Eine allgemeine logarithmische Ungleichung. (A general logarithmic inequality)
type: Item
instance of: scholarly article
title: Eine allgemeine logarithmische Ungleichung. (A general logarithmic inequality)
zbMATH Open document ID: 0607.26010
author: Dieter Rüthing
published in: Elemente der Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/141373
review text: The purpose of this paper is to prove the following. Let \(f:[a,b]\to {\mathbb{R}}(0<a<b)\) be positive and continuous on \([a,b]\) and have a positive derivative on \(]a,b[.\) Then \[ (f(a)f(b))^{1/2}<\frac{f(a)-f(b)}{\ln (f(a)/f(b))}<((f(a)^{1/3}+f(b)^{1/3})/2)^ 3. \] \{The reviewer does not understand what the difference is between this and the case \(f=id\) which, as the editors note in a postscript, \textit{T. P. Lin} [Am. Math. Mon. 81, 879-883 (1974; Zbl 0292.26015)] has proved. He also thinks there are errors or extensive misprints in the present proof.\}
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 3981470
zbMATH Keywords: logarithmic mean
zbMATH Keywords: geometric mean
zbMATH Keywords: root-mean-power
zbMATH Keywords: mean value theorem
zbMATH Keywords: chains of inequalities between mean values
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3981470