Hermite-Hadamard type inequalities for p-convex stochastic processes

Abstract

In this study are investigated p-convex stochastic processes which are extensions of convex stochastic processes. A suitable example is also given for this process. In addition, in this case a p-convex stochastic process is increasing or decreasing, the relation with convexity is revealed. The concept of inequality as convexity has an important place in literature, since it provides a broader setting to study the optimization and mathematical programming problems. Therefore, Hermite-Hadamard type inequalities for p-convex stochastic processes and some boundaries for these inequalities are obtained in present study. It is used the concept of mean-square integrability for stochastic processes to obtain the above mentioned results.

Hermite-Hadamard Type Inequalities for Exponentially p-Convex Stochastic Processes

“…If | | is exponentially -convex stochastic process on [ , ] for , > 1, + = 1, then Remark 3.14. If we take = 0 in Theorem 3.13, we attain Theorem 7 in [9].…”

“…Lemma 2.5. [9] Let : ⊂ (0, ∞) × → ℝ be a mean-square differentiable stochastic process on ∘ (the interior of ) and , ∈ , < and ∈ ℝ\{0}. If is mean-square integrable on [ , ], then the following equality holds almost everywhere: Now we recall the following special functions (see [1]).…”

“…In recent years, there have been many studies on the above mentioned processes. For recent generalizations and improvements on convex stochastic processes, please refer to [2]- [6], [9]- [12], [15], [16].…”

A Stochastic Fractional Calculus with Applications to Variational Principles

“…A stochastic calculus of variations, which generalizes the ordinary calculus of variations to stochastic processes, was introduced in 1981 by Yasue, generalizing the Euler-Lagrange equation and giving interesting applications to quantum mechanics [1]. Recently, stochastic variational differential equations have been analyzed for modeling infectious diseases [2,3], and stochastic processes have shown to be increasingly important in optimization [4].…”

Hermite-Hadamard Type Inequalities for m-Convex and (α, m)-Convex Stochastic Processes

“…In recent years, there have been many studies on the above mentioned processes. For recent generalizations and improvements on convex stochastic processes, please refer to [4]- [8], [11]- [15], [19].…”