1 Introduction
In 1980, Nikodem [11] introduced convex stochastic processes and investigated their regularity properties. In 1992, Skwronski [17] obtained some further results on convex stochastic processes.
Let $(\varOmega ,\mathcal{A},P)$ be an arbitrary probability space. A function $X:\varOmega \to \mathbb{R}$ is called a random variable if it is $\mathcal{A}$-measurable. A function $X:I\times \varOmega \to \mathbb{R}$, where $I\subset \mathbb{R}$ is an interval, is called a stochastic process if for every $t\in I$ the function $X(t,.)$ is a random variable.
Recall that the stochastic process $X:I\times \varOmega \to \mathbb{R}$ is called
(i) continuous in probability in interval I, if for all ${t_{0}}\in I$ we have
where $P\text{-}\lim $ denotes the limit in probability.
(ii) mean-square continuous in the interval I, if for all ${t_{0}}\in I$
where $E[X(t)]$ denotes the expectation value of the random variable $X(t,.)$.
Obviously, mean-square continuity implies continuity in probability, but the converse implication is not true.
Definition 1.
Suppose we are given a sequence $\{{\Delta ^{m}}\}$ of partitions, ${\Delta ^{m}}=\{{a_{m,0}},\dots ,{a_{m,{n_{m}}}}\}$. We say that the sequence $\{{\Delta ^{m}}\}$ is a normal sequence of partitions if the length of the greatest interval in the n-th partition tends to zero, i.e.,
Now we would like to recall the concept of the mean-square integral. For the definition and basic properties see [18].
Let $X:I\times \varOmega \to \mathbb{R}$ be a stochastic process with $E[X{(t)^{2}}]<\infty $ for all $t\in I$. Let $[a,b]\subset I$, $a={t_{0}}<{t_{1}}<{t_{2}}<\cdots <{t_{n}}=b$ be a partition of $[a,b]$ and ${\varTheta _{k}}\in [{t_{k-1}},{t_{k}}]$ for all $k=1,\dots ,n$. A random variable $Y:\varOmega \to \mathbb{R}$ is called the mean-square integral of the process X on $[a,b]$, if we have
\[ \underset{n\to \infty }{\lim }E\Bigg[{\Bigg({\sum \limits_{k=1}^{n}}X({\varTheta _{k}})({t_{k}}-{t_{k-1}})-Y\Bigg)^{2}}\Bigg]=0\]
for all normal sequences of partitions of the interval $[a,b]$ and for all ${\varTheta _{k}}\in [{t_{k-1}},{t_{k}}]$, $k=1,\dots ,n$. Then, we write
For the existence of the mean-square integral it is enough to assume the mean-square continuity of the stochastic process X.Throughout the paper we will frequently use the monotonicity of the mean-square integral. If $X(t,\cdot )\le Y(t,\cdot )$ (a.e.) in some interval $[a,b]$, then
Of course, this inequality is an immediate consequence of the definition of the mean-square integral.
Definition 2.
We say that a stochastic processes $X:I\times \varOmega \to \mathbb{R}$ is convex, if for all $\lambda \in [0,1]$ and $u,v\in I$ the inequality
is satisfied. If the above inequality is assumed only for $\lambda =\frac{1}{2}$, then the process X is Jensen-convex or $\frac{1}{2}$-convex. A stochastic process X is concave if $(-X)$ is convex. Some interesting properties of convex and Jensen-convex processes are presented in [11, 18].
(1)
\[ X\big(\lambda u+(1-\lambda )v,\cdot \big)\le \lambda X(u,\cdot )+(1-\lambda )X(v,\cdot )\hspace{5pt}\text{(a.e.)}\]Now, we present some results proved by Kotrys [6] about Hermite–Hadamard inequality for convex stochastic processes.
Lemma 1.
If $X:I\times \varOmega \to \mathbb{R}$ is a stochastic process of the form $X(t,\cdot )=A(\cdot )t+B(\cdot )$, where $A,B:\varOmega \to \mathbb{R}$ are random variables, such that $E[{A^{2}}]<\infty ,E[{B^{2}}]<\infty $ and $[a,b]\subset I$, then
Proposition 1.
