Weighted integral inequalities with the geometric mean operator
type: Item
instance of: scholarly article
title: Weighted integral inequalities with the geometric mean operator
zbMATH Open document ID: 1024.26008
DOI: 10.1155/S1025583402000371
author: Lars-Erik Persson
author: V. D. Stepanov
published in: Journal of Inequalities and Applications
publication date: 2003-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/122998
review text: The geometric mean operator is defined by \(Gf(x)=\exp\left( \frac{1}{x} \int_{0}^{x}\log f(t)dt\right) \) for locally integrable \(f\geq 0.\) Given two weights \(u,v\geq 0\) on \((0,\infty)\) and two indices \(0<p,q<\infty,\) one associates to them the function \(w=\left( G(1/v)\right) ^{q/p}u.\) The paper under review contains two-sided estimates (in both cases \(p\leq q\) and \(q<p\)) of the best constant \(C\) such that \[ \left( \int_{0}^{\infty}(Gf)^{q}udx\right) ^{1/q}\leq C\left( \int _{0}^{\infty}f^{p}v dx\right) ^{1/p} \] for all \(f\in L^{p}(vdx),\) \(f\geq 0.\) For example, if \(p\leq q\) and \[ D=\sup_{t>0}t^{-1/p}\left( \int_{0}^{t}w(x)dx\right) ^{1/q}<\infty, \] then a best constant \(C\) exists and \(D\leq C\leq e^{1/p}D.\) The factor \(e^{1/p}\) is best possible for \(p=q\) and attains for \(u=v=1.\)
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 1910309
zbMATH Keywords: geometric mean operator
zbMATH Keywords: Hardy inequalities
zbMATH Keywords: weighted inequalities
MaRDI profile type: Publication
OpenAlex ID: W2112841484
description: scientific article; zbMATH DE number 1910309