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∈Rⁿ 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⊆Cⁿ→R with f(z)=∑_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.