| Cauchy不等式和Kantorovich不等式的推广 |
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| 引用本文： | 刘建忠.Cauchy不等式和Kantorovich不等式的推广[J].河北大学学报(自然科学版),2004,24(3):240-242. |
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| 作者姓名： | 刘建忠 |
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| 作者单位： | 宁夏大学,数学与计算机学院,宁夏,银川,750021 |
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| 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.
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| 关 键 词： | Cauchy不等式 Kantorovich不等式 矩阵函数 |
| 文章编号： | 1000-1565(2004)03-0240-03 |
| 修稿时间： | 2003年12月31 |
Generaligations of Cauchy Inequality and Kantorovich Inequality |
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| LIU Jian-zhong.Generaligations of Cauchy Inequality and Kantorovich Inequality[J].Journal of Hebei University (Natural Science Edition),2004,24(3):240-242. |
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| Authors: | LIU Jian-zhong |
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| Abstract: | 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. |
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| Keywords: | Cauchy inequality Kantorovich inequality matrix function |
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