Equality of two-variable functional means generated by different measures
type: Item
instance of: scholarly article
title: Equality of two-variable functional means generated by different measures
zbMATH Open document ID: 1234.39003
author: László Losonczi
author: Zsolt Páles
published in: Aequationes Mathematicae
publication date: 2011-03-31T00:00:00Z
review text: Two-variable functional means, \(M_{f,g;\mu}\) defined by \[ M_{f,g;\mu}(x,y)= \Biggl({f\over g}\Biggr)^{-1}\left({\int_{[0,1]} f(tx+ (1- t)y)\,d\mu(t)\over \int_{[0,1]} g(tx+ (1- t)y)\,d\mu(t)}\right) \] for real-valued continuous functions \(f\) and \(g\) defined on a real open interval \(I\), \(\mu\) a measure on the Borel sets of \([0,1]\), \(g\) a positive function, and \(f\) strictly \(g\) monotonic, are considered. Note that \(M\) generalizes quasi-arithmetic, Bajraktarević, Lagrangian, and Cauchy means. The main results concern the functional equation \(M_{t,g;\mu}= M_{t,g;\nu}\) for unknown functions \(f\) and \(g\) and with given measures \(\mu\) and \(\nu\). The original functional equation is re-written as \[ \int_{[0,1]}\,\int_{[0,1]} \left|\begin{matrix} f(tx+ (1-t)y)\;& f(sz+ (1- s)y)\\ g(tx+ (1-t)y)\;& g(sx+ (1-s)y)\end{matrix}\right|\,d\mu(t)d\nu(s)= 0.\tag{\(*\)} \] This form of the functional equation allows significant relaxation of the conditions imposed on \(f\), \(g\), \(\mu\), and \(\nu\) in earlier work; thus \((*)\) is studied under minimal assumptions on the two functions and the two measures. Since \((*)\) is trivially satisfied if the two functions are linearly dependent or if the two measures are linearly dependent, the results here concern the linearly independent cases. Necessary and sufficient conditions on the two functions and on finite Borel measures \(\mu\) and \(\nu\) are given for solutions of \((*)\). The particular cases of linearly independent finite Borel measures which are either both symmetric or both antisymmetric are also studied. In addition, the symmetry equatio \(M_{f,g;\mu}(x,y)= M_{f,g;\mu}(y,x)\) is discussed. The paper concludes with three particular applications of \((*)\) obtained from various choices of measures of discrete support. The resulting functions equations (i) \([f(x)- 2f({x+y\over 2})+ f(y)][g(x)+ g(y)]= [g(x)- 2g({x+iy\over 2})+ g(y)][f(x)+ f(y)]\), (ii) \([f(x)- 2f({x+y\over 2})+ f(y)][g(x)- g(y)]…
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 5872782
zbMATH Keywords: functional means
zbMATH Keywords: functional equation
zbMATH Keywords: equality problem for means
zbMATH Keywords: symmetric means
zbMATH Keywords: analytic solutions
MaRDI profile type: Publication
full work available at URL: s00010-010-0059-5
OpenAlex ID: W2085455545
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DOI: 10.1007/S00010-010-0059-5
description: scientific article; zbMATH DE number 5872782