The sum of squared logarithms inequality in arbitrary dimensions

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)