On Some Recent Applications of Stochastic Convex Ordering Theorems to Some Functional Inequalities for Convex Functions: A Survey

Abstract

This is a survey paper concerning some theorems on stochastic convex ordering and their applications to functional inequalities for convex functions. We present the recent results on those subjects.

Ohlin-Type Theorem for Convex Set-Valued Maps

“…It has been rediscovered by Rajba [14], who found its various applications to the theory of functional inequalities. In [13,15,18], the Ohlin lemma is used, among others, to get a simple proof of the known Hermite-Hadamard inequalities, as well as to obtaining new Hermite-Hadamard type inequalities.…”

“…The next two corollaries are counterparts for convex set-valued maps of the following inequalities concerning convex functions f : I → R (cf. [5,15]…”

The Family of Alpha,[a,b] Stochastic Orders: Risk vs. Expected Value

“…Convex functions are fundamental in proving many well-known inequalities. The convex stochastic order is a powerful tool for proving inequalities that involve convex functions (see Rajba (2017) for a survey). In this section we prove Hermite-Hadamard type inequalities for convex functions that belong to the set D 2,[a,b] using the 2, [a, b]-convex stochastic order.…”