Schur-convexity of the complete symmetric function

This paper investigates Schur-convexity of the complete symmetric function c(r)(x) = Sigma(i1+...+in=r)x(1)(i1)...x(n)(in) and the function phi(r)(x) = C-r(x)/Cr-1(x), where i(1),...i(n) are non-negative integers and r >= 1. Some inequalities, including Ky Fan type inequality, are established by use of the theory of majorization. It is also concerned with an open problem proposed by Menon [1].