一类对称函数的Schur凸性及其应用

对x=(x_1,…,x_n)∈0,1)~n∪(1,+∞o)~n,定义对称函数■其中r∈N,i_1,i_2,…,i_n为非负整数.研究了F_n(x,r)的Schur凸性、Schur乘性凸性和Schur调和凸性.作为应用,用控制理论建立了一些不等式,特别地,给出了高维空间的一些新的几何不等式.

Schur Convexity of a Class of Symmetric Functions with Its Applications

For $x=(x_1, \cdots, x_n)\in 0, 1)^n\cup (1, +\infty)^n$, the symmetric function $F_n(x,r)$ is defined by $$ {F_n}(x,r)={F_n}(x_1, x_2, \cdots, x_n; r)=\sum_{i_1+i_2+\cdots+i_n=r}\Big({\frac{1+x_1}{1-x_1}}\Big)^{i_1} \Big({\frac{1+x_2}{1-x_2}}\Big)^{i_2}\cdots\Big({\frac{1+x_n}{1-x_n}}\Big)^{i_n}, $$ where $r\in\mathbb{N}$, and $i_1, i_2, \cdots , i_n$ are non-negative integers. In this paper, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of ${F_n}(x,r)$ are investigated. As applications, the authors establish some inequalities by use of the theory of majorization. In particular, the authors give some new geometric inequalities in the $n$-dimensional space.