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
1Morsbacher Street 10, Waldbröl 51545, Germany
2Departamento de Matemática, Universidade de Coimbra, Coimbra 3001-454, Portugal
Received 15 September 2004; Accepted 19 September 2004
Copyright © 2006 Horst Alzer and Alexander Kovačec. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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−min1≤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.