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 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.