An invariance of the geometric mean with respect to Stolarsky mean-type mappings
type: Item
instance of: scholarly article
title: An invariance of the geometric mean with respect to Stolarsky mean-type mappings
zbMATH Open document ID: 1040.26013
DOI: 10.1007/BF03322720
author: J. Błasińska-Lesk
author: D. Głazowska
author: Janusz Matkowski
published in: Results in Mathematics
publication date: 2003-11-10T00:00:00Z
review text: Let \(E_{r,s} (x,y)\) be the Stolarsky mean of \(x,y> 0\). Let \(G\) be the geometric mean. The authors prove that \(G(E_{r, s}(x,y), E_{k,m} (x, y))= G(x,y)\) if and only if one of the following conditions occur: (i) \(k=m =r= s= 0\); (ii) \(k= -r\), \(m= s= 0\); (iii) \(k= m\neq 0\), \(r= s\neq 0\) and \(k=- r\); (iv) \(rk\neq 0\), \(r\neq s\), \(k\neq m\) and either \(r=-s\) and \(k= -m\), or \(k=-r\) and \(m=-s\), or \(k=-s\) and \(m=-r\). As an application, they study the functional equation \(F(x,y) =F(M(x,y), N(x,y))\), where \(M\) and \(N\) are some means, a special case of which was a problem posed by H. Haruki and Th. M. Rassias.
reviewed by: József Sándor
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 2002339
zbMATH Keywords: Stolarsky mean
zbMATH Keywords: functional equations
zbMATH Keywords: iterates
MaRDI profile type: Publication
cites work: A new analogue of Gauss' functional equation
cites work: Q4947993
cites work: Generalizations of the Logarithmic Mean
Recommended article: Some Cauchy mean-type mappings for which the geometric mean is invariant
Recommended article: Q5421138
Recommended article: An invariance of geometric mean with respect to generalized quasi-arithmetic means
Recommended article: An invariance of geometric mean with respect to Lagrangian means
Recommended article: Invariance of the arithmetic mean with respect to special mean-type mappings
Recommended article: Invariant means and iterates of mean-type mappings
Recommended article: Hermite-Hadamard-type inequalities for geometrically \(r\)-convex functions in terms of Stolarsky's mean with applications to means
Recommended article: Stolarsky-type means related to generalizations of Steffensen's and Gauss' inequality
Recommended article: Q4947993
Recommended article: Mean-type mappings and invariance principle
description: scientific article; zbMATH DE number 2002339