Hermite-Hadamard type inequalities for p-convex stochastic processes

Article history: Received: 10 April 2018 Accepted: 26 January 2019 Available Online: 19 March 2019 In this study are investigated pconvex stochastic processes which are extensions of convex stochastic processes. A suitable example is also given for this process . In addition, in this case a pconvex stochastic process is increasing or decreasi ng, the relation with convexity is revealed. The concep t of inequality as convexi ty has an important place in literature, since it provides a broader setting to study the optimization and mathematical programming probl ems. Therefore, HermiteHadamard type inequalities for pconvex stochastic processes and some boundaries for these inequalities are obtained in p resent study. It is used the concept of mean-square integrability for stochastic processes to obtain the above mentioned results.


Introduction and preliminaries
The convexity for stochastic processes is of great importance in optimization, especially in optimal designs, and also useful for numerical approximations when there exist probabilistic quantities in the literature. In 1980 Nikodem defined convex stochastic processes and gave some properties which are also known for classical convex functions. Some types of convex stochastic processes were introduced by Skowronski in 1992. In 2012 Kotrys obtained the classical Hermite-Hadamard inequality to convex stochastic processes. There are many studies in recent years on the above mentioned processes. A lot of definitions of various convexity and some new inequalities were for these stochastic processes in the literature [7][8][9][10][11][12][13]. The author's findings led to our motivation to build our work.The main goal is to introduce p-convex stochastic processes. Moreover, we prove Hermite-Hadamard type inequalities for p-convex stochastic processes and obtain some important results for these processes. Let us show the definition of a stochhastic process: Definition 1 ( [5]). The process ሼܺሺ‫ݐ‬ሻ: ‫ݐ‬ ∈ ‫ܫ‬ሽ is a parameterized collection of random variables defined on a common probability spaceሺߗ, Ա, ܲሻ. Its parameter t is considered to be time. Then ܺሺ‫ݐ‬ሻ, which can also be shown as ܺሺ‫,ݐ‬ ߱ሻ for ߱ ∈ ߗ, is considered to be state or position of the process at time t. For any fixed outcome ω of sample spaceߗ, the deterministic mapping ‫ݐ‬ → ܺሺ‫,ݐ‬ ߱ሻ denotes a realization, trajectory or sample path of the process.
For any particular ‫ݐ‬ ∈ ‫ܫ‬the mapping depends ω alone, i.e., then we obtain a random variable. It can be said that, ܺሺ‫,ݐ‬ ߱ሻ changes in time in a random manner. We restrict our attention to continuous time stochastic processes, i.e., index set is ‫:ܫ‬ ሾ0, ∞ሻ. The concept "mean-square convergence" is used as the statement "almost everywhere" in this paper.

Main results
The main subject of this paper is to adapt some wellknown related results p-convex functions on p-convex stochastic processes. Also, we purpose to obtain Hermite-Hadamard type inequalities for p-convex stochastic processes. Definition 7. Let ‫ܫ‬ be a p-convex set. The process ܺ: ‫ܫ‬ ൈ ߗ → Թ is called a p-convex stochastic process, if the following inequality holds almost everywhere: Thus, we can also define p-convex stochastic processes as follows: Definition 8. The process ܺ: ‫ܫ‬ ൈ ߗ → Թ is called a pconvex stochastic process, if the following inequality holds almost everywhere: for all ‫,ݑ‬ ‫ݒ‬ ∈ ‫ܫ‬ ⊂ ሺ0, ∞ሻ,ߣ ∈ ሾ0,1ሿ,‫‬ ∈ Թ\ሼ0ሽ. If the inequality in Eq. (1) is reversed, then the process ܺ is called p-concave.

Multiplying by
Proof. Using the power mean integral inequality and Lemma 2, then we have Proof. If ‫‬ ൌ െ1 in Theorem 4, then the proof of Corollary 5 is completed.   (6) and (7) in Eq. (5), then the proof of Theorem 7 is completed.

Conclusion
In this paper, we considered an important extension of convexity for stochastic processes which is called pconvex stochastic processes and obtained new Hermite-Hadamard inequalities for these processes. In the future, new inequalities for the other convex stochastic processes can be obtained using similar methods in this study.