace{2mm}

\textbf{Gabil Adilov}\vspace*{-2mm}\\
{\small Akdeniz University} \vspace{2mm}

\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, several integral
inequalities for the functions whose derivatives are $s$-convex functions in
the fourth sense are obtained by means of Caputo fractional derivative and
Caputo-Fabrizio integral operator. Also the Hermite-Hadamard type inequalities
for $s$-convex functions and their products are stated via Caputo-Fabrizio
integral operator. }

{\footnotesize \noindent\textbf{AMS Subject Classification:} 26A33; 26D07;
26D10; 26D15; 26A51 }

{\footnotesize \noindent\textbf{Keywords and Phrases:} Convex functions,
s-convexity, Hermite-Hadamard type inequality, Caputo fractional
\ derivatives, Caputo-Fabrizio integral operator. }
\end{quotation}

\section{Introduction}

The complexity of the problems countered in the ongoing scientific and
technological progresses, which cannot be easily modelled and solved via
ordinary calculus, push forward the usage of the fractional calculus, which is
based on the existence of the fractional order derivative. The first track of
it was seen in the studies of Abel and Lacroix in early 1800s \cite{Lacro}.
Since the extension of the order of the derivative from integer to fraction
yields to fractional differential equations, the fractional derivatives and
integrals have found a lot of applications area from medicine to engineering
\cite{Ray}.

There have been defined various fractional derivative and integral operators.
Some of them are the Riemann-Liouville fractional derivative and integral, the
Hadamard fractional derivative and integral, Gr\"{u}nwald-Letnikov fractional
derivative, the Caputo fractional derivative and the others (see
\cite{srivastava} and the references therein). The Caputo-Fabrizio fractional
derivative is one of the most useful among them due to the fact that it has no
singular kernel \cite{caputo, caputo2, caputo2015}. Also the Caputo-Fabrizio
integral operator is defined and used as a fractional integral operation in
many works, although it is shown that this operator is not a fractional
derivative or integral operator in the context of the classical definitions of
fractional calculus, but just an auxiliary operator to simplify the problem
solutions \cite{cfnd, nfd}. Nevertheless, this misusage continues in a lot of papers.

Besides the applications in different disciplines, one favor of it is the
generalization of the inequalities in mathematics \cite{lemma2,farid1,farid2,
mg, product3}. The inequalities arising from the convex functions are the
prominent ones such as the Jensen inequality, the Hermite-Hadamard inequality
\cite{opt3, opt1, KTA,opt2, ineq5, YA1,YA2,Y2}. In this study, we focus on the
obtaining the Hermite-Hadamard type inequalities for $s$-convex functions via
the Caputo fractional derivative operator and the Caputo-Fabrizio integral operator.

$s$-convexity is one of the generalizations of convexity and it has some
applications in fractal theory \cite{Kwun,Kilicman}. Its origin is based on
the studies on modular spaces and Orlicz spaces \cite{orc}. In literature,
essentially, four different types of $s$-convex functions are introduced
\cite{s4, s-convexity, khan, s4HH, product3}. In this paper, we state new
inequalities including the Hermite-Hadamard type inequalities for $s$-convex
functions in the fourth senses via Caputo derivative and Caputo-Fabrizio
integral operator.

The organization of paper is as follows. The main result section is separated
into two subsections. The first includes the inequalities by the
Caputo-Fabrizio fractional derivative for the functions whose $n$th
derivatives are $s$-convex in the fourth sense. The second is allocated to the
inequalities by the Caputo-Fabrizio integral operator for the functions whose
first and second derivatives and itself are $s$-convex in the fourth sense,
respectively. Also some inequalities for the product of this kind of functions
are presented.

In the following, some definitions and theorems that are needed throughout the
paper are presented.

\begin{definition}
\cite{s4}\label{s4} Let $s\in(0,1]$ and $U$ be a convex set on vector space
$X$. A function $\psi:U\rightarrow{\mathbb{R}}$ is said to be $s$-convex
function in the fourth sense if the following inequality holds%
\[
\psi(\lambda x+(1-\lambda)y)\leq\lambda^{\frac{1}{s}}\psi(x)+(1-\lambda
)^{\frac{1}{s}}\psi(y)
\]
for all $x,y\in U$ and $\lambda\in\lbrack0,1]$.
\end{definition}

The class of these functions is denoted by $K_{s}^{^{4}}$. Using both
definitions for $\lambda=\frac{1}{2}$ and $x=y,$ we easily see that if
$\psi\in K_{s}^{4},$ then $\psi$ is not positive.

\begin{theorem}
\cite{s4HH}\label{hhfors4} Let $\psi:\mathbb{R\rightarrow R}_{-}$ be
$s$-convex function in the fourth sense where $s\in\left(  0,1\right]  $ and
integrable on $\left[  a,b\right]  \subseteq\mathbb{R}$, then the following
inequality holds%
\[
2^{\frac{1}{s}-1}\psi\left(  \frac{a+b}{2}\right)  \leq\frac{1}{b-a}%
\int\limits_{a}^{b}\psi(x)dx\leq\frac{s\left[  \psi(a)+\psi(b)\right]  }%
{s+1}.
\]

\end{theorem}

\begin{definition}
\cite{caputo,caputo2} Let $AC^{n}[a,b]$ be a space of functions having $n$th
derivatives absolutely continuous, $\psi\in AC^{n}[a,b]$ ,$\alpha
\notin\left\{  1,2,3,...\right\}  $ and $n=[\alpha]+1$ where $[.]$ is denoted
floor function. The right side Caputo fractional derivative is as follows%
\[
\left(  ^{C}D_{a^{+}}^{\alpha}\psi\right)  (x)=\frac{1}{\Gamma(n-\alpha)}%
\int\limits_{a}^{x}\frac{\psi^{(n)}(t)}{\left(  x-t\right)  ^{\alpha-n+1}%
}dt,\text{ \ \ }x>a,
\]
the left side Caputo fractional derivative is as follows%
\[
\left(  ^{C}D_{b^{-}}^{\alpha}\psi\right)  (x)=\frac{\left(  -1\right)  ^{n}%
}{\Gamma(n-\alpha)}\int\limits_{x}^{b}\frac{\psi^{(n)}(t)}{\left(  t-x\right)
^{\alpha-n+1}}dt,\text{\ \ \ }x<b.
\]

