Characterization of higher-order monotonicity via integral inequalities

Abstract

The Hermite-Hadamard inequality not only is a consequence of convexity but also characterizes it: if a continuous function satisfies either its left-hand side or its right-hand side on each compact subinterval of the domain, then it is necessarily convex. The aim of this paper is to prove analogous statements for the higher-order extensions of the Hermite-Hadamard inequality. The main tools of the proofs are smoothing by convolution and the support properties of higher-order monotone functions.

Support-type properties of generalized convex functions

“…For (u 0 , u 1 )-convex functions see [3,9]. For higher-order convexity see [10]. Till now such a characterization of U -convexity for non-polynomial T-systems consisting of more than two functions is not known.…”

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

“…In [32,41,42], the authors used the Levin-Stečkin theorem [25] to study Hermite-Hadamard type inequalities. Many results on higher order generalizations of the Hermite-Hadamard type inequality one can found, among others, in [1,2,3,4,5,14,36,37]. In recent papers [36,37] the theorem of M. Denuit, C.Lefèvre and M. Shaked [13] was used to prove Hermite-Hadamard type inequalities for higher-order convex functions.…”

An application of the Choquet theorem to the study of randomly-superinvariant measures

“…Let us mention that the theory of integral equations and inequalities has many useful applications in describing numerous events and problems in the real word. Various types of integral operators were investigated in several papers (see [1][2][3][4]). We first recall some definitions and give some preliminary results.…”