On the equality of two-variable general functional means

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 $$M_{f,g;\mu }(x,y):=(\frac{f}{g}{)}^{-1}\left(\frac{{\int }_0^{1}f(tx+(1-t)y)d\mu (t)}{{\int }_0^{1}g(tx+(1-t)y)d\mu (t)}\right)(x,y\in I).$$ This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević 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 $$M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y)(x,y\in I)$$ 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 Bajraktarević means and of Cauchy means.