Polarization of an inequality

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 $\log (1+t\mathrm{\mid }f{\mathrm{\mid }}^2)$ with $\log ({\sum }_{k=1}^K{t}_k\mathrm{\mid }{f}_k{\mathrm{\mid }}^2)$. The proof occurs on the level of finite Toeplitz matrices, where it reduces to an inequality between their polarized determinants (or “mixed discriminants”).