On Hilbert's inequality and its applications
type: Item
instance of: scholarly article
title: On Hilbert's inequality and its applications
zbMATH Open document ID: 0890.26011
author: Mingzhe Gao
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-03-30T00:00:00Z
review text: The following Hilbert's inequality is well-known: \[ \sum_{m=0}^\infty\sum_{n=0}^\infty a_mb_n/(m+n+1)\leq\pi \Biggl(\sum_{m=0}^\infty a_m^2\Biggr)^{1/2} \Biggl(\sum_{n=0}^\infty b_n^2\Biggr)^{1/2} \] [see, for example, \textit{G. H. Hardy, J. E. Littlewood} and \textit{G. Pólya}: ``Inequalities'', 2nd ed. (1952; Zbl 0047.05302, reprint 1988; Zbl 0634.26008, 1st ed. 1934; Zbl 0010.10703)]. The author converts the above \(\ell_2\) norms into weighted \(\ell_2\) norms and gives a corresponding integral version. In both cases, the weights are explicitly given and the inequalities are sharp.
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1053566
zbMATH Keywords: Hilbert inequality
reviewed by: Peng-Yee Lee
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5490
OpenAlex ID: W2136506999
cites work: Q5812325
cites work: Q4311688
DOI: 10.1006/JMAA.1997.5490
description: scientific article; zbMATH DE number 1053566