\end{definition}

If $\alpha=n\in\left\{  1,2,3,...\right\}  $ and usual derivative
$\psi^{\left(  n\right)  }(x)$ of order $n$ exists, then Caputo fractional
$\left(  ^{C}D_{a^{+}}^{n}\psi\right)  (x)$ coincides with $\psi^{\left(
n\right)  }(x)$ whereas $\left(  ^{C}D_{b^{-}}^{n}\psi\right)  (x)$ coincides
with $\psi^{\left(  n\right)  }(x)$ with exactness to a constant multiplier
$(-1)^{n}.$ In particular we have
\[
\left(  ^{C}D_{a^{+}}^{0}\psi\right)  (x)=\left(  ^{C}D_{b^{-}}^{0}%
\psi\right)  (x)=\psi(x)
\]
where $n=1,$ $\alpha=0.$

\begin{definition}
\cite{caputo2015} Let $H^{1}(a,b)$ be the Sobolev space of order one defined
as follows
\[
H^{1}(a,b)=\left\{  u\in L^{2}(a,b):u^{\prime}\in L^{2}(a,b)\right\}  .
\]
Let $\psi\in H^{1}(a,b),$ $a<b$, $\alpha\in\lbrack0,1],$ then the definition
of the left derivative in the sense of Caputo-Fabrizio becomes%
\[
\left(  _{a}^{CFD}D^{\alpha}\psi\right)  (x)=\frac{B(\alpha)}{1-\alpha}%
\int\limits_{a}^{x}\psi\prime(t)e^{\frac{-\alpha(x-t)^{\alpha}}{1-\alpha}%
}dt,\text{ \ \ \ }x>a,
\]
and the associated integral operator is%
\[
\left(  _{a}^{CF}I^{\alpha}\psi\right)  (x)=\frac{1-\alpha}{B(\alpha)}%
\psi(x)+\frac{\alpha}{B(\alpha)}\int\limits_{a}^{x}\psi(t)dt,
\]
where $B(\alpha)>0$ is a normalization function satisfying $B(0)=B(1)=1.$ In
the cases $\alpha=0$ and $\alpha=1,$ left derivative is defined as follows%
\[
\left(  _{a}^{CFD}D^{0}\psi\right)  (x)=\psi^{\prime}(x)\text{ \ and
\ }\left(  _{a}^{CFD}D^{1}\psi\right)  (x)=\psi(x)-\psi(a).
\]
For the right derivative operator we have
\[
\left(  _{b}^{CFD}D^{\alpha}\psi\right)  (x)=\frac{-B(\alpha)}{1-\alpha}%
\int\limits_{x}^{b}\psi^{\prime}(t)e^{\frac{-\alpha(t-x)^{\alpha}}{1-\alpha}%
}dt,\text{\ \ \ }x<b,
\]
and the associated integral operator is
\[
\left(  ^{CF}I_{b}^{\alpha}\psi\right)  (x)=\frac{1-\alpha}{B(\alpha)}
\psi(x)+\frac{\alpha}{B(\alpha)}\int\limits_{x}^{b}\psi(t)dt
\]
where $B(\alpha)>0$ is a normalization function satisfying $B(0)=B(1)=1.$
\end{definition}

\section{Main results}

This section contains two subsections. First one includes some new integral
inequalities obtained for the functions whose $n$th derivatives are $s$-convex
functions in the fourth sense via the Caputo fractional derivative. Second one
displays the results obtained by means of Caputo-Fabrizio integral operator,
which covers a generalization of Hermite-Hadamard type inequalities for the
$s$-convex functions of fourth sense and the products of them. Also, in this
subsection, owing to some integral identities, new inequalities involving
Caputo-Fabrizio operators are presented for the functions whose first and
second derivatives are $s$-convex functions.

\subsection{Some new integral inequalities with the Caputo fractional
derivative}

In this subsection, some inequalities are obtained for the functions whose $n
$th order derivative is $s$-convex function in the fourth sense via the Caputo
fractional derivative. The first theorem sets an inequality associated with
the all values of an interval over which function is integrated.

\begin{theorem}
\label{thm6} Let $\psi:[a,b]\subset\left[  0,\infty\right)  \rightarrow
\mathbb{R}$ be $n$-times differentiable function where $n$ is a positive
integer. If $\psi^{\left(  n\right)  }$ is $s$-convex function in the fourth
sense, then for $\alpha,\beta>1$ with $n>\max\left\{  \alpha,\beta\right\}  ,$
the following inequality for Caputo fractional derivatives holds%
\begin{align*}
&  \Gamma\left(  n-\alpha+1\right)  \left(  ^{C}D_{a^{+}}^{\alpha-1}%
\psi\right)  (x)+\Gamma\left(  n-\beta+1\right)  \left(  ^{C}D_{b^{-}}%
^{\beta-1}\psi\right)  (x)\\
& \hspace{4cm} \leq\frac{s\left[  (x-a)^{n-\alpha+1}\psi^{\left(  n\right)  }%
(a)+(-1)^{n}\left(  b-x\right)  ^{n-\beta+1}\psi^{\left(  n\right)
}(b)\right]  }{1+s}\\
& \hspace{4cm} +\psi^{\left(  n\right)  }(x)\frac{s\left[  (x-a)^{n-\alpha+1}%
+(-1)^{n}\left(  b-x\right)  ^{n-\beta+1}\right]  }{1+s}.
\end{align*}
for $x\in\lbrack a,b]$
\end{theorem}

