The sum of squared logarithms inequality in arbitrary dimensions

Abstract

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.