On the equality of two-variable general functional means

TL;DR

This paper investigates when two-variable generalized functional means, including quasiarithmetic, Cauchy, and Bajraktarević means, are equal, providing necessary conditions and complete characterizations for specific measures.

Contribution

It characterizes the equality of two-variable generalized means for fixed measures, extending understanding of their functional relationships and offering complete solutions for particular cases.

Findings

Derived necessary conditions for mean equality under differentiability assumptions.

Provided complete characterizations for specific probability measures.

Established eight equivalent conditions for Bajraktarević and Cauchy mean equality.

Abstract

Given two functions $f,g:I\to\mathbf{R}$ and a probability measure $\mu$ on the Borel subsets of $[0,1]$, the two-variable mean $M_{f,g;\mu}:I^2\to I$ is defined by $$ M_{f,g;\mu}(x,y) :=\bigg(\frac{f}{g}\bigg)^{-1}\left( \frac{\int_0^1 f\big(tx+(1-t)y\big)d\mu(t)} {\int_0^1 g\big(tx+(1-t)y\big)d\mu(t)}\right) \qquad(x,y\in I). $$ This class of means includes quasiarithmetic as well as Cauchy and Bajraktarevi\'c means. The aim of this paper is, for a fixed probability measure $\mu$, to study their equality problem, i.e., to characterize those pairs of functions $(f,g)$ and $(F,G)$ such that $$ M_{f,g;\mu}(x,y)=M_{F,G;\mu}(x,y) \qquad(x,y\in I) $$ holds. Under at most sixth-order differentiability assumptions for the unknown functions $f,g$ and $F,G$, we obtain several necessary conditions for the solutions of the above functional equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarevi\'c means and of Cauchy means.