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\begin{frontmatter}

\dochead{Research Article}
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%\titulo{El t\'itulo en espa\~ol}
\titulo{Desigualdades del tipo Hermite-Hadamard para procesos estoc\'{a}sticos $ (m,h_{1},h_{2})- $convexos usando la integral fraccional de Katugampola.}
\title{Hermite - Hadamard type inequalities for $ (m,h_{1},h_{2})- $convex stochastic processes
using Katugampola fractional integral.}

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\author[J.Hernandez]{Jorge Eliecer Hern\'{a}ndez Hern\'{a}ndez }
\author[J. Gomez]{Juan Francisco Gomez}
\address[J.Hernandez]{Universidad Centroccidental Lisandro Alvarado \\ Decanato de Ciencias Econ\'{o}micas y Empresariales \\ Departamento de T\'{e}cnicas Cuantitativas, Barquisimeto, Venezuela.}
\address[J. Gomez]{Universidad Centroccidental Lisandro Alvarado \\ Decanato de Ciencias Econ\'{o}micas y Empresariales \\ Direcci\'{o}n del Centro de Investigaciones de DCEE, Barquisimeto, Venezuela.}
\email{jorgehernandez@ucla.edu.ve and juanfranciscogomez@ucla.edu.ve }

\begin{abstract}
In this article some inequalities of the Hermite-Hadamard type are presented for $ (m, h_ {1}, h_ {2}) - $convex stochastic processes using the fractional integral of Katugampola, and from these results specific cases are deduced for other stochastic processes with generalized convexity properties using the Riemann-Liouville fractional integral and the Riemann integral. 
\end{abstract}

\begin{keyword}
Hermite-Hadamard Inequalities, $ (m, h_ {1}, h_ {2}) - $convex stochastic processes, Katugampola Fractional integral

\MSC 35A23, 60E15, 26A33
\end{keyword}
\begin{resumen}
En este artículo se presntan algunas desigualdades del tipo Hermite-Hadamard para procesos estoc\'{a}sticos $ (m,h_{1},h_{2})- $con-vexos usando la integral fraccional de Katugampola, y a partir de estos resultados se deducen casos espec\'{i}ficos para otros procesos estoc\'{a}sticos con propiedades de convexidad generalizada usando la integral fraccional de Riemann-Liouville y la cl\'{a}sica integral de Riemann. 
\end{resumen}
\begin{palabras}
Desigualdad de Hermite-Hadamard, Procesos estoc\'{a}sticos $ (m,h_{1},h_{2})- $convexos, Integral fraccional de Katugampola

\MSC 35A23, 60E15, 26A33
\end{palabras}

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\end{frontmatter}
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\section{Introduction}
        The study of convex functions has been of interest for mathematical analysis
        based on the properties that are deduced from this concept. Due to generalization
        requirements of the convexity concept, in order to obtain new
        applications, in the last years great efforts have been made in the study
        and investigation of this topic.

        A function $f:I\rightarrow \mathbb{R}$ is said to be convex if for all $%
        x,y\in I$ and $t\in \left[ 0,1\right] $ the inequality%
        \begin{equation*}
            f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)
        \end{equation*}%
        holds.

        Numerous works of investigation have been realized extending results on
	    inequalities for convex func-tions towards others much more generalized,
	    using new concepts such as $E-$convexity (\cite{You1999}), quasi-convexity (\cite%
		{Set2016}), $s-$convexity (\cite{Alo2010}), logarithmically convexity (\cite%
		{Alo2009}), and others.

		A compendium about the history of the Hermite Hadamard inequality can be
		found in the work of D.S. Mitrinovic and I.B. Lackovic \cite{Mitri1985}. The
		formulation of this result is as follows:

		\textit{(Hermite-Hadamard Inequality)}. Let $f:I\rightarrow \mathbb{R}$ be
		a convex function, and $a,b\in I$ with $a<b$, then
		\begin{equation*}
			f\left( \frac{a+b}{2}\right) \leq \frac{1}{b-a}\int_{a}^{b}f(x)dx\leq \frac{f(a)+f(b)}{2}.
		\end{equation*}%
		The inequality of Hermite Hadamard has become a very useful tool in the
		Theory of Probability and Optimization (See \cite{Kum2002}).

		The study on convex stochastic processes began in 1974 when B. Nagy in \cite%
		{Nag1974}, applied a characterization of measurable stochastic processes to
		solving a generalization of the (additive) Cauchy functional equation. In
		1980 Nikodem \cite{Nick1980} considered convex stochastic processes. In 1992 and 1995
		Skowronski \cite{Skr1992,Skr1995} obtained some further results on convex stochastic
		processes which generalize some known properties.

		In the year 2014, E. Set et. al. in \cite{Set2014} investigated
		Hermite-Hadamard type inequalities for stochastic processes in the second
		sense. For other results related to stochastic processes see \cite{Bai2009},%
		\cite{Dev2015},\cite{Mik2010},\cite{Shak1985}, \cite{Shy2013}, where
		further references are given.

		Also, Fractional calculus \cite{Gor1997,Mill1993} was introduced at the end of the nineteenth century
		by Liouville and Riemann, the subject of which has become a rapidly growing
		area and has found applications in diverse fields ranging from physical sciences and
		engineering to biological sciences and economics.

		In 2011, U. Katugampola presented  a new fractional integral operator in \cite{Katugampola0} which generalizes the 		
		Riemann-Liouville and the Hadamard integrals into a single form, and various researchers   have made use of this 	
		result in the field of convexity, generalized convexity and others (\cite{Chen2017,Dahmani0,Dahmani1,Thaiprayoon}).

		Recently, several Hermite-Hadamard type inequalities \cite{Hernandez2018,Hernandez20182,Liu2016,Tunc} associated with
		fractional integrals have been investigated. Here, it is established several generalized Hermite-Hadamard type 
		integral inequalities for Stochastic processes using Katugampola fractional  integral operator.

\section{Preliminaries}


        The following notions can be found in some text books and articles. The reader can be see 
         \cite{Kot2012,Kot2013,Mik2010,Skr1992,Skr1995}.
        
       	Let $(\Omega ,\mathcal{A},\mu)$ be an arbitrary probability space. A function $%
        X:\Omega \rightarrow \mathbb{R}$ is called a random variable if it is $%
        \mathcal{A}$-measurable. Let $ I \subset \mathbb{R} $ be time.  A function $
        X:I \times \Omega \rightarrow \mathbb{R}$ is called stochastic process, if for all
        $ u \in I $ the function $X(u, \cdot)$ is a random variable.

        A stochastic process $X:I\times \Omega \rightarrow
        \mathbb{R}$ is called continuous in probability in the interval $I$ if for all $t_{0}\in I$ it is had that
        \begin{equation*}
         	\mu-\underset{t\rightarrow t_{0}}{\lim }X(t,\cdot )=X(t_{0},\cdot ),
        \end{equation*}
        where $\mu-\lim$ denotes the limit in probability, and it is called mean-square continuous 
        in the interval $I$ if for all $t_{0}\in I$ 
        \begin{equation*}
        	 \mu-\underset{t\rightarrow t_{0}}{\lim }\mathbb{E}(X(t,\cdot )-X(t_{0},\cdot))=0,
        \end{equation*}
        where $\mathbb{E}(X(t,\cdot ))$ denote the expectation value of the random
        variable $X(t,\cdot )$.
        
        Also, the monotony property it is attained. A stochastic process is called increasing (decreasing)
        if for all $u,v\in I$ such that $t<s$, 
        \begin{equation*}
         	X(u,\cdot )\leq X(v,\cdot ),\hspace{1cm}(X(u,\cdot )\geq X(v,\cdot )) \hspace{0.5cm}(a.e.)
        \end{equation*}
        respectively, and is called monotonic if it's increasing or decreasing.
         
