On Symmetrized Stochastic Harmonically Convexity and Hermite–Hadamard Type Inequalities

: Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given as well.


Introduction
Nikodem [1] proposed the concept of convex stochastic processes in 1980. Skowroński [2] then extended the well-known characteristics of convex functions to convex stochastic processes. Kotrys [3] proved the Hermite-Hadamard inequality using convex stochastic processes in his study. A number of scholars have examined numerous integral inequalities in recent papers on convex stochastic processes. In [4], Agahi and Babakhani studied fractional inequalities related to the Hermite-Hadamard and Jensen types for convex stochastic processes. Kotrys [3] obtained the Hermite-Hadamard inequality for convex stochastic processes. In another study, Kotrys [5] discussed properties of strongly convex stochastic processes. Li and Hao [6] acquired the Hermite-Hadmard inequality for h-convex stochastic processes. Dragomir [7] defined symmetrized convex functions and highlighted their several features. Haq and Kotrys [8] introduced the concept of symmetrized convex stochastic processes and analyzed the Hermite-Hadmard-type inequalities in the perspective of the preceding papers. Additionally, Haq and Kotrys [8] addressed various ways of characterizing symmetrized convex stochastic processes. Okur et al. [9] extended a well-known work on harmonically convex functions to harmonically convex stochastic processes. In addition, the authors intended to find Hermite-Hadmard-type inequalities for harmonically convex stochastic processes. Following the prior studies on this topic, we introduce the concept of symmetrized harmonically convex stochastic processes and investigate the Hermite-Hadmard-type inequalities for symmetrized harmonically convex functions as well as their applications. In this study, we also describe a number of characterizations of harmonic symmetrized convex stochastic processes.

Preliminaries Section
Let (Λ, F , P) be an arbitrary probability space. A function H : Λ → R is a random variable if it is F -measurable. Let I ⊂ R be an interval. A function H : I × Λ → R is a stochastic process if the function H(ν, ·) is a random variable for all ν ∈ I. Definition 1 ([8]). A stochastic process H : I × Λ → R is said to be continuous in probability in I, if for all ν 0 ∈ I P − lim where P − lim denotes the limit in probability.

Definition 2 ([8]).
A stochastic process H : I × Λ → R is said to be mean-square continuous in I, if for every ν 0 ∈ I lim where E[H(ν)] denotes the expectated value of the random variable H(ν, ·).
To be clear, it is important to note that mean-square continuity of H : I × Λ → R implies probability continuity but the converse does not hold true.
To refresh our memory, let us have a look at the mean-square integral.

Definition 3 ([8]
). For any normal sequence of partitions [α 1 , α 2 ] ⊂ I, a random variable Y : Λ → R is called the mean-square integral of the stochastic process H : In this case, we can write

Remark 1.
A stochastic process must have mean-square continuity in order for the mean-square integral to exist. The following inference follows directly from the concept of a mean-square integral.
The monotonicity of mean-square integrals and the positivity of stochastic processes will be used extensively throughout this paper.
The proof of the following Lemma exists in [8].
We also need the following lemma to prove our results.
We recall the definition of a convex stochastic process and ν 1 , ν 2 ∈ I the following inequality holds If the above inequality (3) holds for every ν 1 , ν 2 ∈ I and σ = 1 2 , then H is known as Jensen-convex or 1 2 -convex. A stochastic process H is said to be concave if (−H) is convex.
Haq and Kotrys [8] defined the symmetrical form of a stochastic process as follows: The notion of a symmetrized convex stochastic process is given in the definition below: It is observed that every convex stochastic process is symmetrized convex, but there exists a stochastic process H which is not convex on [α 1 , α 2 ], whereas its symmetrical form is convex (see for instance [7]).
The well-known Hermite-Hadmard integral inequality for convex stochastic processes was proved by Kotrys in [3]: ). If H : I × Λ → R is Jensen-convex and mean square continuous in the interval I × Λ, then for any α 1 , α 2 ∈ I with α 1 < α 2 we have Haq and Kotrys [8] investigated with a counterpart of the Hermite-Hadmard inequality for symmetrized convex stochastic processes.

Theorem 2 ([8]
). If H : I × Λ → R is be a symmetrized convex and mean-square continuous stochastic process, then the inequality holds

Hermite-Hadmard-Type Inequalities and Symmetrized Harmonic Convex Stochastic Process
It is our primary goal to discuss both the notion of the harmonic symmetrized form of stochastic processes and the inequalities of the Hermite-Hadmard-type that we will obtain as an application of the harmonic symmetrized stochastic processes. We also discuss the separation theorem for harmonically convex stochastic processes and the Hyers-Ulam stability of these stochastic processes as a result of the separation theorem. We construct Hyers-Ulam stability conditions for symmetrized harmonically convex stochastic processes by making use of this separation theorem.
Okur et al. [9] extended some results concerning harmonically convex functions to harmonically convex stochastic processes and obtained Hermite-Hadmard-type inequalities for harmonically convex stochastic processes.
for all ν 1 , ν 2 ∈ I and σ ∈ [0, 1]. If the inequality above is reversed, then H is said to be harmonically concave.
The following result of the Hermite-Hadmard-type inequalities holds.
The discussion presented above leads us to introduce the following definition of the symmetrized harmonically convex stochastic process. We conclude, that every harmonically convex stochastic process is symmetrized harmonically convex. The following example illustrates that there are stochastic processes which are not harmonically convex, but their symmetrical form is harmonically convex. Example 1. Let H : [1,2] × Λ → R be defined as H(u, ·) = u 3 , then H(u, ·) is not harmonically convex on [1,2], whereas its symmetrical form H : [α 1 , α 2 ]\{0} × Λ → R given by is harmonically convex as evident from the Figures 1 and 2.
By Lemma 2 and the basic properties of the mean-square integral, we obtain Thus, we obtain The proof is thus accomplished.
Similarly, as in the real function case, we can prove the following result.
We also observe that holds; thus, it is proved the left-hand side of the inequality (10) is valid. We can write any ν ∈ [α 1 , α 2 ]\{0} as follows By using the harmonic convexity of H The theorem is thus accomplished.

Hyers-Ulam Type Stability and Harmonic Convexity
González et al. [10] demonstrated a separation theorem for convex stochastic processes and subsequently analyzed their Hyers-Ulam stability. Using this separation concept, the Hyers-Ulam stability criterion for symmetrized harmonically convex processes can be derived. See [10] for the usual result of the Hyers-Ulam stability. Let us recall the following definition of a ε-convex stochastic process.
Haq and Kotrys [8] introduced the definition of ε-symmetrized convex stochastic processes and showed a Hyers-Ulam-type stability result for ε-symmetrized convex stochastic processes.
As an immediate consequence of the above theorem, we obtain the following Hyers-Ulam-type stability results for harmonically convex stochastic processes. For the classical Hyers-Ulam theorem, see [12].
Firstly, we introduce an ε-harmonically convex stochastic process and ε-symmetrized harmonically convex stochastic processes and establish a Hyers-Ulam-type stability result for ε-symmetrized harmonic stochastic processes.
Let us prove a Hyers-Ulam-type stability result for ε-symmetrized stochastic processes.