两类对称函数的Schur凸性

本文用一种新方法研究两类对称函数的Schur凸性.首先,对x=(x1,...,xn)∈(-∞,1)n∪(1,+∞)n和r∈{1,2,...,n},讨论Guan(2007)定义的对称函数Fn(x,r)=Fn(x1,x2,...,xn;r)=∑1≤i1≤i2≤···≤ir≤n r∏j=1xij/(1-xij)的Schur凸性,其中i1,i2,...,in为正整数;推广褚玉明等人(2009)的主要结果,因而用新方法推广并解决Guan(2007)提出的一个公开问题.然后,对x=(x1,...,xn)∈(-∞,1)n∪(1,+∞)n和r∈{1,2,...,n},研究本文定义的对称函数Gn(x,r)=Gn(x1,x2,...,xn;r)=∑1≤i1≤i2≤···≤ir≤n(r∏j=1xij/(1-xij))1/r的Schur凸性、Schur乘性凸性和Schur调和凸性,其中i1,i2,...,in为正整数.作为应用,用Schur凸函数自变量的双射变换得到其他几类对称函数的Schur凸性,用控制理论建立一些不等式,特别地,由此给出Sharpiro不等式和Ky Fan不等式一个共同的推广,导出Safta猜想在高维空间的推广.

The Schur convexity for two classes of symmetric functions

The main aim of this paper is to investigate the Schur convexity for two classes of symmetric functions by using a new method. Firstly, for x =（x1,..., xn） ∈（-∞, 1）n∪（1, ＋∞）n and r ∈ {1, 2,..., n}, we discuss the Schur convexity of the symmetric functions Fn（x, r）, which is defined by Guan（2007） as Fn（x,r）=Fn（x1,x2,...,xn;r）=∑1≤i1≤i2≤···≤ir≤n r∏j=1xij/（1-xij）where i1, i2,..., in are positive integers, generalize the main results of Chu et al.（2009）, thus by using a new method, generalize and solve an open problem proposed by Guan（2007）. Then, for x =（x1,..., xn） ∈（-∞, 1）n ∪ （1, ＋∞）n and r ∈ {1, 2,..., n}, we investigate the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of the symmetric function Gn（x, r）, which is defined by where i1, i2,..., in are positive integers. As applications, by a bijective transformation of independent variable for a Schur convex function, we obtain the Schur convexity for some other symmetric functions, and establish some inequalities by use of the theory of majorization, in particular, we give from our results a common generalization of the Sharpiro’s inequality and the Ky Fan’s inequality, and derive a generalization of Safta’s conjecture in the n-dimensional space and others.