        About differentiability it is said that a stochastic processes is differentiable at a point $t\in I$ 
        if there is a random variable $X^{\prime }(t,\cdot )$ such that
        \begin{equation*}
         	 X^{\prime }(t,\cdot )=\mu-\lim_{t\rightarrow t_{0}}{\frac{X(t,\cdot)-X(t_{0},\cdot )}{t-t_{0}}.}
        \end{equation*}

        Let $[a,b]\subset I,a=t_{0}<t_{1}<...<t_{n}=b$ be a partition of $[a,b]$ and $\theta _{k}\in \lbrack t_{k-1},t_{k}]$
        for $k=1,2,...,n$. Let $ X $ be a stochastic process such that $ \mathbb{E}(X(u,\cdot)^{2}) < \infty $. A random 
        variable $Y:\Omega \rightarrow \mathbb{R}$ is called mean-square integral of the process $X(t,\cdot )$ on $[a,b]$ 
        if the following identity holds:
        \begin{equation*}
           \lim_{n\rightarrow \infty }{E[X(\theta _{k},\cdot)(t_{k}-t_{k-1})-Y(\cdot)]^{2}}=0
        \end{equation*}%
        Then
        \begin{equation*}
           \int_{a}^{b}X(t,\cdot )dt=Y(\cdot )\hspace{1cm}(a.e.).
        \end{equation*}
        The book of K. Sobczyk \cite{Sobczyk} contains basic properties of mean-square integral.

        Important theorems as the mean value theorem for mean square derivatives and integrals for stochastic processes have 
        been proved in the work of J.C. Cort\'{e}z et. al. The reader can find these results in \cite[Lemma 3.1,Theorem 3.2]
        {Cortez2007}.

        In 1980 K. Nikodem introduced the following definition \cite{Nick1980}.
        \begin{definition} \label{def:ConvSP}
        Set $(\Omega ,\mathcal{A},P)$ be a probability space and $I\subset \mathbb{R}
        $ be an interval. The stochastic process $X:I\times \Omega
        \rightarrow \mathbb{R}$ is said to be convex stochastic process if 
        \begin{equation}
               X(\lambda u+(1-\lambda )v,\cdot )\leq \lambda X(u,\cdot )+(1-\lambda )X(v,\cdot )  \label{psc1}
        \end{equation}%
        holds almost everywhere for all $u,v\in I$ and $\lambda \in \lbrack 0,1]$.
        \end{definition}

      	Using Definition \ref{def:ConvSP}, D. Kotrys presented, in 2012, the Hermite-Hadamard integral 
        inequality version for Stochastic Processes \cite{Kot2012}.
        \begin{theorem}
        If $X:I\times \Omega \rightarrow \mathbb{R}$ is convex and mean
        square continuous in the interval $T\times \Omega $, then for any $u,v\in T$, the inequality 
        \begin{equation*}
          X\left( \frac{u+v}{2},\cdot \right) \leq \frac{1}{u-v}\int_{u}^{v}X(t,\cdot)dt\leq \frac{X(u,\cdot )+X(v,\cdot )}{2}
        \end{equation*}
        holds almost everywhere.
        \end{theorem}

       	There is also a generalization of the concept of convexity associated with stochastic processes. In \cite{Set2014}
      	 we find the following definition.

       	\begin{definition}
        Let 0 $< s < 1 $. A stochastic processes $X :I \times \Omega \rightarrow \mathbb{R}$ is said to be $s-$convex  
        stochastic processes in the second sense if 
        \begin{equation}
           X(ta+(1-t)b,\cdot) \leq t^{s}X(a,\cdot)+ (1-t)^{s}X(b,\cdot)
        \end{equation}
        holds almost everywhere for any $a, b \in I$ and all $t \in [0, 1]$.
        \end{definition}

       	Also, Hern\'{a}ndez Hern\'{a}ndez J.E. and G\'{o}mez, J.F.  introduced the definition of $ P- $convex Stochastic 
       	processes and $ (m,h_{1},h_{2})- $convex  stochastic processes in \cite{Hernandez2018} .

      	\begin{definition}
        A stochastic processes $X :I \times \Omega \rightarrow \mathbb{R}$ is said to be $P-$convex stochastic
       	processes if
       	\begin{equation}
        	 X(ta+(1-t)b,\cdot) \leq X(a,\cdot)+ X(b,\cdot)
       	\end{equation}
       	holds almost everywhere for any $a, b \in I$ and all $t \in [0, 1]$.
      	\end{definition}

      	\begin{definition}
       	Let $ m \in (0,1] $ and $ h_{1},h_{2}: [0,1] \rightarrow \mathbb{R}^{+} $ be functions. A stochastic processes $X :I  
       	\times \Omega \rightarrow \mathbb{R}$ is said to be $ (m,h_{1},h_{2})- $convex stochastic
       	processes if
       	\begin{equation}
          X(ta+m(1-t)b,\cdot) \leq h_{1}(t)X(a,\cdot)+ mh_{2}(t)X(b,\cdot)
       	\end{equation}
       	holds almost everywhere for any $a, b \in I$ and all $t \in [0, 1]$.
      	\end{definition}

      	Before establishing the main results, it will be given some necessary notions and
      	mathematical preliminaries of fractional calculus theory which are used
      	further in this paper. For more details, one can consult \cite{Gor1997,Kilbas,Mill1993,Pod1999}.

      	\begin{definition}
      	\label{DRL} 
      	Let$f\in L_{1}\left( \left[ a,b\right] \right) .$ The
      	Riemann-Liouville integrals $J_{a+}^{\alpha }$ and $J_{b-}^{\alpha }$ of
      	order $\alpha >0$ with $a\geq 0$ are defined by%
      	\begin{equation*}
      		  J_{a+}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int_{a}^{x}\left( x-t\right)^{\alpha -1}f(t)dt
      	\end{equation*}%
      	and%
      	\begin{equation*}
       		 J_{b-}^{\alpha }f(x)=\frac{1}{\Gamma (\alpha )}\int_{x}^{b}\left( t-x\right)^{\alpha -1}f(t)dt
      	\end{equation*}%
      	respectively, where $\Gamma (\alpha )$ is the Euler's Gamma function defined by
      	\begin{equation*}
       		 \Gamma (\alpha )=\int_{0}^{\infty }t^{\alpha -1}e^{-t}dt.
       	\end{equation*}
      	\end{definition}

     	Note that $J_{a+}^{0}f(x)=J_{b-}^{0}f(x)=f(x).$

     	Using the Riemann-Liouville fractional integral, Sarikaya et. al. \cite%
      	{Sar2013}, established the Hermite-Hadamard inequalities version.

     	\begin{theorem}
      	Let $f:\left[ a,b\right] \rightarrow \mathbb{R}$ be a positive function with
      	$a<b$ and $f\in L_{1}\left( \left[ a,b\right] \right) .$ If $f$ is a convex
      	function on $\left[ a,b\right] $ then%
        \begin{equation*}
          f\left( \frac{a+b}{2}\right) \leq \frac{\Gamma \left( \alpha +1\right) }{%
          \left( b-a\right) ^{\alpha }}\left( J_{a+}^{\alpha }f(b)+J_{b-}^{\alpha
          }f(a)\right) \leq \frac{f(a)+f(b)}{2}
        \end{equation*}%
      	with $\alpha >0.$
      	\end{theorem}
      
      	For stochastic processes,  Agahi H. and Babakhani A. established this inequality type in \cite{AgaBak}.
      	\begin{theorem}
      	Let $X : I \times \Omega \rightarrow \mathbb{R}$ be a Jensen-convex stochastic process that is meansquare continuous 
      	in the interval $I$ . Then for any $u, v \in I$ with $ u < v $, the following Hermite–Hadamard inequality
       	\begin{equation*}
         X\left( \frac{u+v}{2}, \cdot \right) \leq \frac{\Gamma(\alpha +1)}{2(v-u)^{\alpha}} \left( J_{a+}^{\alpha }X(b,\cdot)
         +J_{b-}^{\alpha}X(a,\cdot)\right)  \leq \frac{X(u,\cdot)+X(v,\cdot)}{2} \text{ \ \ } (a.e)
       	\end{equation*}
       	holds, where $ \alpha > 0 $.
      	\end{theorem}

      	Also, J. Hadamard in 1892 introduced the following fractional integral ope-rator (\cite{Hadamard}).
      	\begin{definition}  
      	Let $ \alpha > 0 $ with $ n-1< \alpha <n $ , $ n \in \mathbb{N} $, and $ a < x < b$. The left- and   
      	right-side Hadamard fractional integrals of order $ \alpha >0 $ of a function $ f $, are given by
       	\begin{eqnarray*}
          H_{a+}^{\alpha}f(t)= \frac{1}{\Gamma(\alpha)} \int_{a}^{x} \left(\ln \frac{x}{t} \right)^{\alpha -1} \frac{f(t)}{t}
          dt
       	\end{eqnarray*}
      	and
       	\begin{eqnarray*}
         H_{b-}^{\alpha}f(t)= \frac{1}{\Gamma(\alpha)} \int_{x}^{b} \left(\ln \frac{x}{t} \right)^{\alpha -1} \frac{f(t)}{t}dt
       	\end{eqnarray*}
      	respectively.
      	\end{definition}

      	As it was mentioned in the introductory section,  Katugampola introduced a new fractional
      	integral that generalizes the Riemann-Liouville and  Hadamard fractional integrals into a single
      	form (see \cite{Katugampola0,Katugampola1,Katugampola2}).

