Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 493759, 15 pages
Research Article

Schur-Convexity for a Class of Symmetric Functions and Its Applications

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

Received 16 May 2009; Accepted 14 September 2009

Academic Editor: Jozef Banas

Copyright © 2009 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)R+n, the symmetric function ϕn(x,r) is defined by ϕn(x,r)=ϕn(x1,x2,,xn;r)=1i1<i2<irn(j=1r(xij/(1+xij)))1/r, where r=1,2,,n and i1,i2,,in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of ϕn(x,r) are discussed. As applications, some inequalities are established by use of the theory of majorization.