Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 46782, 25 pages

The optimization for the inequalities of power means

Jiajin Wen and Wan-Lan Wang

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.


Let Mn[t](a) be the tth power mean of a sequence a of positive real numbers, where a=(a1,a2,,an),n2, and α,λ++m,m2,j=1mλj=1,min{α}θmax{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.