\begin{proof}
For $t\in\lbrack a,x],$ $x\in\left[  a,b\right]  ,$ $n\in N^{+}$and $n>\alpha$
the following inequality holds%
\begin{equation}
\left(  x-t\right)  ^{n-\alpha}\leq\left(  x-a\right)  ^{n-\alpha}.
\label{thm6_1}%
\end{equation}
Since\ $\psi^{\left(  n\right)  }\in K_{s}^{4},$ we have%
\begin{equation}
\psi^{\left(  n\right)  }(t)\leq\left(  \frac{x-t}{x-a}\right)  ^{\frac{1}{s}%
}\psi^{\left(  n\right)  }(a)+\left(  \frac{t-a}{x-a}\right)  ^{\frac{1}{s}%
}\psi^{\left(  n\right)  }(x). \label{thm6_2}%
\end{equation}
Multiplying (\ref{thm6_1}) and (\ref{thm6_2}), then integrating with respect
to over $\left[  a,x\right]  $ we get%
\[
\int\limits_{a}^{x}(x-t)^{n-\alpha}\psi^{\left(  n\right)  }(t)dt
$$\leq\left(
x-a\right)  ^{n-\alpha-\frac{1}{s}}\left[  \psi^{\left(  n\right)  }%
(a)\int\limits_{a}^{x}(x-t)^{\frac{1}{s}}dt+\psi^{\left(  n\right)  }%
(x)\int\limits_{a}^{x}\left(  t-a\right)  ^{\frac{1}{s}}dt\right] $$ .
\]
We can write the same inequality as follows,%
\[
\int\limits_{a}^{x}\frac{\psi^{\left(  n\right)  }(t)}{(x-t)^{\alpha-n}}%
dt\leq\frac{s}{1+s}\left(  x-a\right)  ^{^{n-\alpha+1}}\left[  \psi^{\left(
n\right)  }(a)+\psi^{\left(  n\right)  }(x)\right]  .
\]
From here, the following inequality is obtained for Caputo fractional
derivative;%
\begin{equation}
\Gamma\left(  n-\alpha+1\right)  \left(  ^{C}D_{a^{+}}^{\alpha-1}\psi\right)
(x)\leq\frac{s}{1+s}\left(  x-a\right)  ^{n-\alpha+1}\left[  \psi^{\left(
n\right)  }(a)+\psi^{\left(  n\right)  }(x)\right]  . \label{thm6_5}%
\end{equation}
Now we consider $t\in\lbrack x,b]$ and $n>\beta$ the following inequality
holds%
\begin{equation}
\left(  t-x\right)  ^{n-\beta}\leq(b-x)^{n-\beta}. \label{thm6_3}%
\end{equation}
Since\ $\psi^{\left(  n\right)  }\in K_{s}^{4},$ then we have%
\begin{equation}
\psi^{\left(  n\right)  }(t)\leq\left(  \frac{t-x}{b-x}\right)  ^{\frac{1}{s}%
}\psi^{\left(  n\right)  }(b)+\left(  \frac{b-t}{b-x}\right)  ^{\frac{1}{s}%
}\psi^{\left(  n\right)  }(x). \label{thm6_4}%
\end{equation}
Multiplying inequalities (\ref{thm6_3}) and (\ref{thm6_4}), then integrating
with respect to $t$ over $[x,b]$ we have%
\[
\int\limits_{x}^{b}(t-x)^{n-\beta}\psi^{\left(  n\right)  }(t)dt\leq\left(
b-x\right)  ^{n-\beta-\frac{1}{s}}\left[  \psi^{\left(  n\right)  }%
(b)\int\limits_{x}^{b}(t-x)^{\frac{1}{s}}dt+\psi^{\left(  n\right)  }%
(x)\int\limits_{x}^{b}\left(  b-t\right)  ^{\frac{1}{s}}dt\right]  .
\]
We can write the same inequality as follows,%
\[
\int\limits_{x}^{b}\frac{\psi^{\left(  n\right)  }(t)}{(t-x)^{\beta-n}}%
dt\leq\frac{s}{1+s}\left(  b-x\right)  ^{n-\beta-\frac{1}{s}}\left[
\psi^{\left(  n\right)  }(b)(b-x)^{\frac{1}{s}+1}+\psi^{\left(  n\right)
}(x)\left(  b-x\right)  ^{\frac{1}{s}+1}dt\right]  .
\]
Multiplying both sides of the inequality above with $(-1)^{n}$ by taking into
account the oddness and evenness of $n$ and adding to (\ref{thm6_5}) side by
side, we have the desired inequality.
\end{proof}


\begin{corollary}
If we take $\alpha=\beta$ in Theorem \ref{thm6}, then we have the following
fractional derivative inequality%
\begin{align*}
&  \Gamma\left(  n-\alpha+1\right)  \left[  \left(  ^{C}D_{a^{+}}^{\alpha
-1}\psi\right)  (x)+\left(  ^{C}D_{b^{-}}^{\alpha-1}\psi\right)  (x)\right] \\
&  \hspace{4cm}\leq\frac{s\left[  (x-a)^{n-\alpha+1}\psi^{\left(  n\right)  }%
(a)+(-1)^{n}\left(  b-x\right)  ^{n-\alpha+1}\psi^{\left(  n\right)
}(b)\right]  }{1+s}\\
&  \hspace{4cm}+\psi^{\left(  n\right)  }(x)\frac{s\left[  (x-a)^{n-\alpha+1}%
+(-1)^{n}\left(  b-x\right)  ^{n-\alpha+1}\right]  }{1+s}.
\end{align*}

\end{corollary}

\begin{corollary}
\label{corol4}If we take $\alpha=\beta,$ $s=1$ Theorem \ref{thm6}, then we
have the following fractional derivative inequality in \cite{farid2}%
\begin{align*}
&  \Gamma\left(  n-\alpha+1\right)  \left[  \left(  ^{C}D_{a^{+}}^{\alpha
-1}\psi\right)  (x)+\left(  ^{C}D_{b^{-}}^{\alpha-1}\psi\right)  (x)\right] \\
& \hspace{4cm} \leq\frac{(x-a)^{n-\alpha+1}\psi^{\left(  n\right)  }(a)+(-1)^{n}\left(
b-x\right)  ^{n-\alpha+1}\psi^{\left(  n\right)  }(b)}{2}\\
& \hspace{4cm} +\psi^{\left(  n\right)  }(x)\frac{(x-a)^{n-\alpha+1}+(-1)^{n}\left(
b-x\right)  ^{n-\alpha+1}}{2}.
\end{align*}

\end{corollary}

\begin{remark}
When $n$ is even number in Corollary \ref{corol4}, the inequality in Corollary
2.1 in \cite{farid2} is obtained.
\end{remark}

The following theorem sets an inequality involving Caputo fractional
derivative, which is stated for only the end points of the interval over which
function is integrated.