      	In the following $\ $\ will denote the space $X_{c}^{p}\left( a,b\right)
      	,\left( c\in \mathbb{R},1\leq p\leq \infty \right) $ of those complex valued
      	Lebesgue measurable functions $f$ on $\left[ a,b\right] $ for which $%
      	\left\Vert f\right\Vert _{X_{c}^{p}}<\infty $ where%
      	\begin{equation*}
       		 \left\Vert f\right\Vert _{X_{c}^{p}}=\left( \int_{a}^{b}\left\vert t^{c}f(t)\right\vert ^{p}\frac{dt}{t}\right) 
       		 ^{1/p}.
      	\end{equation*}%
      	Katugampola in \cite{Katugampola1} established the following definition and
      	property.

      	\begin{definition} \label{Katdef}
        Let $\left[ a,b\right] \subset \mathbb{R}$ be a finite interval. The left
        side and right side Katugampola fractional integral of order $\alpha >0$ of $%
        f\in X_{c}^{p}\left( a,b\right) $ are defined by
        \begin{equation*}
          ^{\rho }I_{a+}^{\alpha }f(x)=\frac{\rho ^{1-\alpha }}{\Gamma (\alpha )}%
          \int_{a}^{x}\frac{t^{\rho -1}}{\left( x^{\rho }-t^{\rho }\right) ^{1-\alpha }%
          }f(t)dt\text{ }
        \end{equation*}%
       	and
        \begin{equation*}
          ^{\rho }I_{b^{-}}^{\alpha }f(x)=\frac{\rho ^{1-\alpha }}{\Gamma (\alpha )}%
          \int_{x}^{b}\frac{t^{\rho -1}}{\left( t^{\rho }-x^{\rho }\right) ^{1-\alpha }%
          }f(t)dt,
        \end{equation*}%
       	with $a<x<b$ and $\rho >0,$ if the integrals exist.
      	\end{definition}

      	\begin{theorem}
      	\label{Thelp}
      	Let $\alpha >0$ and $\rho >0$. For $x>a$
      	\begin{equation*}
        	 \lim_{\rho \rightarrow 1}\text{ }^{\rho }I_{a+}^{\alpha }f(x)=J_{a+}^{\alpha}f(x)
       	\end{equation*}%
      	and
       	\begin{equation*}
         	\lim_{\rho \rightarrow 0^{+}}\text{ }^{\rho }I_{a+}^{\alpha}f(x)=H_{a+}^{\alpha }f(x).
       	\end{equation*}
     	Similar results also hold for the right-sided operators.
     	\end{theorem}

     	For this kind of fractional integral operator is proved the following Theorem which establish the Hermite-Hadamard    
     	inequality \cite{Hernandez20182}.
     	
     	\begin{theorem}
     	\label{T00}
     	Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }%
     	\right] \times \Omega \rightarrow \mathbb{R}$ be a positive stochastic process with $0\leq a<b$
     	and $X(t,\cdot )\in X_{c}^{p}\left( a^{\rho },b^{\rho }\right) .$ If $%
     	X\left( t,\cdot \right) $ is convex, the following inequality holds
     	almost everywhere:
     	\begin{equation*}
      		X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right) \leq \frac{\Gamma \left(
       		\alpha +1\right) }{2\rho ^{-\alpha }\left( b^{\rho }-a^{\rho }\right)
       		^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho
       		},\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho
       		},\cdot \right) \right) \leq \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right)
       		}{2\rho \alpha }
     	\end{equation*}%
     	\end{theorem}

     	In the same reference is proved the following Lemma.
     	\begin{lemma} 
     	\label{lem:l2}
     	Let $X:[a^{\rho },b^{\rho }]\times \Omega \rightarrow \mathbb{R}$ be a mean
     	square differentiable stochastic process; then the following equality holds:%
     	\begin{eqnarray*}
       		\hspace{-3cm}\frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right) }{2}-%
       		\frac{ \rho ^{\alpha }\Gamma \left( \alpha +1\right) }{2\left( b^{\rho
       		}-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
       		}X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
       		}X\left( b^{\rho },\cdot \right) \right)
     	\end{eqnarray*}%
     	\begin{equation*}
       		=\frac{\rho(b^{\rho }-a^{\rho })}{2}\int_{0}^{1}\left[ (1-t^{\rho })^{\alpha
       		}-t^{\rho \alpha }\right] t^{\rho -1}X^{\prime }\left( t^{\rho }a^{\rho
       		}+(1-t^{\rho })b^{\rho },\cdot \right) dt.
     	\end{equation*}
     	\end{lemma}

     	The purpose of this paper is to derive some inequalities of type Hermite-Hadamard for $ (m,h_{1},h_{2})-$convex  
     	stochastic processes using the Katugampola fractional integrals.

