Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 21572, 7 pages
doi:10.1155/JIA/2006/21572

The inequality of Milne and its converse II

Horst Alzer1 and Alexander Kovačec2

 1Morsbacher Street 10, Waldbröl 51545, Germany
2Departamento de Matemática, Universidade de Coimbra, Coimbra 3001-454, Portugal
Received 15 September 2004; Accepted 19 September 2004

Abstract

We prove the following let α,β,a>0, and b>0 be real numbers, and let wj(j=1,…,n;n≥2) be positive real numbers with w1+…+wn=1. The inequalities α∑j=1nwj/(1−pja)≤∑j=1nwj/(1−pj)∑j=1nwj/(1+pj)≤β∑j=1nwj/(1−pjb) hold for all real numbers pj∈[0,1)(j=1,…,n) if and only if α≤min⁡(1,a/2) and β≥max⁡(1,(1−min⁡1≤j≤nwj/2)b). Furthermore, we provide a matrix version. The first inequality (with α=1 and a=2) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.