\begin{theorem}
\label{thm4} Let $\psi:[a,b]\subseteq\mathbb{R}\rightarrow\mathbb{R}$ be a
function, $n$-times differentiable function where $n$ is a positive integer.
If $\psi^{\left(  n\right)  }$ is $s$-convex function in the fourth sense and
integrable on $[a,b]$, then for $s\in(0,1],$ the following inequality for
Caputo fractional derivatives holds%
\begin{align}
\frac{2^{\frac{1}{s}}}{n-\alpha}\psi^{\left(  n\right)  }(\frac{a+b}{2})  &
\leq\frac{\Gamma(n-\alpha)}{(b-a)^{n-\alpha}}\left[  \left(  ^{C}D_{a^{+}%
}^{\alpha}\psi\right)  \left(  b\right)  +\left(  -1\right)  ^{n}\left(
^{C}D_{b^{-}}^{\alpha}\psi\right)  \left(  a\right)  \right] \nonumber\\
&  \leq\left(  \psi^{\left(  n\right)  }(a)+\psi^{\left(  n\right)
}(b)\right)  \left[  \frac{s}{ns-\alpha s+1}+\frac{\Gamma(1+\frac{1}{s}%
)\Gamma(n-\alpha)}{\Gamma\left(  n-\alpha+\frac{1}{s}+1\right)  }\right]
\label{thm4_0}%
\end{align}
or%
\begin{align*}
\frac{2^{\frac{1}{s}}}{n-\alpha}\psi^{\left(  n\right)  }(\frac{a+b}{2})  &
\leq\frac{\Gamma(n-\alpha)}{(b-a)^{n-\alpha}}\left[  \left(  ^{C}D_{a^{+}%
}^{\alpha}\psi\right)  \left(  b\right)  +\left(  -1\right)  ^{n}\left(
^{C}D_{b^{-}}^{\alpha}\psi\right)  \left(  a\right)  \right] \\
&  \leq\left(  \psi^{\left(  n\right)  }(a)+\psi^{\left(  n\right)
}(b)\right)  \left[  \frac{s}{ns-\alpha s+1}+B\left(  1+\frac{1}{s}%
,n-\alpha\right)  \right]
\end{align*}

\end{theorem}

\begin{proof}
Since\ $\psi^{\left(  n\right)  }\in K_{s}^{4},$ for $x,y\in\lbrack a,b]$ we
have%
\begin{equation}
\psi^{\left(  n\right)  }(\frac{x+y}{2})\leq\frac{\psi^{\left(  n\right)
}(x)+\psi^{\left(  n\right)  }(y)}{2^{\frac{1}{s}}}. \label{thm4_1}%
\end{equation}
Let $x=ta+(1-t)b,$ $y=(1-t)a+tb$ for $t\in\left[  0,1\right]  $. Then
inequality (\ref{thm4_1}) gives%
\begin{equation}
2^{\frac{1}{s}}\psi^{\left(  n\right)  }(\frac{a+b}{2})\leq\psi^{\left(
n\right)  }(ta+(1-t)b)+\psi^{\left(  n\right)  }((1-t)a+tb). \label{thm4_2}%
\end{equation}
Multiplying both sides of inequality (\ref{thm4_2}) by $t^{n-\alpha-1}$, then
integrating with respect to $t$ over $[0,1]$ we get%
\begin{align*}
& 2^{\frac{1}{s}}\psi^{\left(  n\right)  }(\frac{a+b}{2})\int\limits_{0}%
^{1}t^{n-\alpha-1}dt \\
& \hspace{3cm} \leq\int\limits_{0}^{1}t^{n-\alpha-1}\psi^{\left(
n\right)  }(ta+(1-t)b)dt +\int\limits_{0}^{1}t^{n-\alpha-1}\psi^{\left(  n\right)  }((1-t)a+tb)dt.
\end{align*}
Thus, we have the following inequalities,%
\begin{equation}
\frac{2^{\frac{1}{s}}}{n-\alpha}\psi^{\left(  n\right)  }(\frac{a+b}{2}%
)\leq\frac{\Gamma(n-\alpha)}{(b-a)^{n-\alpha}}\left[  \left(  ^{C}D_{a^{+}%
}^{\alpha}\psi\right)  \left(  b\right)  +\left(  -1\right)  ^{n}\left(
^{C}D_{b^{-}}^{\alpha}\psi\right)  \left(  a\right)  \right]  . \label{thm4_6}%
\end{equation}
On the other hand convexity of $\psi^{\left(  n\right)  }$ gives%
\begin{equation}
\psi^{\left(  n\right)  }(ta+(1-t)b)+\psi^{\left(  n\right)  }((1-t)a+tb)\leq
\left(  t^{\frac{1}{s}}+(1-t)^{\frac{1}{s}}\right)  \left(  \psi^{\left(
n\right)  }(a)+\psi^{\left(  n\right)  }(b)\right)  . \label{thm4_4}%
\end{equation}
Multiplying both sides of inequality (\ref{thm4_4}) by $t^{n-\alpha-1}$, then
integrating with respect to $t$ over $[0,1]$ we get%
\begin{align*}
&  \int\limits_{0}^{1}t^{n-\alpha-1}\psi^{\left(  n\right)  }%
(ta+(1-t)b)dt+\int\limits_{0}^{1}t^{n-\alpha-1}\psi^{\left(  n\right)
}((1-t)a+tb)dt\\
&\hspace{5cm}   \leq\int\limits_{0}^{1}t^{n-\alpha-1}\left(  t^{\frac{1}{s}}+(1-t)^{\frac
{1}{s}}\right)  \left(  \psi^{\left(  n\right)  }(a)+\psi^{\left(  n\right)
}(b)\right)  dt,
\end{align*}
from which one can have%
\begin{align}
& \frac{\Gamma(n-\alpha)}{(b-a)^{n-\alpha}}\left[  \left(  ^{C}D_{a^{+}}%
^{\alpha}\psi\right)  \left(  b\right)  +\left(  -1\right)  ^{n}\left(
^{C}D_{b^{-}}^{\alpha}\psi\right)  \left(  a\right)  \right] \nonumber \\
& \hspace{3cm} \leq\left(
\psi^{\left(  n\right)  }(a)+\psi^{\left(  n\right)  }(b)\right)  \left[
\frac{s}{ns-\alpha s+1}+\frac{\Gamma(1+\frac{1}{s})\Gamma(n-\alpha)}%
{\Gamma\left(  n-\alpha+\frac{1}{s}+1\right)  }\right]  \label{thm4_5}%
\end{align}
Inequalities (\ref{thm4_6}) and (\ref{thm4_5}) give the inequality
(\ref{thm4_0}).
\end{proof}


\subsection{Various inequalities involving the Hermite-Hadamard type
inequalities via Caputo-Fabrizio integral operator}

In this subsection, Hermite-Hadamard type inequalities for $s$-convex
functions of the fourth sense and their products are obtained in terms of
Caputo-Fabrizio integral operators \ Also some inequalities are stated via
integral identities involving Caputo-Fabrizio integral operators are given for
the functions whose second and first derivatives are $s$-convex functions.

