Schur Convexity with Respect to a Class of Symmetric Functions and Their Applications

For x = (x1, x2, · · · , xn) ∈ R+, the symmetric function φn(x, r) is defined by φn(x, r) = φn(x1, x2, · · · , xn; r) = ∏ 1≤i1<i2···<ir≤n  r ∑ j=1 1 + xij xij  , where r = 1, 2, · · · , n, and i1, i2, · · · , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn(x, r) are discussed. As applications, some inequalities, including Weierstrass inequality, are established by use of the theory of majorization.