华罗庚不等式的上界与下界的研究

The Generalization of Hua Loo-Keng Inequalities of Determinants

在多复变分析的研究中,华罗庚发现并证明了行列式不等式det(I-AAH)det(I-BBH)≤|det(I-ABH)|2,其中n×n复矩阵A,B满足I-AAH,I-BBH都是Hermitian正定矩阵.本文从一个矩阵恒等式的应用出发,给出了较为精细的华罗庚不等式的新的上界和下界:det(I-AAH)det(I-BBH)+|det(A-B)|2+(2n-2)|det(A-B)|[det(I-AAH)det(I-BBH)]21≤|det(I-ABH)|2≤det(I+AAH)det(I+BBH)+(22n-1-2n+1+1)|det(A+B)|2-(2n-2)|det(A+B)[(22(n-1)-2n)|det(A+B)|2+det(I+AAH)det(I+BBH)]21.

In the study of the functions of several complex variables, Hua Loo-Keng discovered and proved the following determinant inequality： If A,B are n× n complex matrices and I-AA^H and I-BB^H are Hermitian positive definite matrices,then det（I-AA^H）det（I-BB^H）≤｜det（I-AB^H）｜^2 From an application of a matrix identity,newly tight upper bound and lower bound were presented for Hua Loo -Keng inequalities of determinants： det（I-AA^H）det（I-BB^H）＋｜det（A-B）｜^2＋（2^n-2）｜det（A-B）｜[det（I-AA^H）det（I-BB^H）]^1/2≤｜det（I-AB^H）｜^2≤det（I＋AA^H）det（I＋BB^H）＋（2^2n-1 -2^n＋1 ＋1）｜det（A＋B）｜^2-（2^n -2）｜det（A＋B）[（2^2（n-1） -2^n）｜det（A＋B）｜^2＋det（I＋AA^H）det（I＋BB^H）]^1/2.