In the following theorem, the Hermite-Hadamard inequalities and related
inequalities are obtained for the $s$-convex functions in the fourth sense by
means of Caputo-Fabrizio type integral operators.

\begin{theorem}
\label{thm2}Let $\psi:[a,b]\subseteq\mathbb{R}\rightarrow\mathbb{R}$ be $s
$-convex function in the fourth sense and integrable on $[a,b].$ The following
double inequality holds for $\alpha\in\lbrack0,1],$%
\begin{align*}
& 2^{\frac{1}{s}-1}\psi\left(  \frac{a+b}{2}\right)  \leq\frac{B(\alpha)}%
{\alpha(b-a)}\left[  \left(  _{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(
^{CF}I_{b}^{\alpha}\psi\right)  (x)-\frac{2\left(  1-\alpha\right)  }%
{B(\alpha)}\psi(x)\right] \\
& \hspace {9cm}\leq\frac{s\left(  \psi(a)+\psi(b)\right)  }{s+1}
\end{align*}
where $x\in\lbrack a,b],$ $B(\alpha)>0$ is normalization function.
\end{theorem}

\begin{proof}
From Theorem \ref{hhfors4}$,$ we have%
\begin{equation}
2^{\frac{1}{s}-1}\psi\left(  \frac{a+b}{2}\right)  \leq\frac{1}{b-a}%
\int\limits_{a}^{b}\psi(t)dt\leq\frac{s\left(  \psi(a)+\psi(b)\right)  }{s+1}.
\label{thm2_2}%
\end{equation}
By multiplying both sides of (\ref{thm2_2}) with $\frac{\alpha(b-a)}%
{B(\alpha)}$ we get%
\[
\frac{\alpha(b-a)}{B(\alpha)}2^{\frac{1}{s}-1}\psi\left(  \frac{a+b}%
{2}\right)  \leq\frac{\alpha}{B(\alpha)}\int\limits_{a}^{b}\psi(t)dt\leq
\frac{\alpha(b-a)}{B(\alpha)}\frac{s\left(  \psi(a)+\psi(b)\right)  }{s+1},
\]
and adding $\frac{2(1-\alpha)}{B(\alpha)}\psi(x)$ we get%
\begin{align}
\frac{\alpha(b-a)}{B(\alpha)}2^{\frac{1}{s}-1}\psi\left(  \frac{a+b}%
{2}\right)  +\frac{2(1-\alpha)}{B(\alpha)}\psi(x)  &  \leq\frac{\alpha
}{B(\alpha)}\int\limits_{a}^{b}\psi(t)dt+\frac{2(1-\alpha)}{B(\alpha)}%
\psi(x)\nonumber\\
&  \leq\frac{\alpha(b-a)}{B(\alpha)}\frac{s\left(  \psi(a)+\psi(b)\right)
}{s+1}+\frac{2(1-\alpha)}{B(\alpha)}\psi(x). \label{thm2_3}%
\end{align}
Considering the left side of inequality (\ref{thm2_3}), we have%
\begin{align}
\frac{\alpha(b-a)}{B(\alpha)}2^{\frac{1}{s}-1}\psi\left(  \frac{a+b}%
{2}\right)  +\frac{2(1-\alpha)}{B(\alpha)}\psi(x)  &  \leq\frac{\alpha
}{B(\alpha)}\int\limits_{a}^{x}\psi(t)dt+\frac{(1-\alpha)}{B(\alpha)}%
\psi(x)
\nonumber\\
&  +\frac{\alpha}{B(\alpha)}\int\limits_{x}^{b}\psi(t)dt+\frac{(1-\alpha
)}{B(\alpha)}\psi(x)\nonumber\\
&  =\left(  _{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha
}\psi\right)  (x). \label{thm2_4}%
\end{align}
Considering the right side of inequality (\ref{thm2_3}), we have%
\begin{align}
\frac{\alpha(b-a)}{B(\alpha)}\frac{s\left(  \psi(a)+\psi(b)\right)  }%
{s+1}+\frac{2(1-\alpha)}{B(\alpha)}\psi(x)  &  \geq\frac{\alpha}{B(\alpha
)}\int\limits_{a}^{x}\psi(t)dt+\frac{(1-\alpha)}{B(\alpha)}\psi(x)\nonumber\\
&  +\frac
{\alpha}{B(\alpha)}\int\limits_{x}^{b}\psi(t)dt+\frac{(1-\alpha)}{B(\alpha
)}\psi(x)\nonumber\\
&  =\left(  _{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha
}\psi\right)  (x). \label{thm2-5}%
\end{align}
If $\frac{2(1-\alpha)}{B(\alpha)}\psi(x)$ is subtracted from both sides of the
inequalities (\ref{thm2_4}) and (\ref{thm2-5}), divided by both sides by
$\frac{\alpha(b-a)}{B(\alpha)}$, and the two inequalities are combined, the
proof is complete.
\end{proof}


In the following theorems, using integral operators of Caputo-Fabrizio type,
we obtain the Hermite-Hadamard type inequalities for the product of $s$-convex
functions in the fourth sense.

