Entropic means
type: Item
instance of: scholarly article
title: Entropic means
zbMATH Open document ID: 0675.26007
DOI: 10.1016/0022-247X(89)90128-5
author: Aharon Ben-Tal
author: A. Charnes
author: Marc Teboulle
published in: Journal of Mathematical Analysis and Applications
publication date: 1989-01-01T00:00:00Z
review text: With a somewhat unusual notation, the authors denote by \[ (E_{d_{\phi}})\quad \min \{\sum^{n}_{i=1}w_ ia_ i\phi (x/a_ i):\quad X\in {\mathbb{R}}_+\} \] the problem of determining that positive number x (as function of the positive \(a_ 1,...,a_ n)\) for which \(\sum^{n}_{i=1}w_ ia_ i\phi (x/a_ i)\) is minimum \((w_ 1,...,w_{n-1},w_ n=1-\sum^{n-1}_{j=1}w_ j\) are given positive numbers, \(\phi\) : \({\mathbb{R}}_+\to {\mathbb{R}}^ a \)given strictly convex differentiable function with \((0,1]\subset dom \phi\) and such that \(\phi (1)=\phi '(1)-0),\) with reference to \textit{I. Csiszár} [Stud. Sci. Math. Hung. 2, 299-318 (1967; Zbl 0157.25802)]. Then they define similarly \[ (E_{d_ h})\quad \min \{h(x)-x\sum^{n}_{i=1}w_ ih'(a_ i):\quad x\in {\mathbb{R}}_+\} \] (h again strictly convex differentiable). They call the first (or both?) an entropic mean (''entropic'' not in the sense of \textit{I. M. E. Etherington} [e.g. Proc. R. Soc. Edinb., Sect. A 65, 84-108 (1958; Zbl 0228.17002)]. Then, as examples of one or the other, the power means, the geometric, the Lehmer means [\textit{D. H. Lehmer}, J. Math. Anal. Appl. 36, 183-200 (1971; Zbl 0222.26018), but see \textit{E. F. Beckenbach}, Am. Math. Mon. 57, 1-6 (1950; Zbl 0035.15704) or even \textit{K. Goldziher}, Skand. Aktuarietidskr. 1920, 72-96 (1920)], the Gini means [\textit{C. Gini}, Metron 13, No. 2, 3-22 (1938; Zbl 0018.41404)] and the quasilinear means [here called ''HLP-means'' for \textit{G. H. Hardy, J. E. Littlewood} and \textit{G. Pólya}, Inequalities (1952; Zbl 0047.05302; 1st ed. 1934; Zbl 0010.10703), who quote \textit{B. de Finetti}, Giorn. Ist. Ital. Attuari 2, 369-296 (1931; Zbl 0002.27802)], among others, are enumerated. For the first, \(''(E_{d_{\phi}})''\) kind of means a sufficient, but not necessarily necessary, condition of comparability and an asymptotic property is offered, which generalizes that of \textit{L. Hoehn} and \textit{I. Niven} [Math. Mag. 58, 151-156 (1985; Zbl 0601.26011), cf. \te…
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 4105270
zbMATH Keywords: geometric mean
zbMATH Keywords: Stolarski mean
zbMATH Keywords: strictly convex differentiable function
zbMATH Keywords: entropic mean
zbMATH Keywords: power means
zbMATH Keywords: Lehmer means
zbMATH Keywords: Gini means
zbMATH Keywords: quasilinear means
zbMATH Keywords: asymptotic property
zbMATH Keywords: Integral means
zbMATH Keywords: quasi-arithmetic integral mean
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4105270