On generalized stochastic fractional integrals and related inequalities

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 $X$ are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite--Hadamard inequality is establish via generalized stochastic fractional integrals.


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 (Ω, A, P ) be an arbitrary probability space. A function X : Ω → R is called a random variable if it is A-measurable. A function X : I × Ω → R, where I ⊂ R is an interval, is called a stochastic process if for every t ∈ I the function X(t, .) is a random variable.
Recall that the stochastic process X : I × Ω → R is called (i) continuous in probability in interval I, if for all t 0 ∈ I we have Plim t→t0 X(t, .) = X(t 0 , .), where Plim denotes the limit in probability.
(ii) mean-square continuous in the interval I, if for all t 0 ∈ 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. Now we would like to recall the concept of the mean-square integral. For the definition and basic properties see [18].
Let X : for all normal sequences of partitions of the interval [a, b] and for all Θ k ∈ [t k−1 , t k ], k = 1, . . . , 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, ·) ≤ Y (t, ·) (a.e.) in some interval [a, b], then b a X(t, ·)dt ≤ b a Y (t, ·)dt (a.e.).
Of course, this inequality is an immediate consequence of the definition of the meansquare integral. Definition 2. We say that a stochastic processes X : I × Ω → R is convex, if for all λ ∈ [0, 1] and u, v ∈ I the inequality is satisfied. If the above inequality is assumed only for λ = 1 2 , then the process X is Jensen-convex or 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]. Now, we present some results proved by Kotrys [6] about Hermite-Hadamard inequality for convex stochastic processes.
Proposition 1. Let X : I × Ω → R be a convex stochastic process and t 0 ∈ intI. Then there exists a random variable A : That is for all t ∈ I. Theorem 1. Let X : I × Ω → R be a Jensen-convex, mean-square continuous in the interval I stochastic process. Then for any u, v ∈ I we have In [7], Kotrys introduced the concept of strongly convex stochastic processes and investigated their properties.
Definition 3. Let C : Ω → R denote a positive random variable. The stochastic process X : I × Ω → R is called strongly convex with modulus C(·) > 0, if for all λ ∈ [0, 1] and u, v ∈ I the inequality is satisfied. If the above inequality is assumed only for λ = 1 2 , then the process X is strongly Jensen-convex with modulus C(·).
In [5], Hafiz gave the following definition of stochastic mean-square fractional integrals.
Definition 4. For the stochastic proces X : I × Ω → R, the concept of stochastic mean-square fractional integrals I α u+ and I α v+ of X of order α > 0 is defined by Using this concept of stochastic mean-square fractional integrals I α a+ and I α b+ , Agahi and Babakhani proved the following Hermite-Hadamard type inequality for convex stochastic processes: Theorem 2. Let X : I × Ω → R be a Jensen-convex stochastic process that is meansquare continuous in the interval I. Then for any u, v ∈ I, the following Hermite-Hadamard inequality For more information and recent developments on Hermite-Hadamard type inequalities for stochastic process, please refer to [2-4, 9-11, 14, 16, 15, 20, 19].

Main results
In tis section, we introduce the concept of the generalized mean-square fractional integrals J σ ρ,λ,u+;ω and J σ ρ,λ,v−;ω of the stochastic process X. In [13], Raina studied a class of functions defined formally by where the cofficents σ(k) (k ∈ N 0 = N∪{0}) make a bounded sequence of positive real numbers and 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.
Many useful generalized mean-square fractional integrals can be obtained by specializing the coefficients σ(k). Here, we just point out that the stochastic mean-square fractional integrals I α a+ and I α b+ can be established by coosing λ = α, σ(0) = 1 and w = 0. Now we present Hermite-Hadamard inequality for generalized mean-square fractional integrals J σ ρ,λ,a+;ω and J σ ρ,λ,b−;ω of X. Theorem 3. Let X : I × Ω → R be a Jensen-convex stochastic process that is meansquare continuous in the interval I. For every u, v ∈ I, u < v, we have the following Hermite-Hadamard inequality a.e.
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 ∈ intI. Let us take a support at t 0 = u+v 2 , then we have Multiplying both sides of (8) by , then integrating the resulting inequality with respect to t over [u, v], we obtain Calculating the integrals, we have where σ 1 (k) = σ(k) ρk+λ+1 , k = 0, 1, 2, . . .. Using the identities (10) and (11) in (9), we obtain That is, which completes the proof of the first inequality in (7).