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- Title
On the equality of two-variable general functional means.
- Authors
Losonczi, László; Páles, Zsolt; Zakaria, Amr
- Abstract
Given two functions f , g : I → R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean M f , g ; μ : I 2 → I is defined by M f , g ; μ (x , y) : = (f g ) - 1 ∫ 0 1 f (t x + (1 - t) y) d μ (t) ∫ 0 1 g (t x + (1 - t) y) d μ (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 M f , g ; μ (x , y) = M F , 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.
- Subjects
BOREL subsets; ORDINARY differential equations; PROBABILITY measures; BOREL sets; FUNCTIONAL equations
- Publication
Aequationes Mathematicae, 2021, Vol 95, Issue 6, p1011
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-020-00755-w