Steffensen pairs and associated inequalities
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.
scientific article; zbMATH DE number 1437136