On new generalizations of Hardy's integral inequality
type: Item
instance of: scholarly article
title: On new generalizations of Hardy's integral inequality
zbMATH Open document ID: 0893.26008
author: Bicheng Yang
author: Zhuohua Zeng
author: Lokenath Debnath
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-08-03T00:00:00Z
review text: The well-known Hardy's inequality states that \[ \int^\infty_0 \Biggl[x^{-1} \int^x_0 f(t)dt\Biggr]^p dx< q^p \int^\infty_0 f^p(t)dt\tag{1} \] provided that \(p\in (1,\infty)\), \(q= p/(p- 1)\), \(f>0\) and \(f\not\equiv 0\). The inequality (1) implies that for any \(a\) and \(b\) satisfying \(0<a<b<\infty\), \[ \int^b_a \Biggl[x^{-1} \int^x_a f(t)dt\Biggr]^p< q^p \int^b_a f^p(t)dt,\quad f>0,\quad f\not\equiv 0.\tag{2} \] One of the main results of the paper shows that the constant \(q^p\) in (2) can be replaced by \(q^p[1-(a/b)^{1/q}]^p\). The rest of the paper represents other variants of (1).
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1115472
zbMATH Keywords: Hardy inequality
zbMATH Keywords: integral inequality
reviewed by: Bohumír Opic
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5758
OpenAlex ID: W2055750450
cites work: Q5812325
cites work: Q3999482
cites work: Analytic Inequalities
DOI: 10.1006/JMAA.1998.5758
description: scientific article; zbMATH DE number 1115472