Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 543250, 12 pages
Research Article

On Schur Convexity of Some Symmetric Functions

1School of Teachers Education, Huzhou Teachers College, Huzhou 313000, China
2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 20 November 2009; Accepted 3 March 2010

Academic Editor: Shusen Ding

Copyright © 2010 Wei-Feng Xia and Yu-Ming Chu. 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.


For x=(x1,x2,,xn)(0,1)n and r{1,2,,n}, the symmetric function Fn(x,r) is defined as Fn(x,r)=Fn(x1,x2,,xn;r)=1i1<i2<irnj=1r((1+xij)/(1-xij)), where i1,i2,,in are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of Fn(x,r) are discussed. As consequences, several inequalities are established by use of the theory of majorization.