Fractional version of Ostrowski-type inequalities for strongly p-convex stochastic processes via a k-fractional Hilfer–Katugampola derivative

In the present research, we introduce the notion of convex stochastic processes namely; strongly p-convex stochastic processes. We establish a generalized version of Ostrowski-type integral inequalities for strongly p-convex stochastic processes in the setting of a generalized k-fractional Hilfer–Katugampola derivative by using the Hölder and power-mean inequalities. By using our main results, we derived some known results as special cases and many well-known existing results are also recaptured. It is assumed that this research will offer new guidelines in fractional calculus.


Introduction and preliminaries
The theory of inequalities has undergone rapid developments because of its widespread use in pure and applied mathematics. Recently, the role of fractional calculus made this area more interesting for researchers (see [1][2][3]). As classical convexity is being used in less applied problems, it is always appropriate to explore new versions of convexity. A consensus of the history of the Hermite-Hadamard integral inequality can be found in the literature [4]. In optimization and probability theory, the Hermite-Hadamard inequality has become a helpful tool [5].
In 1971, the research on convex stochastic processes began when Nagy [6] employed a characterization of measurable stochastic processes to resolve a generalization of the Cauchy functional equation. At the end of the twentieth century, Nikodem introduced the convex stochastic processes, and Skowronski derived several advanced results on convex stochastic processes that generalize further well-known properties, [7][8][9]. Further, Kotrys [10] presented the Hermite-Hadamard inequality for convex stochastic processes in 2012. Many studies have been done by several researchers on different classes of convex stochastic processes and also Hermite-Hadamard inequalities for convex stochastic processes in the literature [11][12][13]. The well-reputed Hermite-Hadamard inequality for convex stochastic processes is defined as follows: Let η : I × → R be Jensen-convex and mean-square continuous in I × , then for any c 1 , d 1 ∈ I, c 1 < d 1 .
In 2014, the authors of [14] considered Hermite-Hadamard integral-type inequalities for stochastic processes. Katugampola, considered a fractional integral operator that generalizes the Hadamard and Riemann-Liouville integrals into a single form and many authors use these results in the area of convexity, generalized convexity, and so on (see [15,16]). Recently, different Hermite-Hadamard-type inequalities via fractional integrals have been presented [17][18][19].
The purpose of this article is to develop some integral inequalities of Ostrowski-type for a strongly p-convex stochastic process using the generalized k-fractional Hilfer-Katugampola derivative. Definition 1.1 ([19, 30]) A stochastic process is a family of random variables η(κ) parameterized by κ ∈ I, where I ⊂ R. When I = {1, 2, . . .}, then η(κ) is known as a stochastic process in discrete time. When I = [0, ∞), then η(κ) is a stochastic process in continuous time.
For any ϑ ∈ the function I κ − → η(κ, ϑ) is termed a path of η(κ). 19,30]) A family G k of α-fields on parameterized by k ∈ I, where I ⊂ R, is said to be a filtration if for any ρ, κ ∈ I such that ρ ≤ κ.
where P -lim represents the limit in probability.
4) It is said to be monotonic if it is increasing or decreasing. 5) If there exists a random variable η (κ, ·) : I × → R then it is differentiable at a point κ ∈ I, such that A stochastic process X : I × → R is known as continuous (differentiable) if it is continuous (differentiable) at every point of I.

Definition 1.5 ([19, 31]) Let ( , A, P) be a probability space and
A mean-square integral operator is increasing, thus, Definition 1.6 ( [7,19]) Let ( , A, P) be a probability space and I ⊆ R. A stochastic process η : I × → R is said to be a convex stochastic process, if holds for all b 1 , b 2 ∈ I and κ ∈ [0, 1].
It is natural to view the new versions of convexity in stochastic processes settings, hence we introduce the strongly p-convex stochastic processes as follows: Definition 1.7 Let c : → R be a positive random variable. A stochastic process η : I × → R is said to be strongly p-convex with modulus c(·), if holds for all b 1 , b 2 ∈ I and κ ∈ [0, 1]. Remark 1.8 1) Taking p = 1 in the above definition, we obtain a strongly convex stochastic process [11].

Definition 1.10 ([34, 35])
The k-Gamma function is defined as; where, y, κ > 0. We can observe that and (1.6) Definition 1.12 ([34, 37]) The left-hand and right-hand generalized k-fractional derivatives having order w are defined as; Then, the left-hand and right-hand generalized k-fractional Hilfer-Katugampola derivatives are defined as; where I is the integral presented in definition (1.6).
Thus, the generalized k-fractional Hilfer-Katugampola derivative can be presented as: (1.13) Definition 1.15 ([34]) The beta function denoted by B is defined as; 14) and the Gaussian hypergeometric function denoted by F 1 is defined as;

Ostrowski-type inequalities for a strongly p-convex stochastic process
The Ostrowski-type inequality for a strongly p-convex stochastic process in the setting of a generalized k-fractional Hilfer-Katugampola derivative is established in this section.
, then the following inequality holds almost everywhere: (2.1) Proof Integrating by parts, we can write For both sides of (2.2) and (2.3), multiplying by From (2.4) and (2.5), we obtain the inequality (2.1).

Conclusion
In the present note, we introduced the notion of a strongly p-convex stochastic process. We established Ostrowski-type inequalities for a strongly p-convex stochastic process. Also, we established some integral inequalities of Ostrowski-type via the generalized kfractional Hilfer-Katugampola derivative.