Homogeneous mean values: weights and asymptotics
type: Item
instance of: scholarly article
title: Homogeneous mean values: weights and asymptotics
zbMATH Open document ID: 0614.26002
DOI: 10.1016/0022-247X(87)90308-8
author: J. L. Brenner
author: B. C. Carlson
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: The authors prove the following. Let M be a reflexive \((M(\lambda,...,\lambda)=\lambda)\) homogeneous mean (presumably on the Cartesian power of a real interval going to \(\infty)\), with continuous second derivatives in each variable in a neighborhood of (1,...,1) and \(q_ k=(\partial M/\partial x_ k)(1,...,1)\quad (k=1,...,n).\) Then \[ M(a_ 1+x,...,a_ n+x)=x+\sum^{n}_{k=1}q_ ka_ k+0(1/x)\quad as\quad x\to \infty. \] (For the history of the problem see the preceding review.) Several means satisfying these conditions (and one which does not) are enumerated. One of them, the hypergeometric mean [\textit{B. C. Carlson}, Proc. Am. Math. Soc. 16, 759-766 (1965; Zbl 0137.268)] is further investigated for infinite sequences.
Mathematics Subject Classification ID: 26A12
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 3993886
zbMATH Keywords: asymptotic formula
zbMATH Keywords: Dresher means
zbMATH Keywords: mixed means
zbMATH Keywords: logarithmic mean
zbMATH Keywords: arithmetic mean
zbMATH Keywords: geometric mean
zbMATH Keywords: harmonic mean
zbMATH Keywords: root-mean-square
zbMATH Keywords: power means of all orders
zbMATH Keywords: weighted means
zbMATH Keywords: Muirhead mean
zbMATH Keywords: symmetric mean
zbMATH Keywords: Stirling numbers
zbMATH Keywords: generating functions
zbMATH Keywords: ordo
zbMATH Keywords: homogeneous mean
zbMATH Keywords: hypergeometric mean
MaRDI profile type: Publication
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full work available at URL: 0022-247x(87)90308-8
OpenAlex ID: W2085671382
description: scientific article; zbMATH DE number 3993886