New versions of weighted multidimensional functional and Stolarsky means
type: Item
instance of: scholarly article
title: New versions of weighted multidimensional functional and Stolarsky means
zbMATH Open document ID: 1374.26079
author: László Horváth
author: J. Pečarić
published in: Acta Mathematica Hungarica
publication date: 2017-03-31T00:00:00Z
review text: In this article, the authors provide an elegant introduction of various multidimensional means, and study their properties. Let us cite a couple of definitions. Definition (doubly weighted multidimensional functional means). Let \(I\subset{\mathbb R}\) be an interval, \({\mathbf x}:=\left(x_1,\dots,x_n\right)\in I^n\) (\(n\geq 2\)), \({\mathbf p}:=\left(p_1,\dots,p_n\right)\) a discrete distribution, \(h,g:\,I\to{\mathbb R}\) be continuous and strictly monotone functions, and let \(\mu\) be a probability measure on the Borel \(\sigma\)-algebra of the simplex \[ E_{n-1}=\left\{\left(t_1,\dots,t_{n-1}\right)\in{\mathbb R}^{n-1}\mid t_1,\dots,t_{n-1}\geq 0,\,\sum_{j=1}^{n-1}t_j\leq 1\right\}. \] Define \[ M\left({\mathbf x},{\mathbf p},h,g,\mu\right):=h^{-1}\left(\int_{E_{n-1}}h\circ g^{-1}\left(\frac{\sum_{j=1}^n t_j p_j g\left(x_j\right)}{\sum_{j=1}^n t_j p_j}\right)\,d\mu\left({\mathbf t}\right)\right), \] where \(t_n:=1-\sum_{j=1}^{n-1}t_j\), and \(d\mu\left({\mathbf t}\right):=d\mu\left(t_1,\dots,t_{n-1}\right)\). Definition (doubly weighted multidimensional Stolarsky means). Let \({\mathbf x}:=\left(x_1,\dots,x_n\right)\in\left(0,\infty\right)^n\) (\(n\geq 2\)), let \({\mathbf p}:=\left(p_1,\dots,p_n\right)\) be a discrete distribution, and let \(\mu\) be a probability measure on the Borel \(\sigma\)-algebra of the simplex \(E_{n-1}\). Define \[ S_w\left({\mathbf x},{\mathbf p},s,r,\mu\right):= \begin{cases} \left(\int_{E_{n-1}}\left(M_r\left({\mathbf x},{\mathbf t}{\mathbf p}\right)\right)^{s-r} \,d\mu\left({\mathbf t}\right)\right)^{\frac{1}{s-r}}, & s\neq r,\\ \exp\left(\int_{E_{n-1}}\ln\left(M_s\left({\mathbf x},{\mathbf t}{\mathbf p}\right)\right)\,d\mu\left({\mathbf t}\right)\right), & s=r, \end{cases} \] where \[ M_r\left({\mathbf x},{\mathbf p}\right):= \begin{cases} \left(\sum_{j=1}^n p_j x_j^r\right)^{1/r}, & r\neq 0,\\ \prod_{j=1}^n x_j^{p_j}, & r=0 \end{cases} \] are weighted multidimensional \(r\)-th power means, and \[ {\mathbf t}{\mathbf p}:=\left(…
reviewed by: Alexander Tovstolis
Mathematics Subject Classification ID: 26E60
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6699216
zbMATH Keywords: functional mean
zbMATH Keywords: Stolarsky mean
zbMATH Keywords: weighted mean
zbMATH Keywords: convexity
MaRDI profile type: Publication
full work available at URL: s10474-015-0538-1
OpenAlex ID: W1068760168
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DOI: 10.1007/S10474-015-0538-1
description: scientific article; zbMATH DE number 6699216