\section{Main Results}

		\begin{theorem}
		\label{T0}
		Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }%
		\right] \times \Omega \rightarrow \mathbb{R}$ be a positive stochastic process with $0\leq a<b$
		and $X(t,\cdot )\in X_{c}^{p}\left( a^{\rho },b^{\rho }\right) .$ If $%
		X\left( t,\cdot \right) $ is $ (m,h_{1},h_{2})- $convex the the following inequalities holds
		almost everywhere
		\begin{equation}
			\label{dt0}
			X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)  \leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) }	
			{\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( h_{1}(1/2)\text{ }^{\rho }I_{b-}^{\alpha }X\left(
			a^{\rho },\cdot \right)+ h_{2}(1/2)\text{ }^{\rho }I_{a+}^{\alpha
			}X\left( \frac{b^{\rho }}{m},\cdot \right) \right) 
		\end{equation}
		and
		\begin{equation*}
			\hspace{-5cm}\frac{\Gamma \left( \alpha \right) }{\rho ^{1-\alpha }\left( b^{\rho
			}-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
			}X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
			}X\left( b^{\rho },\cdot \right) \right)
		\end{equation*}%
		\begin{equation}
			\label{dt0-1}
			\leq \left( X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right)\right)\mathtt{I}(h_{1})
			+ \left( X\left( \frac{a^{\rho }}{m},\cdot \right) +X\left(\frac{b^{\rho }}{m},\cdot \right)\right) \mathtt{I}
			(h_{2}), 
		\end{equation}%
		where
		\begin{equation*}
			\mathtt{I}(h_{1})=\int_{0}^{1}t^{\alpha \rho -1}h_{1}(t^{\rho })dt \text{ \ and  \ } \mathtt{I}(h_{2})=\int_{0}
			^{1}t^{\alpha \rho -1}h_{2}(t^{\rho })dt.
		\end{equation*}
		\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
		\begin{proof}
		\hspace{0.25cm}Let $t\in \left[ 0,1\right] $, and  $u,v \in \left[ a,b \right] $ defined by
		\begin{equation}
			\label{d0}
			u^{\rho }=t^{\rho }a^{\rho }+(1-t^{\rho })b^{\alpha }\text{ \ and \ }v^{\rho
			}=(1-t^{\rho })a^{\rho }+t^{\rho }b^{\rho }.  
			\end{equation}
		then,
		\begin{equation*}
			X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)=X\left( \frac{u^{\rho }+v^{\rho }}{2},\cdot \right) =X\left( 	
			\frac{t^{\rho }a^{\rho }+(1-t^{\rho })b^{\alpha }+(1-t^{\rho })a^{\rho }+t^{\rho }b^{\rho }}{2}\right) .
		\end{equation*}%
		Using the $ (m,h_{1},h_{2})- $convexity of X,
		\begin{equation*}
			\hspace{-10cm}X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)
		\end{equation*}
		\begin{equation}
			\label{d1}
			\leq h_{1}\left( \frac{1}{2}\right) X\left( t^{\rho
			}a^{\rho }+\left( 1-t^{\rho }\right) b^{\alpha },\cdot \right) +mh_{2}\left( \frac{1}{2}\right) X\left(
			\left( 1-t^{\rho }\right) \frac{a^{\rho }}{m}+t^{\rho }\frac{b^{\rho}}{m},\cdot \right). 
		\end{equation}%
		Multiplying both side of $\left( \ref{d1}\right) $ by $t^{\alpha \rho
		-1},(\alpha ,\rho >0)$ and integrating over $t\in \left[ 0,1\right] ,$ it is
		obtained that%
		\begin{equation*}
			\hspace{-10cm}\frac{1}{\alpha \rho }X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)
		\end{equation*}
		\begin{equation*}
			\hspace{-4cm}\leq h_{1}\left( \frac{1}{2}\right)\int_{0}^{1}t^{\alpha \rho -1}X\left( t^{\rho }a^{\rho }+\left(
			1-t^{\rho }\right) b^{\alpha },\cdot \right) dt
		\end{equation*}
		\begin{equation}
			\label{d1.1}
			\hspace{2cm}+mh_{2}\left( \frac{1}{2}\right)\int_{0}^{1}t^{\alpha \rho -1}X\left( \left( 1-t^{\rho
			}\right) \frac{a^{\rho }}{m}+t^{\rho }\frac{b^{\rho}}{m},\cdot \right) dt.  
		\end{equation}
		Now, from $\left( \ref{d0}\right) $ and the Definition \ref{Katdef}, it is obtained that%
		\begin{eqnarray}
			\label{d1.2}
			\hspace{-4cm}\int_{0}^{1}t^{\alpha \rho -1}X\left( t^{\rho }a^{\rho }+(1-t^{\rho
			})b^{\alpha },\cdot \right) dt &=& \frac{1}{\left( b^{\rho }-a^{\rho }\right)
			^{\alpha }}\int_{a}^{b}\frac{u^{\rho -1}}{\left( u^{\rho }-b^{\rho }\right)
			^{1-\alpha }}X\left( u^{\rho },\cdot \right) du \notag \\ &=&\frac{\Gamma \left( \alpha \right) }{\rho ^{1-\alpha 
			}\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\text{ }^{\rho }I_{b-}^{\alpha }X\left(
			a^{\rho },\cdot \right)   
		\end{eqnarray}%
		and%
		\begin{eqnarray}
			\label{d1.3}
			\hspace{-4cm}\int_{0}^{1}t^{\alpha \rho -1}X\left( \left( 1-t^{\rho }\right) \frac{a^{\rho }}{m}+t^{\rho }	
			\frac{b^{\rho}}{m},\cdot \right) dt = \frac{\Gamma \left( \alpha \right) }{\rho ^{1-\alpha }\left(
			b^{\rho }-a^{\rho }\right) ^{\alpha }}\text{ }^{\rho }I_{a+}^{\alpha
			}X\left( \frac{b^{\rho }}{m},\cdot \right) .  
		\end{eqnarray}%
		Replacing \eqref{d1.2} and  \eqref{d1.3} in %
		\eqref{d1.1}, it is attained \eqref{dt0}
		\begin{equation*}
			X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)\leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) }
			{\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( h_{1}(1/2)\text{ }^{\rho }I_{b-}^{\alpha }X\left(
			a^{\rho },\cdot \right)+ h_{2}(1/2)\text{ }^{\rho }I_{a+}^{\alpha
			}X\left( \frac{b^{\rho }}{m},\cdot \right) \right).
		\end{equation*}

		In order to obtain \eqref{dt0-1}
		, it is used the $ (m,h_{1},h_{2})- $convexity property of the stochastic process $X$
		\begin{equation*}
			X\left( t^{\rho }a^{\rho }+(1-t^{\rho })b^{\alpha },\cdot \right) \leq
			h_{1}(t^{\rho })X\left( a^{\rho },\cdot \right) +mh_{2}( t^{\rho }) X\left(
			\frac{b^{\rho }}{m},\cdot \right)
		\end{equation*}%
		\begin{equation*}
			X\left( \left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right)
			\leq h_{1}(t^{\rho }) X\left(b^{\rho },\cdot \right) +mh_{2}(t^{\rho
			})X\left( \frac{a^{\rho }}{m},\cdot \right) ,
		\end{equation*}%
		adding these inequalities it is obtained
		\begin{equation*}
			\hspace{-5cm}X\left( t^{\rho }a^{\rho }+(1-t^{\rho })b^{\alpha },\cdot \right) +X\left(
			\left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right)
		\end{equation*}%
		\begin{equation}
			\label{d2}
			\leq \left( X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right)\right) h_{1}(t^{\rho })
			+ m\left( X\left( \frac{a^{\rho }}{m},\cdot \right) +X\left(\frac{b^{\rho }}{m},\cdot \right)\right) h_{2}(t^{\rho 
			}).
		\end{equation}%
		Multiplying both side of $\left( \ref{d2}\right) $ by $t^{\alpha \rho
		-1},(\alpha ,\rho >0)$ and integrating over $t\in \left[ 0,1\right] ,$ it is
		attained that%
		\begin{equation*}
			\hspace{-5cm}\frac{\Gamma \left( \alpha \right) }{\rho ^{1-\alpha }\left( b^{\rho
			}-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
			}X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
			}X\left( b^{\rho },\cdot \right) \right)
		\end{equation*}%
		\begin{equation*}
			\leq \left( X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right)\right)\mathtt{I}(h_{1})
			+ \left( X\left( \frac{a^{\rho }}{m},\cdot \right) +X\left(\frac{b^{\rho }}{m},\cdot \right)\right) \mathtt{I}
			(h_{2}),
		\end{equation*}%
		where
		\begin{equation*}
		\mathtt{I}(h_{1})=\int_{0}^{1}t^{\alpha \rho -1}h_{1}(t^{\rho })dt \text{ \ and  \ } \mathtt{I}(h_{2})=\int_{0}^{1}
		t^{\alpha \rho -1}h_{2}(t^{\rho })dt.
		\end{equation*}
		The proof is complete.
		\end{proof}

		\begin{remark} \label{rem:r1}
		Letting $ m=1 $ and  $ h_{1}(t)=t, h_{2}(t)=1-t $ for $ t \in [0,1] $ in Theorem \ref{T0}, it is had that
		\begin{eqnarray*}
			\mathtt{I}(h_{1}) =\int_{0}^{1}t^{\alpha \rho -1}t^{\rho}dt = \int_{0}^{1}t^{\rho(\alpha +1) -1}dt  = \frac{1}
			{\rho(\alpha +1)},
		\end{eqnarray*}
		\begin{eqnarray*}
			\mathtt{I}(h_{2}) =\int_{0}^{1}t^{\alpha \rho -1}(1-t^{\rho})dt = \frac{1}{\rho\alpha}-\frac{1}{\rho(\alpha +1)}
		\end{eqnarray*}
		and
		\begin{eqnarray*}
			h_{1}(1/2)=h_{2}(1/2)=\frac{1}{2}
		\end{eqnarray*}
		so, for convex stochastic processes  it is had that
		\begin{equation*}
			X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)  \leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) }
			{2\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( \text{ }^{\rho }I_{b-}^{\alpha }X\left(
			a^{\rho },\cdot \right)+ \text{ }^{\rho }I_{a+}^{\alpha
			}X\left( b^{\rho },\cdot \right) \right) \leq \frac{  X \left( a^{\rho },\cdot \right) +X \left( b^{\rho},\cdot 
			\right) }{2 \rho \alpha}
		\end{equation*}
		almost everywhere, making coincidence with Theorem 3.1 in \cite{Hernandez20182};  using the Theorem \ref{Thelp} it is 			obtained the Hermite-Hadamard inequality version for convex stochastic processes and the  Riemann
		Liouville fractional integral
		\begin{equation}
			\label{dr1}
			X\left( \frac{a+b}{2},\cdot \right) \leq \frac{\Gamma \left( \alpha
			+1\right) }{2\left( b-a\right) ^{\alpha }}\text{ }\left( J_{b-}^{\alpha
			}X\left( a,\cdot \right) +\text{ }J_{a-}^{\alpha }X\left( b,\cdot \right)
			\right) \leq \frac{X\left( a,\cdot \right) +X\left( b,\cdot \right) }{%
			2 \alpha },  
		\end{equation}%
		almost everywhere, making coincidence with the result proved by H. Aghahi and A. Babakhani  in \cite{AgaBak}.
		Letting $ \alpha =1 $ in  \eqref{dr1} it
		is obtained the Hermite Hadamard inequality for the classical
		Riemann integral
		\begin{equation*}
			X\left( \frac{a+b}{2},\cdot \right) \leq \frac{1}{\left( b-a\right) }\text{ }%
			\int_{a}^{b}X\left( t,\cdot \right) dt\leq \frac{X\left( a,\cdot \right)
			+X\left( b,\cdot \right) }{2}
		\end{equation*}%
		almost evrywhere, making coincidence with the result proved by Kotrys in \cite{Kot2012}.
		\end{remark}

