Convexity with respect to families of means

TL;DR

This paper studies the continuity of functions satisfying a generalized Jensen convexity inequality related to means, revealing the existence of discontinuous solutions for rational parameters and conditions for continuity over sets of positive measure.

Contribution

It characterizes the continuity properties of functions satisfying $(p,q)$-Jensen convexity, including the existence of discontinuous solutions for rational $p$, and conditions ensuring continuity.

Findings

Discontinuous multiplicative functions are $(p,p)$-Jensen convex for all positive rational $p$.

If a function is $(p,p)$-Jensen convex for all $p$ in a set of positive measure, then it must be continuous.

The study links convexity properties with measure-theoretic conditions for continuity.

Abstract

In this paper we investigate continuity properties of functions $f:\mathbb{R}_+\to\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity inequality $$ f\big(H_p(x,y)\big)\leq H_q(f(x),f(y)) \qquad(x,y>0), $$ where $H_p$ stands for the $p$th power (or H\"older) mean. One of the main results shows that there exist discontinuous multiplicative functions that are $(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of this result states that if $f$ is $(p,p)$-Jensen convex for all $p\in P\subseteq\mathbb{R}_+$, where $P$ is a set of positive Lebesgue measure, then $f$ must be continuous.