Schur convexity for two classes of symmetric functions and their applications

Abstract

For x = (x ₁, x ₂, ⋯, x _n ) ∈ ℝ ⁿ₊ ∪ ℝ ⁿ₋ , the symmetric functions F _n (x, r) and G _n (x, r) are defined by

$$F_n (x,r) = F_n (x_1 ,x_2 , \cdots ,x_n ;r) = \sum\limits_{1 \leqslant i_1 < i_2 < \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 + x_{i_j } }} {{x_{i_j } }}} }$$

and

$$G_n (x,r) = G_n (x_1 ,x_2 , \cdots ,x_n ;r) = \sum\limits_{1 \leqslant i_1 < i_2 < \cdots < i_r \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - x_{i_j } }} {{x_{i_j } }}} } ,$$

respectively, where r = 1, 2, ⋯, n, and i ₁, i ₂, ⋯, i _n are positive integers. In this paper, the Schur convexity of F _n (x, r) and G _n (x, r) are discussed. As applications, by a bijective transformation of independent variable for a Schur convex function, the authors obtain Schur convexity for some other symmetric functions, which subsumes the main results in recent literature; and by use of the theory of majorization establish some inequalities. In particular, the authors derive from the results of this paper the Weierstrass inequalities and the Ky Fan’s inequality, and give a generalization of Safta’s conjecture in the n-dimensional space and others.