一个多变量对称函数的Schur凸性性质及其应用（英文）

Schur Convexity Properties for A Multivariable Symmetric Function and Its Applications

给出了关于向量x=(x 1,x2,…,xn)∈R_+~n和正整数r∈{1,2,…,n}的多变量对称函数φ_n(x,r)=φ_n(x_1,x_2,…,x_n;r)=∑1≤i_1<i_2<…<i_r≤n((r∏j=11+x_(i_j/x_(i_j)))^(1/r)的Schur凸性,Schur乘法凸性和Schur调和凸性性质,并利用控制理论建立若干新的解析不等式,其中i_1,i_2,…,i_r为正整数.

For x =(x_1,x_2,…,x_n)∈R_+~nand r ∈ {1,2,…,n},the symmetric function φ_n(x,r) is defined as φ_n(x,r)=φ_n(x_1,x_2,…,x_n;r)=∑1≤i_1<i_2<…<i_r≤n((r∏j=11+x_(i_j/x_(i_j)))^(1/r) where i_1,i_2,…,irare positive integers.In this paper,the Schur convexity,Schur multiplicative convexity and Schur harmonic convexity ofφn(x,r)are discussed.As applications,some new analytic inequalities are established by use of the theory of majorization.