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The sum of squared logarithms inequality in arbitrary dimensions
Author(s) -
Borisov Lev,
Neff Patrizio,
Sra Suvrit,
Thiel Christian
Publication year - 2016
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201610321
Subject(s) - mathematics , logarithm , combinatorics , extension (predicate logic) , function (biology) , logarithmic mean , discrete mathematics , mathematical analysis , evolutionary biology , computer science , biology , programming language
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 (log x i ) 2 ≤ ∑ 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 ⊆ ℂ n → ℝ with f ( z ) = ∑ i (log z i ) 2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z . We conclude by providing applications and wider connections of the SSLI. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)