\begin{theorem}
\label{thm5} Let $\psi:[a,b]\subseteq\mathbb{R}\rightarrow\mathbb{R}_{-}$ be
$s_{1}$-convex function in the fourth sense and $\phi:[a,b]\subseteq
\mathbb{R}\rightarrow\mathbb{R}_{-}$ be $s_{2}$-convex function in the fourth
sense. If $\psi\phi$ integrable on $[a,b]$, then the following inequality
holds for $\alpha\in\lbrack0,1],$ $x\in\lbrack a,b]$,%
\begin{align*}
&  \frac{B(\alpha)}{\alpha(b-a)}\left[  \left(  _{a}^{CF}I^{\alpha}\psi
\phi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\phi\right)  (x)-\frac
{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)\right] \\
& \hspace{3cm} \geq\frac{s_{1}s_{2}}{s_{1}+s_{1}s_{2}+s_{2}}M(a,b)+\beta(1+\frac{1}{s_{1}%
},1+\frac{1}{s_{2}})N(a,b)
\end{align*}
where $B(\alpha)>0$ is a normalization function and%
\[
M(a,b)=\psi(a)\phi(a)+\psi(b)\phi(b),\text{ \ \ }N(a,b)=\psi(a)\phi
(b)+\psi(b)\phi(a).
\]

\end{theorem}

\begin{proof}
Since $\psi\in K_{s_{1}}^{4},$ and $\phi\in K_{s_{2}}^{4},$ we have following
inequalities%
\[
\psi(\lambda a+(1-\lambda)b)\leq\lambda^{\frac{1}{s_{1}}}\psi(a)+(1-\lambda
)^{\frac{1}{s_{1}}}\psi(b),
\]
and%
\[
\phi(\lambda a+(1-\lambda)b)\leq\lambda^{\frac{1}{s_{2}}}\phi(a)+(1-\lambda
)^{\frac{1}{s_{2}}}\phi(b).
\]
We know that $\psi$ and $\phi$ are negative functions since these are
$s$-convex functions in the fourth sense. If we multiply these inequalities
side by side and integrate with $\lambda$ over $[0,1]$, we have%
\begin{align*}
& \int\limits_{0}^{1}\psi(\lambda a+(1-\lambda)b)\phi(\lambda a+(1-\lambda
)b)d\lambda\\
& \hspace {3cm}  \geq \int\limits_{0}^{1}\left[  \lambda^{\frac{1}{s_{1}}}%
\psi(a)+(1-\lambda)^{\frac{1}{s_{1}}}\psi(b)\right]  \left[  \lambda^{\frac
{1}{s_{2}}}\phi(a)+(1-\lambda)^{\frac{1}{s_{2}}}\phi(b)\right]  d\lambda\\
& \hspace {3cm}=\frac{s_{1}s_{2}}{s_{1}+s_{1}s_{2}+s_{2}}M(a,b)+\beta(1+\frac{1}{s_{1}%
},1+\frac{1}{s_{2}})N(a,b).
\end{align*}
If we substitute variables $t=\lambda a+(1-\lambda)b,$ this inequality gives
the following inequality,%
\begin{equation}
\frac{1}{b-a}\int\limits_{a}^{b}\psi(t)\phi(t)dt\geq\frac{s_{1}s_{2}}%
{s_{1}+s_{1}s_{2}+s_{2}}M(a,b)+\beta(1+\frac{1}{s_{1}},1+\frac{1}{s_{2}%
})N(a,b). \label{thm5_2}%
\end{equation}
By multiplying both sides of (\ref{thm5_2}) with $\frac{\alpha(b-a)}%
{B(\alpha)}$ and adding $\frac{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)$ we have%
\begin{align*}
&  \frac{\alpha(b-a)}{B(\alpha)}\left[  \frac{s_{1}s_{2}}{s_{1}+s_{1}%
s_{2}+s_{2}}M(a,b)+\beta(1+\frac{1}{s_{1}},1+\frac{1}{s_{2}})N(a,b)\right]
+\frac{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)\\
&  \leq\frac{\alpha}{B(\alpha)}\left[  \int\limits_{a}^{x}\psi(t)\phi
(t)dt+\int\limits_{x}^{b}\psi(t)\phi(t)dt\right]  +\frac{2(1-\alpha)}%
{B(\alpha)}\psi(x)\phi(x)\\
&  =\frac{\alpha}{B(\alpha)}\int\limits_{a}^{x}\psi(t)\phi(t)dt+\frac
{(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)+\frac{\alpha}{B(\alpha)}\int
\limits_{x}^{b}\psi(t)\phi(t)dt+\frac{(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)\\
&  =\left(  _{a}^{CF}I^{\alpha}\psi\phi\right)  (x)+\left(  ^{CF}I_{b}%
^{\alpha}\psi\phi\right)  (x).
\end{align*}
If $\frac{2(1-\alpha)}{B(\alpha)}\psi(x)$ is subtracted from both sides of
this inequality and divided by $\frac{\alpha(b-a)}{B(\alpha)}$, the desired is achieved.
\end{proof}


\begin{corollary}
\label{thm4_3} If we take $s_{1}=s_{2}=1,$ then inequality in Theorem
\ref{thm4} is reduced to the following inequality%
\begin{align*}
&  \frac{B(\alpha)}{\alpha(b-a)}\left[  \left(  _{a}^{CF}I^{\alpha}\psi
\phi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\phi\right)  (x)-\frac
{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)\right] \\
&  \geq\frac{1}{3}M(a,b)+\frac{1}{6}N(a,b).
\end{align*}

\end{corollary}

\begin{theorem}
\label{thm7} Let $\psi:[a,b]\subseteq\mathbb{R}\rightarrow\mathbb{R}_{-}$ be
$s_{1}$-convex function in the fourth sense and $\phi:[a,b]\subseteq
\mathbb{R}\rightarrow\mathbb{R}_{-}$ be $s_{2}$-convex function in the fourth
sense. If $\psi\phi$ integrable on $[a,b]$, then the following inequality
holds for $\alpha\in\lbrack0,1],$ $x\in\lbrack a,b]$%
\begin{align*}
&  \frac{B(\alpha)}{\alpha(b-a)}\left[  \left(  _{a}^{CF}I^{\alpha}\psi
\phi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\phi\right)  (x)-\frac
{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)\right] \\
&  \leq8\psi\left(  \frac{a+b}{2}\right)  \phi\left(  \frac{a+b}{2}\right)
-\frac{1}{6}N(a,b)-\frac{1}{3}M(a,b)
\end{align*}
where $B(\alpha)>0$ is a normalization function and $M(a,b)$ and $N(a,b)$ are
denoted as in Theorem \ref{thm5}$.$
\end{theorem}