		\begin{theorem} \label{thm:T1}
		Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }\right]
		\times \Omega \rightarrow \mathbb{R}$ be a mean square differentiable
		stochastic process with $0\leq a<b$ and $X(t,\cdot )\in X_{c}^{p}\left(
		a^{\rho },b^{\rho }\right) .$ If $\left\vert X^{\prime }\left( t,\cdot
		\right) \right\vert $ is $ (m,h_{1},h_{2})- $convex then the following inequality holds almost
		everywhere%
		\begin{eqnarray*}
			\hspace{-4cm}\left\vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot
			\right) }{2\alpha \rho }-\frac{\Gamma \left( \alpha  \right) }{2\rho
			^{1-\alpha }\left( b^{\rho }-a^{\rho }\right) ^{\alpha }}\text{ }\left(
			^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho },\cdot \right) +\text{ }%
			^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho },\cdot \right) \right)
			\right\vert
		\end{eqnarray*}%
		\begin{equation}
			\label{ineqT1}
			\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{2\alpha } \left( \left( \left\vert X^{\prime
			}\left( a^{\rho }\cdot \right) \right\vert + \left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert
			\right) \mathtt{I}(h_{1})  + \left( \left\vert X^{\prime}\left( \frac{a^{\rho }}{m},\cdot \right) \right\vert + 	
			\left\vert X^{\prime }\left( \frac{b^{\rho }}{m} ,\cdot \right) \right\vert\right) \mathtt{I}(h_{2})  \right) ,
		\end{equation}%
		where
		\begin{equation*}
			\mathtt{I}(h_{1})=\int_{0}^{1}t^{\rho(\alpha +1) -1}h_{1}(t^{\rho })dt \text{ \ and  \ } \mathtt{I}(h_{2})=
			\int_{0}^{1}t^{\rho(\alpha +1) -1}h_{2}(t^{\rho })dt.
		\end{equation*}
		\end{theorem}

		\begin{proof}
		\hspace{0.25cm}From the Definition \ref{Katdef} and a suitable change of variables we get%
		\begin{equation*}
			\hspace{-4cm}\frac{\Gamma \left( \alpha \right) }{\rho ^{1-\alpha }\left( b^{\rho
			}-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
			}X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
			}X\left( b^{\rho },\cdot \right) \right)
		\end{equation*}%
		\begin{equation*}
			\hspace{-2cm}=\int_{0}^{1}t^{\alpha \rho -1}X\left( t^{\rho }a^{\rho }+\left( 1-t^{\rho
			}\right) b^{\alpha },\cdot \right) dt
		\end{equation*}
		\begin{equation}
			\label{d2.1}
			\hspace{2cm}+\int_{0}^{1}t^{\alpha \rho -1}X\left(
			\left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right) dt
		\end{equation}%
		Integrating by parts each of the integrals we have%
		\begin{equation*}
			\hspace{-9cm}\int_{0}^{1}t^{\alpha \rho -1}X\left( t^{\rho }a^{\rho }+\left( 1-t^{\rho
			}\right) b^{\alpha },\cdot \right) dt
		\end{equation*}%
		\begin{equation*}
			=\left. \frac{t^{\alpha \rho }X\left( t^{\rho }a^{\rho }+\left( 1-t^{\rho
			}\right) b^{\alpha },\cdot \right) }{\alpha \rho }\right\vert _{0}^{1} -\frac{%
			\left( a^{\rho }-b^{\rho }\right) }{\alpha }\int_{0}^{1}t^{\alpha \rho +\rho
			-1}X^{\prime }\left( t^{\rho }a^{\rho }+\left( 1-t^{\rho }\right) b^{\alpha
			},\cdot \right) dt
		\end{equation*}%
		\begin{equation*}
			\hspace{-3cm}=\frac{X\left( a^{\rho },\cdot \right) }{\alpha \rho }-\frac{\left( a^{\rho
			}-b^{\rho }\right) }{\alpha }\int_{0}^{1}t^{\rho (\alpha +1)-1}X^{\prime
			}\left( t^{\rho }a^{\rho }+\left( 1-t^{\rho }\right) b^{\alpha },\cdot
			\right) dt
		\end{equation*}%
		and%
		\begin{equation*}
			\hspace{-9cm}\int_{0}^{1}t^{\alpha \rho -1}X\left( \left( 1-t^{\rho }\right) a^{\rho
			}+t^{\rho }b^{\rho },\cdot \right) dt
		\end{equation*}%
		\begin{equation*}
			=\left. \frac{t^{\alpha \rho }X\left( \left( 1-t^{\rho }\right) a^{\rho
			}+t^{\rho }b^{\rho },\cdot \right) }{\alpha \rho }\right\vert _{0}^{1} -\frac{%
			\left( b^{\rho }-a^{\rho }\right) }{\alpha }\int_{0}^{1}t^{\alpha \rho +\rho
			-1}X^{\prime }\left( \left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho
			},\cdot \right) dt
		\end{equation*}%
		\begin{equation*}
			\hspace{-3cm}=\frac{X\left( b^{\rho },\cdot \right) }{\alpha \rho }-\frac{\left( b^{\rho
			}-a^{\rho }\right) }{\alpha }\int_{0}^{1}t^{\rho (\alpha +1)-1}X^{\prime
			}\left( \left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right)
			dt.
		\end{equation*}%
		So%
		\begin{equation*}
			\hspace{-2cm}\frac{\Gamma \left( \alpha \right) }{\rho ^{1-\alpha }\left( b^{\rho
			}-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
			}X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
			}X\left( b^{\rho },\cdot \right) \right) \text{ } =\frac{X\left( a^{\rho
			},\cdot \right) +X\left( b^{\rho },\cdot \right) }{\alpha \rho }
			-\frac{\left( b^{\rho }-a^{\rho }\right) }{\alpha } \times
		\end{equation*}%
		\begin{equation}
			\label{d2.2}
			\left(\int_{0}^{1}t^{\rho (\alpha +1)-1}\left(X^{\prime }\left( t^{\rho }a^{\rho
			}+\left( 1-t^{\rho }\right) b^{\alpha },\cdot \right) -X^{\prime }\left(
			\left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right)
			\right)dt\right)   
		\end{equation}%
		By means of the equality $\left( \ref{d2.2}\right) ,$ the triangular
		inequality and the $ (m,h_{1},h_{2})- $convexity of $\left\vert X^{\prime }\left( t,\cdot
		\right) \right\vert ,$ it is obtained that%
		\begin{eqnarray*}
			\hspace{-4cm}\left\vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot
			\right) }{\alpha \rho }-\frac{\Gamma \left( \alpha  \right) }{\rho
			^{1-\alpha }\left( b^{\rho }-a^{\rho }\right) ^{\alpha }}\text{ }\left(
			^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho },\cdot \right) +\text{ }%
			^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho },\cdot \right) \right)
			\right\vert
		\end{eqnarray*}%
		\begin{equation*}
			\hspace{-2.5cm}\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{\alpha }\int_{0}^{1}t^{\rho
			(\alpha +1)-1}\left\vert X^{\prime }\left( t^{\rho }a^{\rho }+\left(
			1-t^{\rho }\right) b^{\alpha },\cdot \right) -X^{\prime }\left( \left(
			1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right) \right\vert dt
		\end{equation*}%
		\begin{equation*}
			\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{\alpha }\left(
			\int_{0}^{1}t^{\rho (\alpha +1)-1}\left\vert X^{\prime }\left( t^{\rho
			}a^{\rho }+\left( 1-t^{\rho }\right) b^{\alpha },\cdot \right) \right\vert dt%
			 +\int_{0}^{1}t^{\rho (\alpha +1)-1}\left\vert X^{\prime }\left(
			\left( 1-t^{\rho }\right) a^{\rho }+t^{\rho }b^{\rho },\cdot \right)
			\right\vert dt\right)
		\end{equation*}%
		\begin{equation*}
			\hspace{-4cm}\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{\alpha }\left(
			\int_{0}^{1}t^{\rho (\alpha +1)-1}\left( h_{1}(t^{\rho})\left\vert X^{\prime
			}\left( a^{\rho }\cdot \right) \right\vert + mh_{2}(t^{\rho})
			\left\vert X^{\prime }\left( \frac{b^{\rho }}{m}\cdot \right) \right\vert \right)
			dt\right.
		\end{equation*}%
		\begin{equation*}
			\hspace{2cm} \left.+\int_{0}^{1}t^{\rho (\alpha +1)-1}\left( m h_{2}(t^{\rho })
			\left\vert X^{\prime }\left( \frac{a^{\rho }}{m},\cdot \right) \right\vert + h_{1}(t^{\rho
			})\left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert \right)
			dt\right)
		\end{equation*}%
		\begin{equation*}
			\hspace{-5cm}=\frac{\left( b^{\rho }-a^{\rho }\right) }{\alpha } \left( \left( \left\vert X^{\prime
			}\left( a^{\rho }\cdot \right) \right\vert + \left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert
			\right) \int_{0}^{1}t^{\rho (\alpha +1)-1}h_{1}(t^{\rho})dt \right.
		\end{equation*}%
		\begin{equation*}
			\hspace{1cm}\left. + \left( \left\vert X^{\prime
			}\left( \frac{a^{\rho }}{m},\cdot \right) \right\vert + \left\vert X^{\prime }\left( \frac{b^{\rho }}{m} ,\cdot 	
			\right) \right\vert\right) \int_{0}^{1}t^{\rho (\alpha +1)-1}h_{2}(t^{\rho})dt \right) ,
		\end{equation*}
		and doing
		\begin{equation*}
			\mathtt{I}(h_{1})=\int_{0}^{1}t^{\rho(\alpha +1) -1}h_{1}(t^{\rho })dt \text{ \ and  \ } \mathtt{I}(h_{2})=			
			\int_{0}^{1}t^{\rho(\alpha +1) -1}h_{2}(t^{\rho })dt.
		\end{equation*}
		it is attained the desired result.

