New integral representations of n-th order convex functions

TL;DR

This paper introduces new integral representations for n-convex functions, decomposes them into monotone components, and explores their properties including relative and strong n-convexity, providing new insights and characterizations.

Contribution

It provides the first general integral representation of n-convex functions without additional assumptions and characterizes various forms of n-convexity and their relationships.

Findings

Any n-convex function can be expressed as a sum of two (n+1)-times monotone functions and a polynomial.

Decomposition of n-Wright-convex functions generalizes previous results.

Characterizations of relative and strong n-convexity in terms of derivatives and measures.

Abstract

In this paper we give an integral representation of an $n$-convex function $f$ in general case without additional assumptions on function $f$. We prove that any $n$-convex function can be represented as a sum of two $(n+1)$-times monotone functions and a polynomial of degree at most $n$. We obtain a decomposition of $n$-Wright-convex functions which generalizes and complements results of Maksa and Pales (2009). We define and study relative $n$-convexity of $n$-convex functions. We introduce a measure of $n$-convexity of $f$. We give a characterization of relative $n$-convexity in terms of this measure, as well as in terms of $n$th order distributional derivatives and Radon-Nikodym derivatives. We define, study and give a characterization of strong $n$-convexity of an $n$-convex function $f$ in terms of its derivative $f^{(n+1)}(x)$ (which exists a.e.) without additional assumptions on differentiability of $f$. We prove that for any two $n$-convex functions $f$ and $g$, such that $f$ is $n$-convex with respect to $g$, the function $g$ is the support for the function $f$ in the sense introduced by Wasowicz (2007), up to polynomial of degree at most $n$.