A best approximation for constant \(e\) and an improvement to Hardy's inequality
type: Item
instance of: scholarly article
title: A best approximation for constant \(e\) and an improvement to Hardy's inequality
zbMATH Open document ID: 0988.26017
author: Yibing Zhong
author: Zitian Xie
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-01T00:00:00Z
review text: The authors show that \[ e\Big(1-\frac 7{14x+12}\Big) < \Big(1+\frac 1{x}\Big)^x < e \Big(1-\frac 6{12x+11} \Big), \quad x \geq 1, \] and apply this result to prove that \[ \sum^{\infty}_{n=1} \lambda_{n+1} (a^{\lambda_1}_1 a^{\lambda_2}_2 \dots a^{\lambda_n}_n)^{1/\Lambda_n} < e \sum^{\infty}_{n=1} \Big(1-\frac{6\lambda_n}{12\Lambda_n + 11 \lambda_n}\Big) \lambda_n a_n, \leqno(\ast) \] where \(0< \lambda_{n+1} \leq \lambda_n\), \(\Lambda_n = \sum^n_{m=1} \lambda_m\), \(a_n\geq 0\) \((n\in\mathbb N)\), and \(0<\sum^{\infty}_{n=1} \lambda_n a_n < \infty\). Note that \((*)\) yields an improvement of the well-known Carleman inequality.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1563107
zbMATH Keywords: generalized Carleman's inequality
reviewed by: Bohumír Opic
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6981
OpenAlex ID: W2041676614
cites work: Some inequalities involving the constant \(e\), and an application to Carleman's inequality
cites work: On Hardy's inequality
DOI: 10.1006/JMAA.2000.6981
description: scientific article; zbMATH DE number 1563107