Means and divided differences
type: Item
instance of: scholarly article
title: Means and divided differences
zbMATH Open document ID: 0824.39006
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-11-14T00:00:00Z
review text: Let \(f\) be \(C^{2n}\) on the positive half line with nonzero \(2n\)th derivative and let \(L_ n\) be the limit of the \((2n- 1)\)st divided difference when \(n\) variables tend to \(a\) and \(n\) to \(b\). This paper examines which mean values of \(a\) and \(b\), substituted into \((1/(2n- 1)!) f^{(2n- 1)}\), give \(L_ n\). In much of the paper \(f(x)= x^ p\). Among the results offered are the following. The mean is arithmetic for \(p= 2n\), geometric for \(p= -1\), \({1\over 2}\)th power mean for \(p= (2n- 1)/2\) and for no \(p\) harmonic. The limit of \(L_ n\) as \(n\to \infty\) is the geometric mean for each \(x^ p\) but the arithmetic mean for \(e^ x\).
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 755534
zbMATH Keywords: derivatives
zbMATH Keywords: limits integral representation
zbMATH Keywords: series
zbMATH Keywords: Riemann sum
zbMATH Keywords: Stieltjes integral
zbMATH Keywords: inequalities
zbMATH Keywords: divided difference
zbMATH Keywords: mean values
zbMATH Keywords: geometric mean
zbMATH Keywords: arithmetic mean
author: Alan Horwitz
reviewed by: János Aczél
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1152
OpenAlex ID: W1980062868
DOI: 10.1006/JMAA.1995.1152
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description: scientific article; zbMATH DE number 755534