On the equality of two-variable general functional means

László Losonczi,Amr Zakaria,Zsolt Páles

doi:10.1007/s00010-020-00755-w

László Losonczi, Amr Zakaria + Show 1 more

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https://doi.org/10.1007/s00010-020-00755-w

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Abstract

Given two functions f,g:Irightarrow mathbb {R} and a probability measure mu on the Borel subsets of [0, 1], the two-variable mean M_{f,g;mu }:I^2rightarrow 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).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_{f,g;\\mu }(x,y) :=\\bigg (\\frac{f}{g}\\bigg )^{-1}\\left( \\frac{\\int _0^1 f\\big (tx+(1-t)y\\big )d\\mu (t)}{\\int _0^1 g\\big (tx+(1-t)y\\big )d\\mu (t)}\\right) \\quad (x,y\\in I). \\end{aligned}$$\\end{document}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 , 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)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} M_{f,g;\\mu }(x,y)=M_{F,G;\\mu }(x,y) \\quad (x,y\\in I) \\end{aligned}$$\\end{document}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.

Highlights

AEMThroughout this paper I will stand for a nonempty open real interval
Given two continuous functions f, g : I → R with g ∈ CP(I), f /g ∈ CM(I) and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean Mf,g;μ : I2 → I is defined by f −1 Mf,g;μ(x, y) := g
The aim of this paper is to study the equality problem of generalized quasiarithmetic means, i.e., to characterize those pairs of functions (f, g) and (F, G) for which

Summary

Introduction

Throughout this paper I will stand for a nonempty open real interval. In the sequel, the classes of continuous strictly monotone and continuous positive real-valued functions defined on I will be denoted by CM(I) and CP(I), respectively. Given a function φ ∈ CM(I), the two-variable quasiarithmetic mean generated by φ is the function Aφ : I2 → I defined by. The equality problem of means in various classes of two-variable means has been solved. The aim of this paper is to study the equality problem of generalized quasiarithmetic means, i.e., to characterize those pairs of functions (f, g) and (F, G) for which (3). In our final main results we consider the cases when the measure μ is either of the form δ0 +δ1 2 or is the Lebesgue measure on [0, 1] For these two cases, we shall need sixth-order differentiability properties of the unknown functions f, g and F, G

Basic results

Higher-order directional derivatives of generalized quasiarithmetic means

Equality of Bajraktarevic means

Equality of Cauchy means

Conclusion and open problems