Schur convexity properties for a class of symmetric functions with applications

In the article, we prove that the symmetric function Fn(x1,x2,⋯,xn;r)=∑1≤i1 i2 ⋯ ir≤nr∏j=1(1+xij1−xij)1/r is Schur convex, Schur multiplicatively convex and Schur harmonic convex on [ 0 , 1 ) n , and establish several new analytic inequalities by use of the theory of majorization, where r ∈ { 1 , 2 , ⋯ , n } and i 1 , i 2 , ⋯ i n are integers.