Implications between generalized convexity properties of real functions

TL;DR

This paper explores the relationships between various generalized convexity properties of real functions, establishing implications among different $M$-convexity notions using means and their descendants, with broad applicability to quasi-arithmetic and Matkowski means.

Contribution

It introduces the concept of descendants of means and proves that $M$-convexity for a set of means implies $N$-convexity for their descendants, extending known implications among convexity properties.

Findings

Implication from $M$-convexity to $N$-convexity under mean descendants.

Introduction of the descendant mean concept for generalized convexity.

Applicability to weighted arithmetic, quasi-arithmetic, and Matkowski means.

Abstract

Motivated by the well-known implications among $t$-convexity properties of real functions, analogous relations among the upper and lower $M$-convexity properties of real functions are established. More precisely, having an $n$-tuple $(M_1,\dots,M_n)$ of continuous two-variable means, the notion of the descendant of these means (which is also an $n$-tuple $(N_1,\dots,N_n)$ of two-variable means) is introduced. In particular, when all the means $M_i$ are weighted arithmetic, then the components of their descendants are also weighted arithmetic means. More general statements are obtained in terms of the generalized quasi-arithmetic or Matkowski means. The main results then state that if a function $f$ is $M_i$-convex for all $i\in\{1,\dots,n\}$, then it is also $N_i$-convex for all $i\in\{1,\dots,n\}$. Several consequences are discussed.