On generalized stochastic fractional integrals and related inequalities

TL;DR

This paper introduces generalized mean-square fractional integrals for stochastic processes and establishes a fractional Hermite-Hadamard inequality for Jensen-convex and strongly convex stochastic processes.

Contribution

It defines new generalized stochastic fractional integrals and proves a novel fractional Hermite-Hadamard inequality for convex stochastic processes.

Findings

Introduction of generalized mean-square fractional integrals

Establishment of fractional Hermite-Hadamard inequality for convex processes

Application to Jensen-convex and strongly convex stochastic processes

Abstract

The generalized mean-square fractional integrals $\mathcal{J}_{\rho,\lambda,u+;\omega}^{\sigma}$ and $\mathcal{J}_{\rho,\lambda,v-;\omega}^{\sigma}$ of the stochastic process $X$ are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite--Hadamard inequality is establish via generalized stochastic fractional integrals.