Hermite–Hadamard type inequalities for r-convex positive stochastic processes

ABSTRACT In the present work, we aim to find Hermite–Hadamard type inequality for r-convex positive stochastic processes. The case of the product of an r-convex and s-convex stochastic process is also investigated.

Motivated from the above works, we study the Hermite-Hadamard type inequalities for r-convex stochastic processes. We also consider the case of a product of an r-convex and s-convex stochastic process.

Preliminaries
Let ( , F, P) be an arbitrary probability space. A Fmeasurable function X : −→ R is said to be a random variable. Let I ⊂ R be an interval. Then a function X : I × −→ R is said to be the stochastic process if the function X(t, .) is a random variable for all t in I.
Let P − lim and E[X] denote the limit in probability and the expected value of X, respectively. A stochastic process X : I × −→ R is said to be continuous in It is worthy to note that if X : I × −→ R is meansquare continuous, then it is continuous in I but the converse does not hold.
The mean-square integral is defined as: A random variable Y : −→ R is said to be the mean-square integral of the stochastic process X : 2 ] < ∞ ∀t ∈ I, if for every normal sequence of partitions of [a, b], the following relation holds The assumption of the mean-square continuity of the stochastic process is enough for the mean-square integral to exist.
From the definition of mean-square integral, we immediately have the following relation.
That is a mean-square integral is monotonic. Throughout the entire paper, the monotonicity of mean-square integral and positivity of the stochastic process will be frequently used. Now, we define the following.

Definition 2.1: A stochastic process
Note that 0-convex stochastic processes are logarithmic convex (see [9]) and 1-convex stochastic processes are the classical convex stochastic process. Note that if X is r-convex, then X r is convex stochastic process (r > 0).
The above definition is analogue of the r-convex functions in the classical convex functions, see [10,11].
Kotrys [3] studied the following well-known Hermite -Hadamard type inequality where X : I × −→ R is Jensen convex and meansquare continuous stochastic process.
Hermite-Hadamard type inequalities for log-convex functions was investigated by Dragomir and Mond [12].
Pachpatte [13,14] also gave some other refinements of these inequalities related with differentiable logconvex functions.
Tomar et al. [9] proved the following inequalities: Let X : I × −→ (0, ∞) be a log-convex stochastic process. Then for u, v ∈ I with u < v, the following inequalities hold: where L(p, q) (p = q) is the logarithmic mean of real numbers p,q > 0.
We also need the Jensen inequality for convex stochastic process proved by Sarikaya et al. [7].

Main results
Theorem 3.1: Let X : I × −→ [0, ∞) be a r-convex stochastic process with mean-square continuity in I. Then for u, v ∈ I with u < v, the below inequality holds Proof: From Jensen inequality, we obtain Since X r is convex, then Hemite-Hadamard type inequality for convex stochastic processes yields us (see [3]) Hence, This completes the proof.
which completes the proof.
Note that for r = 1, in the above theorem, we have the same inequality again as in Corollary 3.1.

stochastic process with mean-square continuity in I. Then X is s-convex stochastic process.
Proof: To prove this, we need the following inequality for non-negative real numbers x, y λ)x s + λy s ) 1/s , (see [11]) (4) where 0 ≤ λ ≤ 1, 0 ≤ r ≤ s. Since X is r-convex stochastic process, by inequality (4) for all u, v ∈ I, λ ∈ [0, 1], we obtain Hence, X is a s-convex stochastic process.
As a special case of the above theorem, we deduce the following results.

Proof:
The proof follows at once by using Theorem 3.3 and proceeding on similar lines as Theorem 3.1.
Proof: Since X is r-convex stochastic process and Y is s-convex stochastic convex, for all λ ∈ [0, 1], we have Therefore, Now applying Cauchy's inequality, we obtain .). .)), then we obtain the following inequality , (a.e.), which leads us to the required result.
Note: By putting X = Y, r = s = 2, in the above theorem, we have the following inequality The following result can be easily proved by proceeding on similar lines as in the above theorem.