Year: 2020
Type: article
Abstract: Given two functions $$f,g:I\rightarrow \mathbb {R}$$ f,g:I→R and a probability measure $$\mu $$ μ on the Borel subsets of [0, 1], the two-variable mean $$M_{f,g;\mu }:I^2\rightarrow I$$ Mf,g;μ:I2→I is defined by $$\begin{aligned} M_{f,g;\mu }(x,y) :=\bigg (\frac{f}{g}\bigg )^{-1}\left( \frac{\int _0^1 f\big (tx+(1-t)y\big )d\mu (t)}{\int _0^1 g\big (tx+(1-t)y\big )d\mu (t)}\right) \quad (x,y\in I). \end{aligned}$$ 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 $$\mu $$ μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which $$\begin{aligned} M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y) \quad (x,y\in I) \end{aligned}$$ Mf,g;μ(x,y)=MF,G;μ(x,y)(x,y∈I) holds. Under at most sixth-order differentiability assumptions for the unknown functionsf, gandF, 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.
Source: Aequationes Mathematicae
Authors: László Losonczi, Zsolt Páles, Amr Zakaria
Institutions: University of Debrecen, Ain Shams University
Language: Latvian
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,136
OpenAlex ID: W2995603022