On Hardy's inequality
type: Item
instance of: scholarly article
title: On Hardy's inequality
zbMATH Open document ID: 0946.26011
author: Bicheng Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-11-02T00:00:00Z
review text: The following result is proved: Let \(0< b_{n+1}\leq b_n\), \(B_n= \sum^n_{k= 1}b_k\), \(a_n\geq 0\), \(0< \sum^\infty_{n=1} b_na_n< \infty\). Then \[ \sum^\infty_{n= 1} b_{n+ 1}(a^{b_1}_1\cdots a^{b_n}_n)^{1/B_n}< e \sum^\infty_{n= 1} \Biggl[1- {b_n\over 2(B_n+ b_n)}\Biggr] b_na_n. \] For \(b_n\equiv 1\) a refinement of Carleman's inequality follows. The proof uses Pólya's method [see, e.g., \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya}: ``Inequalities'' (1952; Zbl 0010.10703; 2nd ed. 1988; Zbl 0634.26008)] as well as certain inequalities for the number \(e\). Lemma 2 is well known, even in improved forms [see, e.g., \textit{J. Sándor}, Elem. Math. 43, No. 6, 177-180 (1988; Zbl 0702.26016)].
reviewed by: József Sándor
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1334734
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: Carleman's inequality
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6264
OpenAlex ID: W2020756557
cites work: Q3740166
cites work: Q4945929
cites work: Q4391587
cites work: A note on the Hardy-Hilbert inequality
cites work: Q5812325
cites work: Analytic Inequalities
DOI: 10.1006/JMAA.1999.6264
description: scientific article; zbMATH DE number 1334734