Open Access

The optimization for the inequalities of power means

Journal of Inequalities and Applications20062006:46782

DOI: 10.1155/JIA/2006/46782

Received: 14 November 2005

Accepted: 14 July 2006

Published: 19 October 2006

Abstract

Let https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F46782/MediaObjects/13660_2005_Article_1601_IEq1_HTML.gif be the https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F46782/MediaObjects/13660_2005_Article_1601_IEq2_HTML.gif th power mean of a sequence https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F46782/MediaObjects/13660_2005_Article_1601_IEq3_HTML.gif of positive real numbers, where https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F46782/MediaObjects/13660_2005_Article_1601_IEq4_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F46782/MediaObjects/13660_2005_Article_1601_IEq5_HTML.gif . In this paper, we will state the important background and meaning of the inequality https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F46782/MediaObjects/13660_2005_Article_1601_IEq6_HTML.gif ; 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.

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Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Chengdu University

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Copyright

© J.Wen andW.-L.Wang. 2006

This article is published under license to BioMed Central Ltd. 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.