  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").
