The monotonicity of the ratio between generalized logarithmic means
type: Item
instance of: scholarly article
title: The monotonicity of the ratio between generalized logarithmic means
zbMATH Open document ID: 1160.26012
author: Chao-Ping Chen
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-06-24T00:00:00Z
review text: The generalized logarithmic mean \(L_{r}(a,b)\) of two positive numbers \(a,b\) is defined by \(L_{r}(a,a)=a\) and if \(a\neq b\), by \[ L_{r}(a,b)=\left( \frac{b^{r+1}-a^{r+1}}{(r+1)(b-a)}\right) ,\;r\neq0,1;\;L_{-1}(a,b)=\frac{b-a}{\ln b-\ln a},\;L_{0}(a,b)=\frac{1}{e}\left( \frac{b^{b}}{a^{a}}\right) ^{\frac{1}{b-a}}. \] In this paper the following properties are proven: Let \(\delta,\varepsilon\) and \(b\) be positive numbers, then the function \[ \frac{L_{r}(b,b+\varepsilon)}{L_{r}(b+\delta,b+\delta+\varepsilon)} \] is strictly increasing in \(r\in(-\infty,\infty).\) Let \(n\) be a natural number, then, for all \(r\in\mathbb{R}\), \[ \frac{n}{n+1}<\left( \frac{\frac{1}{n}\sum_{i=1}^{n}i^{r}}{\frac{1}{n+1} \sum_{i=1}^{n+1}i^{r}}\right) ^{1/r}<1. \] Moreover, the lower bound \(\frac{n}{n+1}\) and the upper bound \(1\) are the best possible.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5292290
zbMATH Keywords: generalized logarithmic mean
reviewed by: Joaquim Martin
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.03.071
OpenAlex ID: W2080708534
cites work: On an inequality of H. Minc and L. Sathre
cites work: Q5464379
cites work: Monotonicity of sequences involving convex and concave functions
cites work: Q5718601
cites work: Q3420917
cites work: On Alzer's inequality
cites work: Some Inequalities involving (r!)1/r
cites work: Generalization of H. Alzer's inequality
cites work: Monotonicity of sequences involving convex function and sequence
cites work: On an inequality of Alzer
cites work: Generalizations of the Logarithmic Mean
cites work: The Power and Generalized Logarithmic Means
cites work: Q4332276
cites work: A general form of Alzer's inequality
DOI: 10.1016/J.JMAA.2008.03.071
description: scientific article; zbMATH DE number 5292290