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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.