Decomposition of higher-order Wright convex functions revisited

TL;DR

This paper provides a new, elementary proof for the decomposition of higher-order Wright convex functions into a sum of convex functions and polynomials, simplifying previous transfinite-based methods.

Contribution

It introduces an elementary proof for the decomposition theorem of higher-order Wright convex functions, improving accessibility and understanding.

Findings

New elementary proof of the decomposition theorem

Simplifies previous transfinite methods

Confirms the decomposition of Wright convex functions

Abstract

In 2009, Maksa and P\'ales established an extension of the decomposition theorem of Ng in the context of higher-order convexity notions. They proved that a real function is Wright convex of order $n$ if and only if it can be decomposed as the sum of a convex function of order $n$ and a polynomial function of order at most $n$. Their proof was based on transfinite tools in the background. The main purpose of this paper is to adopt the methods of a paper of P\'ales published in 2020 and establish a new and elementary proof for the theorem of Maksa and P\'ales.