Generalized Steffensen inequalities and their optimal constants
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.26014
author: P. J. Rabier
publication date: 2012-08-10T00:00:00Z
full work available at URL: jca19018.htm
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 6064738
zbMATH Keywords: convexity
zbMATH Keywords: Steffensen's inequality
zbMATH Keywords: weighted integrals
MaRDI profile type: Publication
title: Generalized Steffensen inequalities and their optimal constants
published in: Journal of Convex Analysis
review text: Let \(F: [0,\infty)\to\mathbb R\) be convex and continuous, with \(F(0)= 0\). For \(p> 1\) put \(q= p/(p-1)\). It is proved that, NEWLINE\[NEWLINEF\Biggl(\int^\infty_0 f\,dr\Biggr)\leq C.NEWLINE\]NEWLINE \(\int^\infty_0 f(f) F'(r^{1/q})\,dr\) holds for every \(f\geq 0\) in the unit of \(L^p(0,\infty)\) and with the constant \(C= 1\). Here, in general, both sides may be \(\pm\infty\). Related inequalities for \(f\) in \(L^1(\mathbb R^N)\cap L^p(\mathbb R^N)\), \(f\geq 0\) are also derived. The next aim of the paper is to identify the range of admissible constants \(C\) and, in particular, to characterize the optimal constant when \(F\geq 0\) or \(F\leq 0\). The results are a little complicated to be stated here.
reviewed by: József Sándor
description: scientific article; zbMATH DE number 6064738