On the derivatives of unimodular polynomials

Abstract

Let be the open unit disk of the complex plane; its boundary, the unit circle of the complex plane, is denoted by . Let denote the set of all algebraic polynomials of degree at most with complex coefficients. For , let The class is often called the collection of all (complex) unimodular polynomials of degree . Given a sequence of positive numbers tending to , we say that a sequence of polynomials is -ultraflat if Although we do not know, in general, whether or not -ultraflat sequences of polynomials exist for each fixed , we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences of either conjugate, or plain, or skew reciprocal unimodular polynomials such that with is a -ultraflat sequence of polynomials. Bibliography: 18 titles.