## Abstract

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x,y∈R^{n} whose elementary symmetric polynomials satisfy e_{k}(x)≤e_{k}(y) (for 1≤k<n) and e_{n}(x)=e_{n}(y), the inequality ∑_{i}(logx_{i})^{2}≤∑_{i}(logy_{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⊆C^{n}→R with f(z)=∑_{i}(logz_{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.

Original language | English (US) |
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Pages (from-to) | 124-146 |

Number of pages | 23 |

Journal | Linear Algebra and Its Applications |

Volume | 528 |

DOIs | |

State | Published - Sep 1 2017 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

## Keywords

- Algebraic geometry
- Elementary symmetric polynomials
- Fundamental theorem of algebra
- Geodesics
- Hencky energy
- Logarithmic strain tensor
- Matrix analysis
- Polynomials
- Positive definite matrices