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.

Abstract

For x=(x1,x2,…,xn)∈R+n, the symmetric function ϕn(x,r) is defined by ϕn(x,r)=ϕn(x1,x2,…,xn;r)=∏1≤i1<i2⋯<ir≤n(∑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.