Generalized Fractional Inequalities of the Hermite–Hadamard Type for Convex Stochastic Processes

A generalization of the Hermite–Hadamard (HH) inequality for a positive convex stochastic process, by means of a newly proposed fractional integral operator, is hereby established. Results involving the Riemann– Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl and Liouville fractional integrals are deduced as particular cases of our main result. In addition, we also apply some known HH results to obtain some estimates for the expectations of integrals of convex and p-convex stochastic processes. As a side note, we also pointed out a mistake in the main result of the paper [Hermite–Hadamard type inequalities, convex stochastic processes and Katugampola fractional integral, Revista Integración, temas de matemáticas 36 (2018), no. 2, 133–149]. We anticipate that the idea employed herein will inspire further research in this direction.