Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 46782, 25 pages
doi:10.1155/JIA/2006/46782
The optimization for the inequalities of power means
Department of Mathematics and Computer Science, Chengdu University, Chengdu 610106, Sichuan, China
Received 14 November 2005; Revised 5 May 2006; Accepted 14 July 2006
Copyright © 2006 Jiajin Wen and Wan-Lan Wang. 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
Let Mn[t](a) be the tth power mean of a sequence a of positive real numbers, where a=(a1,a2,…,an),n≥2, and α,λ∈ℝ++m,m≥2,∑j=1mλj=1,min{α}≤θ≤max{α}. In this paper, we will state the important background and meaning
of the inequality ∏j=1m{Mn[αj](a)}λj≤(≥)Mn[θ](a); a necessary and
sufficient condition and another interesting sufficient condition
that the foregoing inequality holds are obtained; an open problem
posed by Wang et al. in 2004 is solved and generalized;
a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also
obtained. Our methods used are the procedure of descending
dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.