\begin{proof}
In \cite{kemali}, from Corollary 2 we have the following inequality%
\begin{equation}
\frac{1}{b-a}\int\limits_{a}^{b}\psi(t)\phi(t)dt\leq8\psi\left(  \frac{a+b}%
{2}\right)  \phi\left(  \frac{a+b}{2}\right)  -\frac{1}{3}N(a,b)-\frac{1}%
{6}M(a,b). \label{thm7_2}%
\end{equation}
By multiplying both sides of (\ref{thm7_2}) with $\frac{\alpha(b-a)}%
{B(\alpha)}$ and adding $\frac{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)$ we have%
\begin{align*}
&  \frac{\alpha(b-a)}{B(\alpha)}\int\limits_{a}^{x}\psi(x)\phi(x)dx+\frac
{(1-\alpha)}{B(\alpha)}\psi(x)\phi(x)+\frac{\alpha(b-a)}{B(\alpha)}%
\int\limits_{x}^{b}\psi(x)\phi(x)dx+\frac{(1-\alpha)}{B(\alpha)}\psi
(x)\phi(x)\\
&  \leq\frac{\alpha(b-a)}{B(\alpha)}\left[  8\psi\left(  \frac{a+b}{2}\right)
\phi\left(  \frac{a+b}{2}\right)  -\frac{1}{3}N(a,b)-\frac{1}{6}M(a,b)\right]
+\frac{2(1-\alpha)}{B(\alpha)}\psi(x)\phi(x).
\end{align*}
\end{proof}


In the following result, a generalization of Corollary 8 in \cite{kemali} is
presented via Caputo-Fabrizio integral operators.

\begin{corollary}
Using Theorem \ref{thm7} and Corollary \ref{thm4_3}, we have the following
double inequality%
\begin{align*}
\frac{1}{3}M(a,b)+\frac{1}{6}N(a,b)  &  \leq\frac{B(\alpha)}{\alpha
(b-a)}\left[  \left(  _{a}^{CF}I^{\alpha}\psi\phi\right)  (x)+\left(
^{CF}I_{b}^{\alpha}\psi\phi\right)  (x)-\frac{2(1-\alpha)}{B(\alpha)}%
\psi(x)\phi(x)\right] \\
&  \leq8\psi\left(  \frac{a+b}{2}\right)  \phi\left(  \frac{a+b}{2}\right)
-\frac{1}{6}N(a,b)-\frac{1}{3}M(a,b).
\end{align*}

\end{corollary}

Using the following lemma, we can obtain an inequality via the Caputo-Fabrizio
integral operators for the functions whose second derivative is $s$-convex
function in the fourth sense.

\begin{lemma}
\label{L2}\cite{lemma2} Let $\psi:[a,b]\subseteq\mathbb{R}\rightarrow
\mathbb{R}$ be twice differentiable function on $(a,b).$ If $\psi
^{\prime\prime}\in L_{1}[a,b]$ and $\alpha\in\lbrack0,1],$ the following
equality holds
\begin{align*}
&  \frac{\psi(a)+\psi(b)}{2}-\frac{B(\alpha)}{\alpha(b-a)}\left[  \left(
_{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\right)
(x)\right]  +\frac{2\left(  1-\alpha\right)  }{\alpha(b-a)}\psi(x)\\
&  =\frac{\left(  b-a\right)  ^{2}}{2}\int\limits_{0}^{1}\lambda
(1-\lambda)\psi^{\prime\prime}(\lambda a+(1-\lambda)b)d\lambda
\end{align*}
where $x\in\lbrack a,b]$ and $B(\alpha)>0$ is a normalization function.
\end{lemma}

\begin{theorem}
\label{thm11}Let $\psi:[a,b]\subseteq\mathbb{R}\rightarrow\mathbb{R}_{-}$ be
twice differentiable on $(a,b)$ and $\psi^{\prime\prime}$ be $s$-convex
function in the fourth sense.\ The following inequality holds%
\begin{align*}
&  \frac{\psi(a)+\psi(b)}{2}-\frac{B(\alpha)}{\alpha(b-a)}\left[  \left(
_{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\right)
(x)\right]  +\frac{2\left(  1-\alpha\right)  }{\alpha(b-a)}\psi(x)\\
&  \leq\frac{\left(  s\left(  b-a\right)  \right)  ^{2}}{\left(  1+2s\right)
\left(  1+3s\right)  }\frac{\psi^{\prime\prime}(b)+\psi^{\prime\prime}(a)}{2}%
\end{align*}
where $x\in\lbrack a,b]$ and $\alpha\in\lbrack0,1]$, $B(\alpha)>0$ is a
normalization function.
\end{theorem}

\begin{proof}
From Lemma \ref{L2} and convexity of $\psi^{\prime\prime}$ we have%
\begin{align*}
&  \frac{\psi(a)+\psi(b)}{2}-\frac{B(\alpha)}{\alpha(b-a)}\left[  \left(
_{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\right)
(x)\right]  +\frac{2\left(  1-\alpha\right)  }{\alpha(b-a)}\psi(x)\\
&  \leq\frac{\left(  b-a\right)  ^{2}}{2}\int\limits_{0}^{1}\lambda
(1-\lambda)\left[  \lambda^{\frac{1}{s}}\psi^{\prime\prime}(a)+(1-\lambda
)^{\frac{1}{s}}\psi^{\prime\prime}(b)\right]  d\lambda\\
&  =\frac{\left(  b-a\right)  ^{2}}{2}\int\limits_{0}^{1}\left[  \left(
\lambda^{\frac{1}{s}+1}-\lambda^{\frac{1}{s}+2}\right)  \psi^{\prime\prime
}(a)+\lambda(1-\lambda)^{\frac{1}{s}+1}\psi^{\prime\prime}(b)\right]
d\lambda\\
&  =\frac{\left(  b-a\right)  ^{2}}{2}\frac{s^{2}}{\left(  1+2s\right)
\left(  1+3s\right)  }\left[  \psi^{\prime\prime}(a)+\psi^{\prime\prime
}(b)\right]  .
\end{align*}
\end{proof}


\begin{corollary}
If we choose $s=1$ in Theorem \ref{thm11}, then the following inequality is
obtained%
\begin{align*}
& \frac{\psi(a)+\psi(b)}{2}-\frac{B(\alpha)}{\alpha(b-a)}\left[  \left(
_{a}^{CF}I^{\alpha}\psi\right)  (x)+\left(  ^{CF}I_{b}^{\alpha}\psi\right)
(x)\right]  +\frac{2\left(  1-\alpha\right)  }{\alpha(b-a)}\psi(x)\\
& \leq\left(  b-a\right)  ^{2}\frac{\psi^{\prime\prime}(b)+\psi^{\prime\prime
}(a)}{48}.
\end{align*}

\end{corollary}

\section{Applications}

We consider the applications of some of the results to the special means. Let
us recall the following means for positive real numbers $a,b$.

