The sum of squared logarithms inequality in arbitrary dimensions

Lev Borisov, Patrizio Neff, Suvrit Sra, Christian Thiel

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x,y∈Rn whose elementary symmetric polynomials satisfy ek(x)≤ek(y) (for 1≤k<n) and en(x)=en(y), the inequality ∑i(log⁡xi)2≤∑i(log⁡yi)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⊆Cn→R with f(z)=∑i(log⁡zi)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 languageEnglish (US)
Pages (from-to)124-146
Number of pages23
JournalLinear Algebra and Its Applications
StatePublished - Sep 1 2017

All Science Journal Classification (ASJC) codes

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


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


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