On a discrete norm of the self-inversive unimodular polynomials

Abstract

The quotient of $$\left\| P \right\|_{L_4 } $$ divided by $$\left\| P \right\|_{L_2 } $$ , whereP is a self-inversive and unimodular polynomial of any degree, dominates an absolute constantK>1. A 1989 paper gaveK=1.0252 on which its authors conjetured that the best constant is $$\sqrt[4]{{3/2}}$$ . We supply counter examples to their claim and provide a partial result for whenever theL q norm is replaced by some “discrete” type norm.