Notes on an inequality involving the constant \(e\)
type: Item
instance of: scholarly article
title: Notes on an inequality involving the constant \(e\)
zbMATH Open document ID: 0978.26005
author: Jian-Lin Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-01-13T00:00:00Z
review text: The main objective of this note is to prove that \[ \sum^{\infty}_{n=1} \lambda_{n+1} (a^{\lambda_1}_1 a^{\lambda_2}_2 \dots a_n^{\lambda_n})^{1/\Lambda_n} < e \sum^{\infty}_{n=1} [1+(\Lambda_n/\lambda_n+1/5)^{-1}]^{-1/2} \lambda_n a_n, \] 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 \(\sum^{\infty}_{n=1} \lambda_n a_n < \infty\). This result improves that of \textit{B. Yang} and \textit{L. Debnath} [J. Math. Anal. Appl. 223, No. 1, 347-353 (1998; Zbl 0910.26011)] and gives that of \textit{B. Yan} and \textit{G. Sun} [J. Math. Anal. Appl. 240, No. 1, 290-293 (1999; Zbl 0941.26012)] if \(\lambda_n = 1\) for all \(n\in\mathbb N\).
reviewed by: Bohumír Opic
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1542476
zbMATH Keywords: Carleman-type inequalities
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6979
OpenAlex ID: W1968255905
cites work: Q5812325
cites work: Q5645344
cites work: A strengthened Carleman's inequality
cites work: On Hardy's inequality
cites work: Some inequalities involving the constant \(e\), and an application to Carleman's inequality
DOI: 10.1006/JMAA.2000.6979
description: scientific article; zbMATH DE number 1542476