Let $a,b,p$ be positive number with $a\neq b$ and $p\in Z/\left\{
-1,0\right\}  $,%
\begin{align*}
A(a,b)  &  =\frac{a+b}{2},\\
M_{p}(a,b)  &  =\left(  \frac{a^{p}+b^{p}}{2}\right)  ^{\frac{1}{p}},\\
L_{p}(a,b)  &  =\left\{
\begin{array}
[c]{ll}%
\qquad a\hspace{2cm}, & \text{if }a=b\\
\left(  \frac{a^{p+1}-b^{p+1}}{(p+1)(a-b)}\right)  ^{1/p}, & \text{else }%
\end{array}
\right.
\end{align*}
are called Arithmetic mean, Power mean and Stolarsky mean (Generalized
logarithmic mean) respectively.

\begin{proposition}
\label{innp}Let $a,b\in\mathbb{R}_{+}$ with $a<b.$ The inequality holds,%
\[
2^{1-s}A(a,b)\geq L_{\frac{1+s}{s}}(a,b)\geq\left(  \frac{2s}{s+1}\right)
^{s}M_{\frac{1}{s}}(a,b)
\]
for all $s\in\left(  0,1\right]  .$
\end{proposition}

\begin{proof}
The assertion follows from Theorem \ref{thm2} applied to the $s$-convex
function in the fourth sense $\psi(x)=-x^{\frac{1}{s}},$ $x\in\left[
a,b\right]  $ and $\alpha=1,$ $B(\alpha)=1,$%
\begin{equation}
\frac{\left(  a+b\right)  ^{\frac{1}{s}}}{2}\geq\frac{s\left(  b^{\frac{1}%
{s}+1}-a^{\frac{1}{s}+1}\right)  }{\left(  b-a\right)  \left(  s+1\right)
}\geq\frac{s}{s+1}\left(  a^{\frac{1}{s}}+b^{\frac{1}{s}}\right)  .
\label{inn}%
\end{equation}
If $s$-th power of each side is taken in this inequality, then we get%
\[
\frac{a+b}{2^{s}}\geq\left[  \frac{b^{\frac{1}{s}+1}-a^{\frac{1}{s}+1}%
}{\left(  \frac{1}{s}+1\right)  \left(  b-a\right)  }\right]  ^{s}\geq\left(
\frac{s}{s+1}\right)  ^{s}\left(  a^{\frac{1}{s}}+b^{\frac{1}{s}}\right)
^{s}.
\]
We can write this inequality as follows%
\[
2^{1-s}\frac{a+b}{2}\geq\left[  \frac{b^{\frac{1}{s}+1}-a^{\frac{1}{s}+1}%
}{\left(  \frac{1}{s}+1\right)  \left(  b-a\right)  }\right]  ^{s}\geq\left(
\frac{2s}{s+1}\right)  ^{s}\left(  \frac{a^{\frac{1}{s}}+b^{\frac{1}{s}}}%
{2}\right)  ^{s}.
\]
From this inequality, we get%
\[
2^{1-s}A(a,b)\geq S_{\frac{1+s}{s}}(a,b)\geq\left(  \frac{2s}{s+1}\right)
^{s}M_{\frac{1}{s}}(a,b).
\]
\end{proof}


Using the results, one can obtain some inequalities relevant to special
function expressed by integral. To illustrate, we present one of them.

\begin{proposition}
Let $x\geq3$. Then%
\[
\frac{x}{x-1}\leq\Psi(x)+\gamma\leq\frac{2^{x-1}-1}{2}
\]
where $\Psi(x)$ is digamma function, i.e.%
\[
\Psi(x)=\frac{\Gamma^{\prime}(x)}{\Gamma(x)}\text{ for }x>0
\]
and $\gamma$ is Euler-Mascheroni constant i.e. $\gamma\approx0.5772156649...$
\end{proposition}

\begin{proof}
Let us suppose $a,b$ as the same in Proposition \ref{innp} write $t=\frac
{a}{b}$ and simplify the expression in (\ref{inn}). Then we have%
\[
\frac{\left(  1+t\right)  ^{\frac{1}{s}}}{2}\geq\frac{s}{s+1}\frac
{1-t^{\frac{1}{s}+1}}{1-t}\geq\frac{s}{s+1}\left(  1+t^{\frac{1}{s}}\right)
.
\]
Integrating the inequality with respect to $t$ on $[0,1]$ and simplifying the
expression, we have
\[
2^{\frac{1}{s}}-2^{-1}\geq%
%TCIMACRO{\dint \limits_{0}^{1}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{1}}
%EndExpansion
\frac{1-t^{\frac{1}{s}+1}}{1-t}dt\geq\frac{2s+1}{s+1}
\]
By making use of the following integral representation of digamma function
given in \cite{Dragomirgamma},
\[
\Psi(r)=%
%TCIMACRO{\dint \limits_{0}^{1}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{1}}
%EndExpansion
\frac{1-t^{r-1}}{1-t}dt-\gamma
\]
where $r>0$, it is obtained that%
\[
2^{\frac{1}{s}}-2^{-1}\geq\Psi(2+\frac{1}{s})+\gamma\geq\frac{2s+1}{s+1}.
\]
The substitution $x=2+\frac{1}{s}$ above yields to%
\[
\frac{2^{x-1}-1}{2}\geq\Psi(x)+\gamma\geq\frac{x}{x-1}
\]
for $x\geq3.$
\end{proof}


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\end{thebibliography}


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