Hermite-Hadamard Type Inequalities for Harmonically Convex Stochastic Processes

Abstract

In this study are investigated harmonically convex stochastic processes which are an extensions of convex stochastic processes. Suitable examples are also given for these types of processes. In addition, in this case a harmonic convex stochastic process is increasing or decreasing, the relation with convexity is revealed. The concepts of convexity and inequality have an important place in literature, since it provides a broader setting to study the optimization and mathematical programming problems. The obtained results in this study are necessary to compare the maximum and minimum values of of a stochastic process with the expected value of its which has a probability density function, is particularly harmonic convex. Therefore, Hermite-Hadamard type inequalities for harmonically convex stochastic processes and some boundaries for these inequalities are obtained. There are used as methods concepts of mean-square continuity, mean-square differentiability, mean-square integrability and monotonicity for stochastic processes in this study.

ON (m, h1, h2)-G-CONVEX DOMINATED STOCHASTIC PROCESSES

“…Similarly, some recent studies have been introduced the following concepts: Jconvex [26], Wright-convex [27], strongly convex [9], strongly Wright [10], p-convex [20], harmonically convex [19], s-convex in the first and second sense [12,23] stochastic process.…”

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“…Other authors have obtained some further results in this area. For example, D. Kotrys investigated convex and strongly convex stochastic processes [25][26][27], E. Set E. et al handled s−convex stochatic processes in the second sense [28], S. Maden et al worked on s−convex stochastic processes in the first sense [18], N. Okur et al investigated harmonically convex stochastic processes [29] and M. Tomar et al worked on log-convex stochastic processes [30]. Furthermore, the works of Vivas-Cortez, Hernández Hernández and Gómez [31][32][33][34][35] addressed the (m, h 1 , h 2 )−convex stochastic processes in the setting of fractional calculus.…”

Hermite-Hadamard type inequalities for p-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.…”