Some properties of a class of symmetric functions and its applications

摘要

For x = (x_1, ..., x_n) ∈ [0, 1)~n {union} (1,+∞)~n, the symmetric functions F_n(x, r) are defined by F_n(x, r) = F_n(x_1, x_2, ..., x_n; r) =∑(x_1/1-x_1)~(i_1)(x_2/1-x_2)~(i_2)…(x_n/1-x_n)~(i_n), (i_1+i_2+…i_n=r) where r = 1, 2, ..., n, ..., and i_1, i_2, ..., i_n are non-negative integers. In this paper, the Schur convexity, geometric Schur convexity and harmonic Schur convexity of F_n(x, r) are investigated. As applications, Schur convexity for the other symmetric functions is obtained by a bijective transformation of independent variable for a Schur convex function, some analytic and geometric inequalities are established by using the theory of majorization, in particular, we derive from our results a generalization of Sharpiro's inequality, and give a new generalization of Safta's conjecture in the n-dimensional space and others.