Hermite-Hadamard, Jensen, and Fractional Integral Inequalities for Generalized P -Convex Stochastic Processes

The stochastic process is one of the important branches of probability theory which deals with probabilistic models that evolve over time. It starts with probability postulates and includes a captivating arrangement of conclusions from those postulates. In probability theory, a convex function applied on the expected value of a random variable is always bounded above by the expected value of the convex function of that random variable. The purpose of this note is to introduce the class of generalized p -convex stochastic processes. Some well-known results of generalized p -convex functions such as Hermite-Hadamard, Jensen, and fractional integral inequalities are extended for generalized p -stochastic convexity.

GENERALIZED STOCHASTIC CONVEXITY AND STOCHASTIC ORDERINGS OF MIXTURES

In this paper, a new concept called generalized stochastic convexity is introduced as an extension of the classic notion of stochastic convexity. It relies on the well-known concept of generalized convex functions and corresponds to a stochastic convexity with respect to some Tchebycheff system of functions. A special case discussed in detail is the notion of stochastic s-convexity (s ∈ [real number symbol]), which is obtained when this system is the family of power functions {x0, x1,..., xs−1}. The analysis is made, first for totally positive families of distributions and then for families that do not enjoy that property. Further, integral stochastic orderings, said of Tchebycheff-type, are introduced that are induced by cones of generalized convex functions. For s-convex functions, they reduce to the s-convex stochastic orderings studied recently. These orderings are then used for comparing mixtures and compound sums, with some illustrations in epidemic theory and actuarial sciences.