Open Access

Schur-convexity of the complete elementary symmetric function

Journal of Inequalities and Applications20062006:67624

DOI: 10.1155/JIA/2006/67624

Received: 2 October 2004

Accepted: 27 January 2005

Published: 7 February 2006

Abstract

We prove that the complete elementary symmetric function https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F67624/MediaObjects/13660_2004_Article_1627_IEq1_HTML.gif and the function https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F67624/MediaObjects/13660_2004_Article_1627_IEq2_HTML.gif are Schur-convex functions in https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F67624/MediaObjects/13660_2004_Article_1627_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F67624/MediaObjects/13660_2004_Article_1627_IEq4_HTML.gif are nonnegative integers, https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F67624/MediaObjects/13660_2004_Article_1627_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F67624/MediaObjects/13660_2004_Article_1627_IEq6_HTML.gif . For which, some inequalities are established by use of the theory of majorization. A problem given by K. V. Menon (Duke Mathematical Journal 35 (1968), 37–45) is also solved.

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Authors’ Affiliations

(1)
Department of Mathematics and Physics, Nanhua University

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Copyright

© Kaizhong Guan 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.