The property of a new class of symmetric functions with applications

The main aim of the article is to prove that the symmetric function Φn(x,r)=∏i1+i2+⋯+in=r(x1i1+x2i2+⋯+xnin) is Schur geometrically convex for x∈R++n and fixed r∈N+={1,2,⋯}, where i1,i2,⋯,in are non-negative integers. Further, we obtain Φn(x,r) is also Schur m-power convex for m≤0. As applications, a Klamkin-Newman type inequality is derived. Finally, we list a counter example to illustrate Φn(x,r) is neither Schur convex nor Schur concave.