		The proof is complete.
		\end{proof}

		\begin{remark} \label{rem2}
		Letting $ m=1 $ and  $ h_{1}(t)=t, h_{2}(t)=1-t $ for $ t \in [0,1] $ in Theorem \ref{thm:T1}, it is had that
		\begin{eqnarray*}
			\mathtt{I}(h_{1}) =\int_{0}^{1}t^{\rho(\alpha +1) -1}t^{\rho}dt = \int_{0}^{1}t^{\rho(\alpha +2) -1}dt  = \frac{1}
			{\rho(\alpha +2)}
		\end{eqnarray*}
		and
		\begin{eqnarray*}
			\mathtt{I}(h_{2}) =\int_{0}^{1}t^{\rho(\alpha +1) -1}(1-t^{\rho})dt = \frac{1}{\rho(\alpha +1)}-\frac{1}
			{\rho(\alpha +2)}
		\end{eqnarray*}
		so, for convex stochastic processes  it is had that
		\begin{eqnarray*}
			\left\vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot
			\right) }{2\alpha \rho }-\frac{\rho^{\alpha -1}\Gamma \left( \alpha  \right) }{2\left( b^{\rho }-a^{\rho }\right) 				^{\alpha }}\text{ }\left(
			^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho },\cdot \right) +\text{ }%
			^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho },\cdot \right) \right)
			\right\vert \leq \frac{\left( b^{\rho }-a^{\rho }\right) }{2\rho \alpha (\alpha +1)} \left( \left\vert X^{\prime
			}\left( a^{\rho },\cdot \right) \right\vert + \left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert
			\right) .
		\end{eqnarray*}%
		Using the Theorem \ref{Thelp} it is obtained the Hermite-Hadamard inequality version for convex stochastic processes 	
		and the  Riemann Liouville fractional integral
		\begin{eqnarray*}
			\left\vert \frac{X\left( a,\cdot \right) +X\left( b,\cdot
			\right) }{2\alpha  }-\frac{\Gamma \left( \alpha  \right) }{2\left( b-a \right) ^{\alpha }}\text{ }\left(
			J_{b-}^{\alpha }X\left( a,\cdot \right) +\text{ }%
			J_{a+}^{\alpha }X\left( b,\cdot \right) \right)
			\right\vert \leq \frac{\left( b-a \right) }{ 2\alpha (\alpha +1)} \left( \left\vert X^{\prime
			}\left( a,\cdot \right) \right\vert + \left\vert X^{\prime }\left( b,\cdot \right) \right\vert\right) .
		\end{eqnarray*}
		and if $ \alpha = 1 $ then
		\begin{eqnarray*}
			\left\vert \frac{X\left( a,\cdot \right) +X\left( b,\cdot
			\right) }{2}-\frac{1}{\left( b-a \right)} \int_{a}^{b}X(t,\cdot)dt
			\right\vert \leq \frac{\left( b-a \right) }{4} \left( \left\vert X^{\prime
			}\left( a,\cdot \right) \right\vert + \left\vert X^{\prime }\left( b,\cdot \right) \right\vert\right) .
		\end{eqnarray*}
		\end{remark}


        \begin{theorem} 
        \label{T3}
          Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }\right]
          \times \Omega \rightarrow \mathbb{R}$ be a mean square differentiable
          stochastic process with $0\leq a<b$ and $X(t,\cdot )\in X_{c}^{p}\left(
          a^{\rho },b^{\rho }\right) .$ If $\left\vert X^{\prime }\left( t,\cdot
          \right) \right\vert $ is $ (m,h_{1},h_{2})- $convex then the following 
          inequality holds almost everywhere
           \begin{eqnarray*}
             \hspace{-4cm}\left \vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right) }{2}-%
             \frac{ \rho ^{\alpha }\Gamma \left( \alpha +1\right) }{2\left( b^{\rho
             }-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
             }X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
             }X\left( b^{\rho },\cdot \right) \right) \right \vert
           \end{eqnarray*}%
           \begin{equation*}
             \leq \frac{b^{\rho }-a^{\rho }}{2}\left( \left \vert X^{\prime }\left( a^{\rho
             },\cdot \right) \right\vert S(h_{1})+m\left\vert X\left( b^{\rho }/m, \cdot \right) \right \vert S(h_{2}) \right)
           \end{equation*}
          where
           \begin{equation*}
             S(h_{1})=\int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
             } -  t^{\rho \alpha } \right)  t^{\rho -1}  h_{1}(t^{\rho})dt + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho   
             \alpha } - (1-t^{\rho })^{\alpha} \right)  t^{\rho -1} h_{1}(t^{\rho})dt
           \end{equation*}
          and
           \begin{equation*}
             S(h_{2})=\int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
             } -  t^{\rho \alpha } \right)  t^{\rho -1}  h_{2}(t^{\rho})dt + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho  
             \alpha } - (1-t^{\rho })^{\alpha} \right)  t^{\rho -1} h_{2}(t^{\rho})dt
           \end{equation*}
          \end{theorem}