Let $X:I\times \varOmega \to \mathbb{R}$ be a convex stochastic process and ${t_{0}}\in intI$. Then there exists a random variable $A:\varOmega \to \mathbb{R}$ such that X is supported at ${t_{0}}$ by the process $A(\cdot )(t-{t_{0}})+X({t_{0}},\cdot )$. That is
for all $t\in I$.
Theorem 1.
Let $X:I\times \varOmega \to \mathbb{R}$ be a Jensen-convex, mean-square continuous in the interval I stochastic process. Then for any $u,v\in I$ we have
In [7], Kotrys introduced the concept of strongly convex stochastic processes and investigated their properties.
Definition 3.
Let $C:\varOmega \to \mathbb{R}$ denote a positive random variable. The stochastic process $X:I\times \varOmega \to \mathbb{R}$ is called strongly convex with modulus $C(\cdot )>0$, if for all $\lambda \in [0,1]$ and $u,v\in I$ the inequality
\[ X\big(\lambda u+(1-\lambda )v,\cdot \big)\le \lambda X(u,\cdot )+(1-\lambda )X(v,\cdot )-C(\cdot )\lambda (1-\lambda ){(u-v)^{2}}\hspace{17.5pt}\text{a.e.}\]
is satisfied. If the above inequality is assumed only for $\lambda =\frac{1}{2}$, then the process X is strongly Jensen-convex with modulus $C(\cdot )$.In [5], Hafiz gave the following definition of stochastic mean-square fractional integrals.
Definition 4.
For the stochastic proces $X:I\times \varOmega \to \mathbb{R}$, the concept of stochastic mean-square fractional integrals ${I_{u+}^{\alpha }}$ and ${I_{v+}^{\alpha }}$ of X of order $\alpha >0$ is defined by
and
Using this concept of stochastic mean-square fractional integrals ${I_{a+}^{\alpha }}$ and ${I_{b+}^{\alpha }}$, Agahi and Babakhani proved the following Hermite–Hadamard type inequality for convex stochastic processes:
Theorem 2.
Let $X:I\times \varOmega \to \mathbb{R}$ be a Jensen-convex stochastic process that is mean-square continuous in the interval I. Then for any $u,v\in I$, the following Hermite–Hadamard inequality
holds, where $\alpha >0$.
(3)
\[ X\bigg(\frac{u+v}{2},\cdot \bigg)\le \frac{\varGamma (\alpha +1)}{2{(v-u)^{\alpha }}}\big[{I_{u+}^{\alpha }}[X](v)+{I_{v-}^{\alpha }}[X](u)\big]\le \frac{X(u,\cdot )+X(v,\cdot )}{2}\hspace{2.5pt}\textit{(a.e.)}\]2 Main results
In tis section, we introduce the concept of the generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X.
In [13], Raina studied a class of functions defined formally by
where the cofficents $\sigma (k)$ $(k\in {\mathbb{N}_{0}}=\mathbb{N}\cup \{0\})$ make a bounded sequence of positive real numbers and $\mathcal{R}$ is the set of real numbers. For more information on the function (4), please refer to [8, 12]. With the help of (4), we give the following definition.
(4)
\[ {\mathcal{F}_{\rho ,\lambda }^{\sigma }}(x)={\mathcal{F}_{\rho ,\lambda }^{\sigma (0),\sigma (1),\dots }}(x)={\sum \limits_{k=0}^{\infty }}\frac{\sigma (k)}{\varGamma (\rho k+\lambda )}{x^{k}}\hspace{7.5pt}\big(\rho ,\lambda >0;|x|<\mathcal{R}\big),\]Definition 5.
Let $X:I\times \varOmega \to \mathbb{R}$ be a stochastic process. The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,a+;\omega }^{\alpha }}$ and ${\mathcal{J}_{\rho ,\lambda ,b-;\omega }^{\alpha }}$ of X are defined by
and
where $\lambda ,\rho >0,\omega \in \mathbb{R}$.
