Flatness of Conjugate Reciprocal Unimodular Polynomials

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) = ∑n j=0 ajz j is called conjugate reciprocal if an−j = aj , 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 that max z∈∂D |f(z)| ≥ (1 + ε) √ 4/3m , for every conjugate reciprocal unimodular polynomial of degree m. We also prove that there is an absolute constant ε > 0 such that There is an absolute constant ε > 0 such that Mq(f ) ≤ exp(ε(q − 2)/q) √ 1/3m, 1 ≤ q < 2 , and Mq(f ) ≥ exp(ε(q − 2)/q) √ 1/3m, 2 < q , for every conjugate reciprocal unimodular polynomial of degree m, where