Some conditions characterizing the ``reverse'' Hardy inequality
type: Item
instance of: scholarly article
title: Some conditions characterizing the ``reverse'' Hardy inequality
zbMATH Open document ID: 1170.26011
author: Elsie Sterbin Gottlieb
published in: Real Analysis Exchange
publication date: 2008-06-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rae/1209398392
review text: The authors obtain necessary and sufficient conditions for the ``reverse'' Hardy inequalities to hold, in the case \( -\infty < q \leq p < 0\). Let us precise that the ``reverse'' Hardy inequalities are given by \[ \begin{aligned} {\left( \int_a^b f^p(x)v(x) dx \right)}^{\frac{1}{p}} &\leq C {\left(\int_a^b {\left( \int_a^x f(t) dt \right)}^q u(x) dx \right)}^{\frac{1}{q}} \\ \noalign{\text{and}} \\ {\left( \int_a^b f^p(x)v(x) dx \right)}^{\frac{1}{p}} &\leq C {\left(\int_a^b {\left( \int_x^b f(t) dt \right)}^q u(x) dx \right)}^{\frac{1}{q}} \end{aligned} \] for \(f \geq 0\), and where the weight functions \(u, \, v\) has been completely characterized by \textit{P. R. Beesack} and \textit{H. P. Heinig} [Proc. Am. Math. Soc. 83, 532--536 (1981; Zbl 0484.26010)] for \(p, q < 1\), and by \textit{D. V. Prokhorov} [Publ. Mat., Barc. 48, No. 2, 423--443 (2004; Zbl 1068.26022)] for \(p, q < 0 \) and \(p, q \in (0,1) \). \textit{A. Kufner} and \textit{K. Kuliev} [Adv. Alg. Anal. 1, No. 3, 219--228 (2006; Zbl 1170.26307)] also obtained conditions in the case \(- \infty < q \leq p < 0 \).
reviewed by: F. Colin
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 5291268
zbMATH Keywords: inequalities
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: weigths
zbMATH Keywords: scales of weight characterizations
zbMATH Keywords: Hardy operator
zbMATH Keywords: continuity
zbMATH Keywords: equivalent integral conditions
zbMATH Keywords: comparisons
MaRDI profile type: Publication
OpenAlex ID: W1565403447
DOI: 10.14321/REALANALEXCH.33.1.0249
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description: scientific article; zbMATH DE number 5291268