(5)
\[ {\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](x)={\int _{u}^{x}}{(x-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(x-t)^{\rho }}\big]X(t,\cdot )dt,\hspace{2.5pt}\hspace{2.5pt}\hspace{2.5pt}\text{(a.e.)}\hspace{2.5pt}\hspace{2.5pt}x>u,\](6)
\[ {\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](x)={\int _{x}^{v}}{(t-x)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-x)^{\rho }}\big]X(s,\cdot )dt,\hspace{2.5pt}\hspace{2.5pt}\hspace{2.5pt}\text{(a.e.)}\hspace{2.5pt}\hspace{2.5pt}x<v,\]Many useful generalized mean-square fractional integrals can be obtained by specializing the coefficients $\sigma (k)$. Here, we just point out that the stochastic mean-square fractional integrals ${I_{a+}^{\alpha }}$ and ${I_{b+}^{\alpha }}$ can be established by coosing $\lambda =\alpha $, $\sigma (0)=1$ and $w=0$.
Now we present Hermite–Hadamard inequality for generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,a+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,b-;\omega }^{\sigma }}$ of X.
Theorem 3.
Let $X:I\times \varOmega \to \mathbb{R}$ be a Jensen-convex stochastic process that is mean-square continuous in the interval I. For every $u,v\in I,\hspace{2.5pt}u<v$, we have the following Hermite–Hadamard inequality
(7)
\[\begin{aligned}{}& X\bigg(\frac{u+v}{2},\cdot \bigg)\\ {} & \hspace{1em}\le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\big[{\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](t)\big]\\ {} & \hspace{1em}\le \frac{X(u,\cdot )+X(v,\cdot )}{2}.\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{}\hspace{2.5pt}\textit{a.e.}\end{aligned}\]Proof.
Since the process X is mean-square continuous, it is continuous in probability. Nikodem [11] proved that every Jensen-convex and continuous in probability stochastic process is convex. Since X is convex, then by Proposition 1, it has a supporting process at any point ${t_{0}}\in intI$. Let us take a support at ${t_{0}}=\frac{u+v}{2}$, then we have
Multiplying both sides of (8) by $[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}[\omega {(v-t)^{\rho }}]+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}[\omega {(t-u)^{\rho }}]]$, then integrating the resulting inequality with respect to t over $[u,v]$, we obtain
(8)
\[ X(t,\cdot )\ge A(\cdot )\bigg(t-\frac{u+v}{2}\bigg)+X\bigg(\frac{u+v}{2},\cdot \bigg).\hspace{7.5pt}\text{a.e.}\](9)
\[\begin{aligned}{}& {\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]X(t,\cdot )dt\\ {} & \hspace{1em}\ge A(\cdot ){\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]\\ {} & \hspace{2em}+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]\bigg(t-\frac{u+v}{2}\bigg)dt\\ {} & \hspace{2em}+X\bigg(\frac{u+v}{2},\cdot \bigg){\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]\\ {} & \hspace{2em}+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt\\ {} & \hspace{1em}=A(\cdot ){\underset{u}{\overset{v}{\int }}}\big[t{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+t{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt\\ {} & \hspace{2em}-A(\cdot )\frac{u+v}{2}{\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt\\ {} & \hspace{2em}+X\bigg(\frac{u+v}{2},\cdot \bigg){\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]\\ {} & \hspace{2em}+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt.\end{aligned}\]Calculating the integrals, we have
and similarly,
where ${\sigma _{1}}(k)=\frac{\sigma (k)}{\rho k+\lambda +1}$, $k=0,1,2,\dots $. Using the identities (10) and (11) in (9), we obtain
(10)
\[\begin{aligned}{}& {\underset{u}{\overset{v}{\int }}}t{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt\\ {} & \hspace{1em}=-{\underset{u}{\overset{v}{\int }}}{(v-t)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt+v{\underset{u}{\overset{v}{\int }}}{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt\\ {} & \hspace{1em}=-{(v-u)^{\lambda +1}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]+v{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\end{aligned}\](11)