         \begin{proof}
           Using Lemma \ref{lem:l2}, the triangular inequality and the $ (m,h_{1},h_{2})- $convexity of $\left\vert X \right     
           \vert$ it is had that
           \begin{equation*}
             \hspace{-3cm}\left \vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot \right) }{2}-%
             \frac{ \rho ^{\alpha }\Gamma \left( \alpha +1\right) }{2\left( b^{\rho
             }-a^{\rho }\right) ^{\alpha }}\text{ }\left( ^{\rho }I_{b^{\rho }-}^{\alpha
             }X\left( a^{\rho },\cdot \right) +\text{ }^{\rho }I_{a^{\rho }+}^{\alpha
             }X\left( b^{\rho },\cdot \right) \right) \right \vert
           \end{equation*}%
           \begin{equation*}
             \hspace{-1cm}\leq \frac{\rho(b^{\rho }-a^{\rho })}{2}\int_{0}^{1}\left \vert (1-t^{\rho })^{\alpha
             }-t^{\rho \alpha }\right \vert t^{\rho -1}\left \vert X^{\prime }\left( t^{\rho }a^{\rho
             }+(1-t^{\rho })b^{\rho },\cdot \right) \right \vert dt
           \end{equation*}
           \begin{equation*}
             \leq \frac{\rho(b^{\rho }-a^{\rho })}{2}\left( \int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
             } -  t^{\rho \alpha } \right)  t^{\rho -1}\left \vert X^{\prime }\left( t^{\rho }a^{\rho
             }+(1-t^{\rho })b^{\rho },\cdot\right) \right \vert dt \right.
           \end{equation*}
           \begin{equation*}
             \hspace{4cm}\left. + \int_{1/2^{\frac{1}{\rho}}}^{1} \left(  t^{\rho \alpha } - (1-t^{\rho })^{\alpha
             }  \right)  t^{\rho -1}\left \vert X^{\prime }\left( t^{\rho }a^{\rho
             }+(1-t^{\rho })b^{\rho },\cdot \right) \right \vert dt \right) 
           \end{equation*}
           \begin{equation*}
             \leq \frac{\rho(b^{\rho }-a^{\rho })}{2}\left( \int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
             } -  t^{\rho \alpha } \right)  t^{\rho -1} \left( h_{1}(t^{\rho})\left \vert X^{\prime }\left( a^{\rho
             },\cdot \right) \right\vert + mh_{2}(t^{\rho })\left\vert X\left( b^{\rho }/m, \cdot\right) \right \vert \right) 
             dt \right.
           \end{equation*}
           \begin{equation*}
             \hspace{2cm}\left. + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho \alpha } - (1-t^{\rho })^{\alpha
             }  \right)  t^{\rho -1} \left( h_{1}(t^{\rho})\left \vert X^{\prime }\left( a^{\rho
             },\cdot \right) \right\vert + mh_{2}(t^{\rho })\left\vert X\left( b^{\rho }/m, \cdot \right) \right \vert \right) 
             dt \right)
           \end{equation*}
           \begin{equation*}
             \hspace{-4cm}\leq \frac{\rho(b^{\rho }-a^{\rho })}{2}\left( \left \vert X^{\prime }\left( a^{\rho
             },\cdot \right) \right\vert S(h_{1})+m\left\vert X\left( b^{\rho }/m, \cdot \right) \right \vert S(h_{2}) \right)
           \end{equation*}
           where
           \begin{equation*}
             S(h_{1})=\int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
             } -  t^{\rho \alpha } \right)  t^{\rho -1}  h_{1}(t^{\rho})dt + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho    
             \alpha } - (1-t^{\rho })^{\alpha}\right)  t^{\rho -1} h_{1}(t^{\rho})dt
           \end{equation*}
           and
           \begin{equation*}
             S(h_{2})=\int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
             } -  t^{\rho \alpha } \right)  t^{\rho -1}  h_{2}(t^{\rho})dt + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho 
             \alpha } - (1-t^{\rho })^{\alpha} \right)  t^{\rho -1} h_{2}(t^{\rho})dt.
           \end{equation*}
           The proof is complete.
         \end{proof}

         \begin{remark} \label{rem3}
          Letting $ m=1 $ and  $ h_{1}(t)=t, h_{2}(t)=1-t $ for $ t \in [0,1] $ in Theorem \ref{T3} it follows that 
          \begin{eqnarray*}
            S(h_{1})&=&\int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha
            } -  t^{\rho \alpha } \right)  t^{\rho -1} t^{\rho}dt + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho \alpha } -            
            (1-t^{\rho })^{\alpha }  \right)  t^{\rho -1} t^{\rho}dt \\
            &=&\frac{1}{\rho (\alpha +1)}\left[ 1-\frac{1}{2^{\alpha }}\right]
          \end{eqnarray*}
          and 
          \begin{eqnarray*}
            S(h_{2})&=&\int_{0}^{1/2^{\frac{1}{\rho}}} \left(  (1-t^{\rho })^{\alpha} -  t^{\rho \alpha } \right)  t^{\rho -1}   
            (1-t^{\rho})dt + \int_{1/2^{\frac{1}{\rho}}}^{1} \left( t^{\rho \alpha } - (1-t^{\rho })^{\alpha}   
            \right) t^{\rho -1} (1-t^{\rho})dt \\
            &=& \frac{1}{\rho (\alpha +1)}\left[1 - \frac{1}{2^{\alpha +1}}\right]
          \end{eqnarray*}
          and in consecuence it follows the Theorem 3.8 in \cite{Hernandez20182} for convex stochastic processes, and in   
          consequence the Remark  3.9 in the same reference.
          \end{remark}

\section{Some Consequences}
          \begin{corollary}
          \label{C1}
            Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }%
            \right] \times \Omega \rightarrow \mathbb{R}$ be a positive stochastic process with $0\leq a<b$
            and $X(t,\cdot )\in X_{c}^{p}\left( a^{\rho },b^{\rho }\right) .$ If $%
            X\left( t,\cdot \right) $ is $ s- $convex in the second sense the following inequalities holds
            almost everywhere
            \begin{equation*}
              X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)  \leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) }
              {2^{s}\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( \text{ }^{\rho }I_{b-}^{\alpha }X\left(
              a^{\rho },\cdot \right)+ \text{ }^{\rho }I_{a+}^{\alpha
              }X\left( b^{\rho },\cdot \right) \right) \leq \frac{ X \left( a^{\rho },\cdot \right) + X \left( b^{\rho },\cdot      
              \right)}{2^{s}} \left( \frac{1}{\alpha \rho +s}+ B(\alpha \rho,s+1) \right)
            \end{equation*}
          \end{corollary}

          \begin{proof}
            Letting $ m=1 $, $ h_{1}(t)=t^{s},h_{2}(t)=(1-t)^{s} $ for all $ t \in [0,1] $ and $ s \in (0,1] $ in Theorem  
            \ref{T0} it is had that
            \begin{equation*}
              \mathtt{I}(h_{1})=\int_{0}^{1}t^{\alpha \rho +s -1}dt = \frac{1}{\alpha \rho +s}, \mathtt{I}(h_{2})=\int_{0}^{1}  
              t^{\alpha \rho  -1}(1-t)^{s}dt =B(\alpha \rho,s+1),
            \end{equation*}
            and
            \begin{equation*}
               h_{1}(1/2)=h_{2}(1/2)=\frac{1}{2^{s}}.
            \end{equation*}
            Therefore
            \begin{equation*}
              X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)  \leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) }   
              {2^{s}\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( \text{ }^{\rho }I_{b-}^{\alpha }X\left(
              a^{\rho },\cdot \right)+ \text{ }^{\rho }I_{a+}^{\alpha
              }X\left( b^{\rho },\cdot \right) \right) \leq \frac{ X \left( a^{\rho },\cdot \right) + X \left( b^{\rho },\cdot    
              \right)}{2^{s}} \left( \frac{1}{\alpha \rho +s}+ B(\alpha \rho,s+1) \right)
            \end{equation*}
            The proof is complete.
          \end{proof}

