Schur-convexity of the complete elementary symmetric function

We prove that the complete elementary symmetric function c(r) = c(r) (x) = C-n([r]) (x) = Sigma i(1+...+in= r) x(1)(i1)...x(n)(in) and the function phi(r)(x) = c(r)(x)/c(r-1)(x) are Schur-convex functions in R-+(n) = {(x(1), x(2),...,x(n)) vertical bar x(i) > 0}, where i(1), i(2),...,i(n) are nonnegative integers, r is an element of N = {1, 2,...}, i = 1, 2,...,n. For which, some inequalities are established by use of the theory of majorization. A problem given by K. V. Menon (Duke Mathematical Journal 35 (1968), 37-45) is also solved. Copyright (C) 2006 Kaizhong Guan. 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.