Matkowski-Sutô type equation on symmetrized weighted quasi-arithmetic means
type: Item
instance of: scholarly article
title: Matkowski-Sutô type equation on symmetrized weighted quasi-arithmetic means
zbMATH Open document ID: 1264.39022
author: Pál Burai
published in: Results in Mathematics
publication date: 2013-02-08T00:00:00Z
review text: Given \(\varphi \in \mathcal{CM}(I)\) (\(\mathcal{CM}(I)\) is the class of continuous and strictly monotone real valued functions defined on the open interval \(I\)), \(A_{\varphi}(x,y;\alpha)\) denotes the weighted quasi-arithmetic mean generated by \(\varphi\) with weight \(\alpha\), \(0<\alpha<1\), i.e., \[ A_{\varphi}(x,y;\alpha)=\varphi^{-1}(\alpha \varphi(x)+(1-\alpha) \varphi(y)), \quad x, y \in I. \] Defined \[ A^*_{\varphi}(x,y;\alpha):=\frac{A_{\varphi}(x,y;\alpha)+A_{\varphi}(x,y;(1-\alpha))}{2}, \] the paper aims to find all pairs of functions \(\varphi, \psi \in \mathcal{CM}(I)\) such that \[ A^*_{\varphi}(x,y;\alpha)+A^*_{\psi}(x,y;\alpha)=x+y, \quad x, y \in I. \tag{E} \] Note that if \(\alpha=1/2\) we have the original Matkowski-Sutô problem. The main result is stated in the following Theorem: Let \((\varphi,\psi) \in \mathcal{CM}(I)^2\) be a solution of equation (E) and assume that there exists a nonempty subinterval \(J\) of \(I\) such that both functions are two times continuously differentiable on \(J\). If at least one of them is six times continuously differentiable on \(J\), then there exists a real number \(p\) such that \[ \varphi(x)=a_1\chi_p(x)+b_1, \quad \psi(x)=a_2\chi_{-p}(x)+b_2, \] for some \(a_1, a_2 \neq 0\) and \(b_1, b_2 \in \mathbb R\), where \[ \chi_r(x)=\begin{cases} e^{rx}, &\text{if }r\neq 0,\\ x, &\text{if }r=0.\end{cases} \]
reviewed by: Gian Luigi Forti
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 6135254
zbMATH Keywords: invariance equation
zbMATH Keywords: Matkowski-Sutô type equation
zbMATH Keywords: functional equations involving means
zbMATH Keywords: weighted quasi-arithmetic mean
MaRDI profile type: Publication
full work available at URL: s00025-011-0204-z
OpenAlex ID: W2090721592
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DOI: 10.1007/S00025-011-0204-Z
description: scientific article; zbMATH DE number 6135254