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.

Abstract

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)=∑1≤i1<i2⋯<ir≤n∏j=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.