\[\begin{aligned}{}& {\underset{u}{\overset{v}{\int }}}t{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]dt\\ {} & \hspace{1em}={\underset{u}{\overset{v}{\int }}}{(t-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]dt+u{\underset{u}{\overset{v}{\int }}}{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]dt\\ {} & \hspace{1em}={(v-u)^{\lambda +1}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]+u{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\end{aligned}\]
\[\begin{aligned}{}& {\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](t)\\ {} & \hspace{1em}\ge A(\cdot )(u+v){(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{2em}-A(\cdot )\frac{u+v}{2}2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{2em}+X\bigg(\frac{u+v}{2},\cdot \bigg)2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{1em}=X\bigg(\frac{u+v}{2},\cdot \bigg)2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big].\end{aligned}\]
That is,
\[\begin{aligned}{}& X\bigg(\frac{u+v}{2},\cdot \bigg)\\ {} & \hspace{1em}\le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\big[{\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](t)\big]\hspace{2.5pt}\text{a.e.,}\hspace{2.5pt}\end{aligned}\]
which completes the proof of the first inequality in (7).By using the convexity of X, we get
\[\begin{aligned}{}X(t,\cdot )& =X\bigg(\frac{v-t}{v-u}u+\frac{t-u}{v-u}v,\cdot \bigg)\le \frac{v-t}{v-u}X(u,\cdot )+\frac{t-u}{v-u}X(v,\cdot )\\ {} & =\frac{X(v,\cdot )-X(u,\cdot )}{v-u}t+\frac{X(u,\cdot )v-X(v,\cdot )u}{v-u}\hspace{5pt}\text{a.e.}\end{aligned}\]
for $t\in [u,v]$. Using the identities (10) and (11), it follows that
\[\begin{aligned}{}& {\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]X(t,\cdot )dt\\ {} & \hspace{1em}\le \frac{X(v,\cdot )-X(u,\cdot )}{v-u}\\ {} & \hspace{2em}\times {\underset{u}{\overset{v}{\int }}}\big[t{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+t{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt\\ {} & \hspace{2em}+\frac{X(u,\cdot )v-X(v,\cdot )u}{v-u}\\ {} & \hspace{2em}\times {\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt\\ {} & \hspace{1em}=\frac{X(v,\cdot )-X(u,\cdot )}{v-u}(u+v){(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{2em}+\frac{X(u,\cdot )v-X(v,\cdot )u}{v-u}2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{1em}=\big[X(u,\cdot )+X(v,\cdot )\big]{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big].\end{aligned}\]
That is,
\[\begin{aligned}{}& \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\big[{\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](t)\big]\\ {} & \hspace{1em}\le \frac{X(u,\cdot )+X(v,\cdot )}{2}\hspace{7.5pt}\text{a.e.,}\hspace{2.5pt}\end{aligned}\]
which completes the proof. □Remark 1.
Theorem 4.
Let $X:I\times \varOmega \to \mathbb{R}$ be a stochastic process, which is strongly Jensen-convex with modulus $C(\cdot )$ and mean-square continuous in the interval I so that $E[{C^{2}}]<\infty $. Then for any $u,v\in I$, we have
\[\begin{aligned}{}& X\bigg(\frac{u+v}{2},\cdot \bigg)\\ {} & \hspace{1em}-C(\cdot )\bigg\{2{(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]-2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{1em}+\big({u^{2}}+{v^{2}}\big){(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]-{\bigg(\frac{u+v}{2}\bigg)^{2}}\bigg\}\\ {} & \hspace{2em}\le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\big[{\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](t)\big]\\ {} & \hspace{2em}\le \frac{X(u,\cdot )+X(v,\cdot )}{2}-C(\cdot )\bigg\{\frac{{u^{2}}+{v^{2}}}{2}+2{(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{1em}\hspace{2em}-2{(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\rho }}\big]\hspace{0.1667em}+\big({u^{2}}\hspace{0.1667em}+\hspace{0.1667em}{v^{2}}\big){(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda \hspace{0.1667em}+\hspace{0.1667em}1}^{\sigma }}\big[\omega {(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\rho }}\big]\bigg\}\hspace{2.5pt}\textit{a.e.}\end{aligned}\]
Proof.