         \begin{corollary}
          \label{C2}
          Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }%
          \right] \times \Omega \rightarrow \mathbb{R}$ be a positive stochastic process with $0\leq a<b$
          and $X(t,\cdot )\in X_{c}^{p}\left( a^{\rho },b^{\rho }\right) .$ If $%
          X\left( t,\cdot \right) $ is $ P- $convex the following inequalities holds
          almost everywhere
          \begin{equation*}
            X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)  \leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) } 
            {\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( \text{ }^{\rho }I_{b-}^{\alpha }X\left(
            a^{\rho },\cdot \right)+ \text{ }^{\rho }I_{a+}^{\alpha
            }X\left( b^{\rho },\cdot \right) \right) \leq \frac{ X \left( a^{\rho },\cdot \right) + X \left( b^{\rho },\cdot 
            \right)}{\alpha \rho}
          \end{equation*}
         \end{corollary}

         \begin{proof}
           Letting $ m=1 $, $ h_{1}(t)= h_{2}(t)= 1$ for all $ t \in [0,1] $  in Theorem \ref{T0} it is had that
           \begin{equation*}
             \mathtt{I}(h_{1})=\mathtt{I}(h_{2})=\int_{0}^{1}t^{\alpha \rho -1}dt = \frac{1}{\alpha \rho }, h_{1}(1/2)=h_{2}
             (1/2)= 1.
           \end{equation*}
          Therefore
           \begin{equation*}
             X\left( \frac{a^{\rho }+b^{\rho }}{2},\cdot \right)  \leq \frac{ \rho^{\alpha}\Gamma \left( \alpha + 1 \right) } 
             {\left( b^{\rho}-a^{\rho }\right) ^{\alpha }}\left( \text{ }^{\rho }I_{b-}^{\alpha }X\left(
             a^{\rho },\cdot \right)+ \text{ }^{\rho }I_{a+}^{\alpha
             }X\left( b^{\rho },\cdot \right) \right) \leq \frac{ X \left( a^{\rho },\cdot \right) + X \left( b^{\rho },\cdot  
             \right)}{\alpha \rho}
           \end{equation*}
         \end{proof}

		\begin{corollary} \label{C3}
		Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }\right]
		\times \Omega \rightarrow \mathbb{R}$ be a mean square differentiable
		stochastic process with $0\leq a<b$ and $X(t,\cdot )\in X_{c}^{p}\left(
		a^{\rho },b^{\rho }\right) .$ If $\left\vert X^{\prime }\left( t,\cdot
		\right) \right\vert $ is $ s- $convex in the second sense then the following inequality holds almost
		everywhere%
		\begin{eqnarray*}
			\hspace{-4cm}\left\vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot
			\right) }{2\alpha \rho }-\frac{\Gamma \left( \alpha  \right) }{2\rho
			^{1-\alpha }\left( b^{\rho }-a^{\rho }\right) ^{\alpha }}\text{ }\left(
			^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho },\cdot \right) +\text{ }%
			^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho },\cdot \right) \right)
			\right\vert
		\end{eqnarray*}%
		\begin{equation*}
			\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{\rho\alpha} \left(  \left\vert X^{\prime
			}\left( a^{\rho }\cdot \right) \right\vert + \left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert
			\right)\left( \frac{1}{(\alpha +s+1)}+B(\alpha ,s+1)\right) 
		\end{equation*}
		\end{corollary}
		
		\begin{proof}
		Letting $ m=1 $, $ h_{1}(t)=t^{s},h_{2}(t)=(1-t)^{s} $ for all $ t \in [0,1] $ and $ s \in (0,1] $ in Theorem \ref{T0} 
		it is had that
		\begin{equation*}
			\mathtt{I}(h_{1})=\int_{0}^{1}t^{\rho(\alpha + s+1) -1}dt = \frac{1}{\rho(\alpha +s+1)}, \mathtt{I}(h_{2})=
			\int_{0}^{1}t^{\alpha \rho  -1}(1-t^{\rho})^{s}dt =\frac{1}{\rho}B(\alpha ,s+1).
		\end{equation*}
		Replacing these values in \eqref{ineqT1} it is attained
		\begin{eqnarray*}
			\hspace{-4cm}\left\vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot
			\right) }{2\alpha \rho }-\frac{\Gamma \left( \alpha  \right) }{2\rho
			^{1-\alpha }\left( b^{\rho }-a^{\rho }\right) ^{\alpha }}\text{ }\left(
			^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho },\cdot \right) +\text{ }%
			^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho },\cdot \right) \right)
			\right\vert
		\end{eqnarray*}
		\begin{equation*}
		\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{\rho\alpha} \left(  \left\vert X^{\prime
		}\left( a^{\rho }\cdot \right) \right\vert + \left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert\right)
		\left( \frac{1}{(\alpha +s+1)}+B(\alpha ,s+1)\right) 
		\end{equation*}
		\end{proof}
		
		\begin{corollary} \label{C4}
		Let $\alpha >0$ and $\rho >0.$ Let $X:\left[ a^{\rho },b^{\rho }\right]
		\times \Omega \rightarrow \mathbb{R}$ be a mean square differentiable
		stochastic process with $0\leq a<b$ and $X(t,\cdot )\in X_{c}^{p}\left(
		a^{\rho },b^{\rho }\right) .$ If $\left\vert X^{\prime }\left( t,\cdot
		\right) \right\vert $ is $ P- $convex then the following inequality holds almost
		everywhere%
		\begin{eqnarray*}
			\hspace{-4cm}\left\vert \frac{X\left( a^{\rho },\cdot \right) +X\left( b^{\rho },\cdot
			\right) }{2\alpha \rho }-\frac{\Gamma \left( \alpha  \right) }{2\rho
			^{1-\alpha }\left( b^{\rho }-a^{\rho }\right) ^{\alpha }}\text{ }\left(
			^{\rho }I_{b^{\rho }-}^{\alpha }X\left( a^{\rho },\cdot \right) +\text{ }%
			^{\rho }I_{a^{\rho }+}^{\alpha }X\left( b^{\rho },\cdot \right) \right)
			\right\vert
		\end{eqnarray*}%
		\begin{equation*}
			\leq \frac{\left( b^{\rho }-a^{\rho }\right) }{\alpha^{2} \rho } \left(  \left\vert X^{\prime
			}\left( a^{\rho }\cdot \right) \right\vert + \left\vert X^{\prime }\left( b^{\rho },\cdot \right) \right\vert
			\right)
		\end{equation*}%
		\end{corollary}
		
		\begin{proof}
		Using the scheme presented in Corollary \ref{C2} it is attained the desired result.
		\end{proof}

\section{Conclusions}
          In the present work some inequalities for $ (m,h_{1},h_{2})- $convex stochastic processes have been established and 
          from these results it has been deduced some previous results found in the works of Agahi H. and Babakhani A.
          \cite{AgaBak}, D. Kotrys \cite{Kot2012} and Hern\'{a}ndez Hern\'{a}ndez J.E and G\'{o}mez J.F
          \cite{Hernandez20182}. Some others inequalities can be found from the Theorems established using the Lemma
          \ref{Thelp} related with the Riemann Liouville fractional integral, Hadamard fractional integral and the
          classical Riemann integral. The authors hope that this work serve to stimulate the advance in this research
          area.

\section*{Acknowledgement}
          The authors would like to thank the Council of Scientific, Humanistic and Technological Development
          (Consejo de Deasarrollo Cient\'{i}fico, Human\'{i}stico y Tecnol\'{o}gico) of the Centroccidental University 
          Lisandro Alvarado (Universidad Centroccidental Lisandro Alvarado, Venezuela) for the technical support provided in
          the preparation of this work (Project RAC-2018-01), as well as the arbitrators appointed for the evaluation
          of this work and the editorial body of the Matua Journal of the Mathematics Program of Atlantic University
         (Revista del Programa de Matem\'{a}ticas de la Universidad del Atl\'{a}ntico, Colombia.
          They also thank Dr. Miguel Vivas (from Pontificia Universidad Católica del Ecuador),
          for his valuable collaboration.
      
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