On the equality of two-variable general functional means

Losonczi, László and Páles, Zsolt and Zakaria, Amr (2021) On the equality of two-variable general functional means. AEQUATIONES MATHEMATICAE, 95 (6). pp. 1011-1036. ISSN 0001-9054

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Abstract

Given two functions f, g : I -> R and a probability measure mu on the Borel subsets of [0, 1], the two-variable mean M-f,(g;mu) : I-2 -> I is defined byM-f,M- g;mu(x, y) := (f/g)(-1) (integral(1)(0) f(tx + (1 - t)y) d mu(t)/(1)(integral 0) g(tx + (1 - t)y)d mu(t) (x, y is an element of I).This class of means includes quasiarithmetic as well as Cauchy and Bajraktarevic 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) for whichM-f,M- g;mu(x, y) = M-F,M- G;mu(x, y) (x, y is an element of 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 Bajraktarevi ' c means and of Cauchy means.

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