On certain inequalities involving the constant \(e\) and their applications
type: Item
instance of: scholarly article
title: On certain inequalities involving the constant \(e\) and their applications
zbMATH Open document ID: 0965.26007
author: József Sándor
author: Lokenath Debnath
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-07-17T00:00:00Z
review text: This is a survey of results, mainly by the first author, some improved, with several further related inequalities and limit values offered. Among the latter the following is typical (and relatively less technical). For all \(y>0,\) \[ \Biggl(1+\frac{1}{y} \Biggr)^{1/2} \exp \Biggl[-\frac{1}{12y^2 +12y}\Biggr]< e\Bigl(1+\frac{1}{y}\Bigr)^{-y}< \Biggl(1+\frac{1}{y}\Biggr)^{1/2} \exp \Biggl[-\frac{1}{12y^2 +12y+3}\Biggr]. \] Connections between different inequalities and limits are emphasized. [Note: The inequality \(\sum_{k=1}^n a_k \log a_k \geq \sum_{k=1}^n a_k\log b_k\) if \(\sum_{k=1}^n a_k = \sum_{k=1}^n b_k\) is mentioned on p. 570; the name ``Shannon inequality'' is not].
reviewed by: János Aczél
Mathematics Subject Classification ID: 26D07
Mathematics Subject Classification ID: 33B10
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1533185
zbMATH Keywords: inequalities
zbMATH Keywords: limits
zbMATH Keywords: mean values
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6911
OpenAlex ID: W2090259575
cites work: On the Functional Inequality f(x + y) ≧f(x)f(y)
cites work: Analytic Inequalities
cites work: Q5645344
cites work: Some inequalities involving the constant \(e\), and an application to Carleman's inequality
cites work: Ungleichungen für Mittelwerte. (Inequalities for means)
cites work: On the identric and logarithmic means
cites work: On certain inequalities for means
cites work: The Power and Generalized Logarithmic Means
cites work: Some integral inequalities
DOI: 10.1006/JMAA.2000.6911
description: scientific article; zbMATH DE number 1533185