On new generalizations of Hilbert's inequality
type: Item
instance of: scholarly article
title: On new generalizations of Hilbert's inequality
zbMATH Open document ID: 0970.26009
author: Bicheng Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-10-02T00:00:00Z
review text: Let \(\lambda>0\) and let \(f_i : (0,\infty)\rightarrow\mathbb R\) \((i=1,2)\) be functions such that \(0<\int^{\infty}_0 t^{1-\lambda} f_i^2 (t) dt<\infty\). Then \[ \int^{\infty}_0 \int^{\infty}_0 \frac{f_1(x)f_2(y)}{(Ax+By)^{\lambda}} dx dy < (AB)^{-\lambda/2} B\Big(\frac{\lambda}{2}, \frac{\lambda}{2}\Big)\prod^2_{i=1} \Big(\int^{\infty}_0 t^{1-\lambda} f^2_i(t) dt\Big)^{1/2} \tag{1} \] and \[ \int^{\infty}_0 y^{\lambda-1} \Big[\int^{\infty}_0 \frac{f_1(x)}{(Ax+By)^{\lambda}} dx\Big]^2 dy < (AB)^{-\lambda} \Big[B\Big(\frac{\lambda}{2}, \frac{\lambda}{2}\Big)\Big]^2 \int^{\infty}_0 t^{1-\lambda} f^2(t) dt, \tag{2} \] where \(B(\cdot,\cdot)\) is the \(\beta\)-function. Moreover, the inequalities (1) and (2) are equivalent and the constants appearing on their right hand sides are the best possible. This is the main result of the paper. The author also presents its discrete analogue.
reviewed by: Bohumír Opic
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1506518
zbMATH Keywords: Hilbert type inequalities
zbMATH Keywords: \(\beta\)-function
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6860
OpenAlex ID: W2040333484
cites work: Q5812325
cites work: On Hilbert's inequality
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cites work: Q4852102
cites work: On Hilbert's integral inequality
cites work: On a new generalization of Hardy-Hilbert's inequality and its applications
DOI: 10.1006/JMAA.2000.6860
description: scientific article; zbMATH DE number 1506518