A Hardy-Littlewood integral inequality on finite intervals with a concave weight
type: Item
instance of: scholarly article
title: A Hardy-Littlewood integral inequality on finite intervals with a concave weight
zbMATH Open document ID: 1363.26024
author: Horst Alzer
author: Man Kam Kwong
published in: Periodica Mathematica Hungarica
publication date: 2016-11-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.00807
review text: The authors derive and prove that for all concave functions \( w:[a,b]\rightarrow [ 0,\infty)\) and for all functions \(f\in C^{2}[a,b] \) with \(f(a)=f(b)=0\) that the following inequality holds \[ \left( \int\nolimits_{a}^{b}w(x)f^{\prime}(x)^{2}dx\right) ^{2}\leq \left( \int\nolimits_{a}^{b}w(x)f(x)^{2}dx\right) \left( \int\nolimits_{a}^{b}w(x)f^{^{\prime \prime}}(x)^{2}dx\right) .\tag{1} \] Furthermore, an example is constructed with the help of MAPLE software to show that the concavity assumption above cannot be dropped and that (1) does not hold for convex weight functions.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 6656709
zbMATH Keywords: integral inequality
zbMATH Keywords: HELP-type inequality
zbMATH Keywords: concave weight function
describes a project that uses: Maple
MaRDI profile type: Publication
OpenAlex ID: W1474687133
arXiv ID: 1507.00807
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DOI: 10.1007/S10998-015-0096-X
description: scientific article; zbMATH DE number 6656709