A bilinear inequality
type: Item
instance of: scholarly article
title: A bilinear inequality
zbMATH Open document ID: 1016.15015
DOI: 10.1016/S0022-247X(02)00104-X
author: Kewei Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: The author establishes that for an \(n\times n\) real symmetric matrix, \(X\neq 0\), the following inequality for the symmetric bilinear form \(a^{T}Xb\) holds: \[ (a^{T}Xb)^{2}\leq \frac{|x_{+}|_{\infty }^{2}+|x_{-}|_{\infty }^{2}}{2} (|a|^{2}|b|^{2}+(a,b)^{2}), \] for all \(a,b\in \mathbb{R}^{n}\), where \(|x_{+}|_{\infty }\) and \( |x_{-}|_{\infty }\) are the largest nonnegative eigenvalues of \(X\) and \(-X\), respectively, and \((a,b)=a^{T}b\) is the Euclidean inner product of \(a\) and \(b \). If \(X=I\), that inequality is equivalent to Cauchy-Schwarz inequality \( |(a,b)|\leq |a||b|\). If \(X\neq I\) and \(X\) is semi-positive definite, the inequality is strictly sharper than \(|a^{T}Xb|\leq |x|_{\infty }|a||b|\). However, when \(X\) is indefinite, for example \(X=\text{diag}(1,-1)\), the inequality might be weaker. The inequality is used to obtain inequalities in real separable Hilbert spaces, in particular, integral inequalities in \(L^{2}(\mathbb{R})\) and \( l^{2}(\mathbb{R})\). The inequality applied to the Hilbert matrix \((1/(i+j-1)) \), that following \textit{W. Magnus} [Am. J. Math. 72, 699-704 (1950; Zbl 0041.23805)] is a nonnegative definite operator on \(l^{2}(\mathbb{R})\) with spectral set \([0,\pi ]\), is the inner product version of Hilbert's inequality that is sharper than the Hilbert inequality. The author also deduces Schur's form of Hilbert's inequality in \(l^{2}(\mathbb{R})\) and the integral form of the inner product version of Hilbert's inequality in \(L^{2}(\mathbb{R })\).
reviewed by: M. Ladra
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B36
zbMATH DE Number: 1836754
zbMATH Keywords: bilinear form
zbMATH Keywords: inner product
zbMATH Keywords: Hilbert' s inequality
zbMATH Keywords: Cauchy-Schwarz inequality
zbMATH Keywords: Lagrangian multiplier
zbMATH Keywords: Schur's form
MaRDI profile type: Publication
cites work: Q4544332
cites work: The initial‐value problem for measure‐valued solutions of a canonical 2 × 2 system with linearly degenerate fields
cites work: Q5653193
cites work: Q5812325
cites work: On the Spectrum of Hilbert's Matrix
cites work: Q3293417
full work available at URL: s0022-247x(02)00104-x
OpenAlex ID: W2070911304
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description: scientific article; zbMATH DE number 1836754