Flatness of conjugate reciprocal unimodular polynomials

Abstract

A polynomial is called unimodular if each of its coefficients is a complex number of modulus 1. A polynomial P of the form P(z)=∑j=0najzj is called conjugate reciprocal if an−j=a‾j, aj∈C for each j=0,1,…,n. Let ∂D be the unit circle of the complex plane. We prove that there is an absolute constant ε>0 such thatmaxz∈∂D⁡|f(z)|≥(1+ε)4/3m1/2, for every conjugate reciprocal unimodular polynomial of degree m. We also prove that there is an absolute constant ε>0 such thatMq(f′)≤exp⁡(ε(q−2)/q)1/3m3/2,1≤q<2, andMq(f′)≥exp⁡(ε(q−2)/q)1/3m3/2,2<q, for every conjugate reciprocal unimodular polynomial of degree m, whereMq(g)=(12π∫02π|g(eit)|qdt)1/q,q>0.