\documentclass[11pt,twoside]{article} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks]{hyperref} \usepackage{float} \usepackage{lipsum} \usepackage{afterpage} \usepackage[labelfont=bf]{caption} \usepackage[nottoc,notlof,notlot]{tocbibind} %\renewcommand\bibname{References} \def\bibname{\Large \bf References} \usepackage{lipsum} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \renewcommand{\headrulewidth}{0pt} \fancyhead[LE,RO]{\thepage} \thispagestyle{empty} %\afterpage{\lhead{new value}} \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} %\topmargin=-1.6cm \textheight 17.5cm% \textwidth 12cm % \topmargin 8mm % \oddsidemargin 20mm % \evensidemargin 20mm % \footskip=24pt % \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} %\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \renewenvironment{proof}{{\bfseries \noindent Proof.}}{~~~~$\square$} \makeatletter \def\th@newremark{\th@remark\thm@headfont{\bfseries}} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % If you want to insert other packages. Insert them here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\long\def\symbolfootnote[#1]#2{\begingroup% %\def\thefootnote{\fnsymbol{footnote}}\footnote[#1]{#2}\endgroup} \def \thesection{\arabic{section}} \begin{document} %\baselineskip 9mm %\setcounter{page}{} \thispagestyle{plain} {\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% \[ \left( ^{C}D_{b^{-}}^{\alpha}\psi\right) (x)=\frac{\left( -1\right) ^{n}% }{\Gamma(n-\alpha)}\int\limits_{x}^{b}\frac{\psi^{(n)}(t)}{\left( t-x\right) ^{\alpha-n+1}}dt,\text{\ \ \ }xa, \] 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{\ \ \ }x0$ 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 $a0 \] 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} \begin{thebibliography}{99} % \bibitem {opt3}G Adilov,G Tinaztepe, R Tinaztepe, On the global minimization of increasing positively homogeneous functions over the unit simplex, {\it International Journal of Computer Mathematics}, 87, (2010), 2733-2746. \bibitem {opt1}Bertsimas, D., Popescu, I., Optimal inequalities in probability theory, {\it A Convex Optimization Approach Sloan} WP \#4025, June 1998. \bibitem {caputo}M Caputo, Linear model of dissipation whose Q is almost frequency independent. 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