It is known that if X is strongly convex process with the modulus $C(\cdot )$, then the process $Y(t,\cdot )=X(t,\cdot )-C(\cdot ){t^{2}}$ is convex [7, Lemma 2]. Appying the inequality (7) for the process $Y(t,.)$, we have
\[\begin{aligned}{}Y\bigg(\frac{u+v}{2},\cdot \bigg)& \le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}{\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]\\ {} & \hspace{1em}+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]Y(t,\cdot )dt\\ {} & \le \frac{Y(u,\cdot )+Y(v,\cdot )}{2}\hspace{130pt}\text{a.e.}\end{aligned}\]
That is
\[\begin{aligned}{}& X\bigg(\frac{u+v}{2},\cdot \bigg)-C(\cdot ){\bigg(\frac{u+v}{2}\bigg)^{2}}\\ {} & \hspace{1em}\le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\Big\{{\underset{u}{\overset{v}{\int }}}\big[{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]\\ {} & \hspace{2em}+{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]X(t,.)dt\\ {} & \hspace{2em}-C(\cdot ){\underset{u}{\overset{v}{\int }}}\big[{t^{2}}{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]+{t^{2}}{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]\big]dt\Big\}\\ {} & \hspace{1em}\le \frac{X(u,\cdot )-C(\cdot ){u^{2}}+X(v,\cdot )-C(\cdot ){v^{2}}}{2}\hspace{130pt}\text{a.e.}\end{aligned}\]
Calculating the integrals, we obtain
\[\begin{aligned}{}& {\underset{u}{\overset{v}{\int }}}{t^{2}}{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt\\ {} & \hspace{1em}={\underset{u}{\overset{v}{\int }}}{t^{2}}{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt+{\underset{u}{\overset{v}{\int }}}{t^{2}}{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt\\ {} & \hspace{2em}+{\underset{u}{\overset{v}{\int }}}{t^{2}}{(v-t)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(v-t)^{\rho }}\big]dt\\ {} & \hspace{1em}={(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]-2v{(v-u)^{\lambda +1}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{2em}+{v^{2}}{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\end{aligned}\]
and similarly,
\[\begin{aligned}{}& {\underset{u}{\overset{v}{\int }}}{t^{2}}{(t-u)^{\lambda -1}}{\mathcal{F}_{\rho ,\lambda }^{\sigma }}\big[\omega {(t-u)^{\rho }}\big]dt\\ {} & \hspace{1em}={(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]+2u{(v-u)^{\lambda +1}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{2em}+{u^{2}}{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big],\end{aligned}\]
where ${\sigma _{2}}(k)=\frac{\sigma (k)}{\rho k+\lambda +2}$, $k=0,1,2,\dots $. Then it follows that
\[\begin{aligned}{}& X\bigg(\frac{u+v}{2},\cdot \bigg)-C(\cdot ){\bigg(\frac{u+v}{2}\bigg)^{2}}\\ {} & \hspace{1em}\le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\big[{\mathcal{J}_{\rho ,\lambda ,a+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,b-;\omega }^{\sigma }}[X](t)\big]\\ {} & \hspace{2em}-C(\cdot )\big[2{(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{2em}-2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]+\big({u^{2}}+{v^{2}}\big){(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]\big]\\ {} & \hspace{1em}\le \frac{X(u,\cdot )+X(v,\cdot )}{2}-C(\cdot )\frac{{u^{2}}+{v^{2}}}{2}\hspace{130pt}\text{a.e.}\end{aligned}\]
Then
\[\begin{aligned}{}& X\bigg(\frac{u+v}{2},\cdot \bigg)\\ {} & \hspace{1em}-C(\cdot )\bigg\{2{(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]-2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{1em}+\big({u^{2}}+{v^{2}}\big){(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v-u)^{\rho }}\big]-{\bigg(\frac{u+v}{2}\bigg)^{2}}\bigg\}\\ {} & \hspace{2em}\le \frac{1}{2{(v-u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}[\omega {(v-u)^{\rho }}]}\big[{\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}[X](t)+{\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}[X](t)\big]\\ {} & \hspace{2em}\le \frac{X(u,\cdot )+X(v,\cdot )}{2}-C(\cdot )\bigg\{\frac{{u^{2}}+{v^{2}}}{2}+2{(v-u)^{\lambda +2}}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{2}}}}\big[\omega {(v-u)^{\rho }}\big]\\ {} & \hspace{1em}\hspace{2em}-\hspace{0.1667em}2{(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda }^{{\sigma _{1}}}}\big[\omega {(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\rho }}\big]\hspace{0.1667em}+\hspace{0.1667em}\big({u^{2}}\hspace{0.1667em}+\hspace{0.1667em}{v^{2}}\big){(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\lambda }}{\mathcal{F}_{\rho ,\lambda +1}^{\sigma }}\big[\omega {(v\hspace{0.1667em}-\hspace{0.1667em}u)^{\rho }}\big]\bigg\}\hspace{2.5pt}\hspace{2.5pt}\text{a.e.}\end{aligned}\]
This completes the proof. □