The Schur harmonic convexity for a class of symmetric functions

In this article, we prove that the symmetric function Fn*(x,r)=∑i1+i2+⋯+in=r(x1i1x2i2⋯xnin)1r is Schur harmonic convex for x∞R+n and r ∞ N = {1, 2, 3,···}. As its applications, some analytic inequalities are established.