Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 21572, 7 pages

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

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.


We prove the following let α,β,a>0, and b>0 be real numbers, and let wj(j=1,,n;n2) be positive real numbers with w1++wn=1. The inequalities αj=1nwj/(1pja)j=1nwj/(1pj)j=1nwj/(1+pj)βj=1nwj/(1pjb) hold for all real numbers pj[0,1)(j=1,,n) if and only if αmin(1,a/2) and βmax(1,(1min1jnwj/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.