On the equality of two-variable general functional means
Losonczi, L.; Pales, Zs.; Amr Zakaria Mohamed Abdelhamed;
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.
Other data
Title | On the equality of two-variable general functional means | Authors | Losonczi, L.; Pales, Zs.; Amr Zakaria Mohamed Abdelhamed | Keywords | Generalized functional mean;Equality problem;System of differential equations;Functional equation | Issue Date | 2020 | Publisher | SPRINGER BASEL AG | Journal | AEQUATIONES MATHEMATICAE | ISSN | 0001-9054 | DOI | 10.1007/s00010-020-00755-w | Scopus ID | 2-s2.0-85091726228 | Web of science ID | WOS:000574059600001 |
Attached Files
File | Description | Size | Format | |
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Losonczi2020_Article_OnTheEqualityOfTwo-variableGen(2).pdf | 462.1 kB | Unknown | View/Open |
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