Steffensen pairs and associated inequalities
type: Item
instance of: scholarly article
title: Steffensen pairs and associated inequalities
zbMATH Open document ID: 0952.26012
DOI: 10.1155/S1025583400000047
author: Hillel Gauchman
published in: Journal of Inequalities and Applications
publication date: 2001-01-14T00:00:00Z
full work available at URL: https://eudml.org/doc/120688
review text: The author introduces the following definition. Let \(f: [c,\infty)\to [0,\infty)\), \(c\geq 0\), and \(g: (0,\infty)\to (0,\infty)\) be two strictly increasing functions. \((f,g)\) is called a \textit{Steffensen pair} on \([c,\infty)\) if the following is satisfied: If \(x_1,\dots, x_n\) are real numbers such that \(x_i\geq c\) for all \(i\), \(A\) and \(B\) are positive real numbers, and \(\sum x_i\leq A\), \(\sum f(x_i)\geq f(B)\), then for any \(k\in \{1,\dots, n\}\) such that \(k\geq g(A/B)\) there are \(k\) numbers among \(x_1,\dots, x_n\) whose sum is \(\geq B\). \textit{J.-C. Evard} and the author [Analysis 17, No. 2, 301-322 (1997; Zbl 0887.26013)] proved that for \(a\geq 2\), \((x^a,x^{1/(a- 1)})\) is a Steffensen pair on \([0,\infty)\). In this paper a general method for constructing Steffensen pairs is described and examples are given.
reviewed by: Ioan Raşa
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1437136
zbMATH Keywords: inequalities
zbMATH Keywords: strictly increasing functions
MaRDI profile type: Publication
OpenAlex ID: W2079487931
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description: scientific article; zbMATH DE number 1437136