On the balancing property of Matkowski means

TL;DR

This paper investigates the balancing property of two-variable means, specifically Matkowski means, and characterizes solutions without assuming differentiability, extending previous results on quasi-arithmetic means.

Contribution

It provides a solution to the balancing property equation for Matkowski means without differentiability assumptions, broadening the understanding of mean characterizations.

Findings

Characterization of solutions to the balancing property for Matkowski means.

Extension of previous results to non-differentiable means.

Identification of conditions under which Matkowski means satisfy the balancing property.

Abstract

Let $I\subseteq\mathbb{R}$ be a nonempty open subinterval. We say that a two-variable mean $M:I\times I\to\mathbb{R}$ enjoys the \emph{balancing property} if, for all $x,y\in I$, the equality \begin{equation}\tag{1} M\big(M(x,M(x,y)),M(M(x,y),y)\big)=M(x,y) \end{equation} holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that $M$ is \emph{analytic}, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes \emph{regular} quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are \emph{continuously differentiable}. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of \emph{Matkowski means}.