An integral inequality of convolution type
type: Item
instance of: scholarly article
title: An integral inequality of convolution type
zbMATH Open document ID: 0787.26015
DOI: 10.1007/BF01834008
author: W. Walter
author: V. Weckesser
published in: Aequationes Mathematicae
publication date: 1994-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/137554
review text: The authors prove that if \(f\) is nondecreasing and nonnegative, \(g\) is convex and the function \(y\mapsto h_ c(y)= g(cy)-\int_ 0^ c k(s) ds g(y)\) is nonnegative and nondecreasing for each \(c\in (0,b]\), or, if \(f\) is nonincreasing and nonnegative, \(g\) is concave and \(y\mapsto h_ c(y)\) is nonnegative and nonincreasing for each \(c\in(0,b]\), then \[ \int^ x_ 0 k(x-s)g(f(s)) ds\leq g\left(\int^ x_ 0 f(s) ds\right),\tag{*} \] when \(0<x\leq b\). Some special cases are considered in a number of corollaries. Furthermore, it is shown that if \(f\) is nonnegative and nondecreasing, \(g\) is nondecreasing and concave with \(\int^ x_ 0 k(s) ds g(y)\leq g(yx)\) and \(\int^ x_ 0 k(s) ds\leq x\) for \(0\leq x\leq 1\) and \(y\geq 0\), then (*) holds when \(0< x\leq 1\).
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 493223
zbMATH Keywords: integral inequality
zbMATH Keywords: convolution
MaRDI profile type: Publication
OpenAlex ID: W1995956260
description: scientific article; zbMATH DE number 493223