The sum of squared logarithms inequality in arbitrary dimensions

被引：4

作者：

Borisov, Lev ^{[1
]}

Neff, Patrizio ^{[2
]}

Sra, Suvrit ^{[3
]}

Thiel, Christian ^{[2
]}

机构：

[1] Rutgers State Univ, Dept Math, 240 Hill Ctr, Newark, NJ 07102 USA

[2] Univ Duisburg Essen, Lehrstuhl Nichtlineare Anal & Modellierung, Fak Math, Thea Leymann Str 9, D-45127 Essen, Germany

[3] MIT, Lab Informat & Decis Syst, 77 Massachusetts Ave, Cambridge, MA 02139 USA

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2017年 / 528卷

关键词：

Elementary symmetric polynomials; Fundamental theorem of algebra; Polynomials; Geodesics; Hencky energy; Logarithmic strain tensor; Positive definite matrices; MINIMIZATION; PROPERTY;

D O I：

10.1016/j.laa.2016.06.026

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y is an element of R-n whose elementary symmetric polynomials satisfy e(k) (x) <= e(k) (V) (for 1 <= k < n) and e(n)(x) = e(n)(y), the inequality Sigma(i)(logx(i))(2) < Sigma i(log y(i))(2) holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M subset of C-n with f(z) = Sigma(i)(log z(i))(2) has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. This property leads to our proof. We conclude by providing applications and wider connections of the SSLI. (C) 2016 Elsevier Inc. All rights reserved.

引用

页码：124 / 146

页数：23

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