On the equality of two-variable general functional means

Abstract

Given two functions f, g: I→ R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean Mf,g;μ: I2→ I is defined by Mf,g;μ(x,y):=(fg)-1(∫01f(tx+(1-t)y)dμ(t)∫01g(tx+(1-t)y)dμ(t))(x,y∈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 μ, to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which Mf,g;μ(x,y)=MF,G;μ(x,y)(x,y∈I)holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above 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.