Journal of Inequalities and Applications 
Volume 5 (2000), Issue 1, Pages 53-61
doi:10.1155/S1025583400000047

Steffensen pairs and associated inequalities

Hillel Gauchman

Department of Mathematics, Eastern Illinois University, Charleston 61920, IL, USA

Received 22 February 1999; Revised 12 April 1999

Abstract

Let x1,…,xn be positive numbers and α≥2. It is known that if ∑i=1nxi≤A, ∑i=1nxiα≥Bα, then for any k such that k≥(A/B)1/(α−1), there are k numbers among x1,…,xn whose sum is bigger than or equal to B. We express this statement saying that a pair of functions (xα,x1/(α−1)) is a Steffensen pair. In this paper we show how to find many Steffensen pairs.