Polarization of an inequality

Ivo Klemes

Published 2009-05-09, updated 2011-01-09Version 2

We generalize a previous inequality related to a sharp version of the Littlewood conjecture on the minimal $L_1$-norm of $N$-term exponential sums $f$ on the unit circle. The new result concerns replacing the expression $\log(1+t|f|^2)$ with $\log (\sum_{k=1}^K t_k|f_{k}|^2)$. The proof occurs on the level of finite Toeplitz matrices, where it reduces to an inequality between their polarized determinants (or "mixed discriminants").