Polarization of an inequality

Abstract

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").

arXiv comment9 pages. Updated references and remarks in section 3. Accepted in Math. Inequalities and Appl