Cauchy不等式和Kantorovich不等式的推广  

Generaligations of Cauchy Inequality and Kantorovich Inequality

作　　者：刘建忠[1] 

机构地区：[1]宁夏大学数学与计算机学院,宁夏银川750021

出　　处：《河北大学学报（自然科学版）》2004年第3期240-242,共3页Journal of Hebei University(Natural Science Edition)

摘　　要：设A为n×n正定Hermite阵,x为n维列向量,λ1≥λ2≥…≥λn>0为A的特征值,得到了Cauchy不等式及Kantorovich不等式的如下推广形式:(x Aα1+α2+…+αkkx)k≤x Aα1xx Aα2x…x Aαkx,其中α1,α2,…,αk为任意实数.(λα+βn)α+β1-λα+β(x Aαx)β(x A-βx)α≤ααββ(α+β)α+β(λ1λn)αβ(λα1-λαn)α(λβ1-λβn)β(x x)α+β.其中α,β为任意正数.Let A be an n×n positive definite Hermite Matrix,x be a vector of dimension n,and λ_1≥λ_2≥ …≥λ_n>0 are eigenevalues of A,the generaliged Cauchy inequality and Kantorovich inequality are obtained as (follows) (x~*A^(α_1+α_2+…+α_kk)x)~k≤x~*A^(α_1)xx~*A^(α_2)x…x~*A^(α_k)x, where α_1,α_2,…,α_k are any real numbers (x~*A~αx)~β(x~*A^(-β)x)~α≤α~αβ~β(α+β)^(α+β)(λ^(α+β)_1-λ^(α+β)_n)^(α+β)(λ_1λ_n)^(αβ)(λ~α_1-λ~α_n)~α(λ~β_1-λ~β_n)~β(x~*x)^(α+β). were α,β are any positive numbers.

关 键 词：CAUCHY不等式 KANTOROVICH不等式 Jti阵函数 

分 类 号：O151.21[理学—数学]