On the derivatives of unimodular polynomials Academic Article uri icon


  • © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. Let D be the open unit disk of the complex plane; its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Pnc denote the set of all algebraic polynomials of degree at most n with complex coefficients. For λ ≥ 0, let (Equation presented) The class Kn0 is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (ϵn) of positive numbers tending to 0, we say that a sequence (Pn) of polynomials Pn ∈ Knλ is {λ, (ϵn)}-ultraflat if (Equation presented) Although we do not know, in general, whether or not {λ, (ϵn)}-ultraflat sequences of polynomials Pn ∈ Knλ exist for each fixed λ > 0, we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences (Pn) of either conjugate, or plain, or skew reciprocal unimodular polynomials Pn ∈ Kn0 such that (Qn) with Qn(z) def// zP′n(z) + 1 is a {1, (ϵn)}-ultraflat sequence of polynomials.

author list (cited authors)

  • Nevai, P., & Erdélyi, T.

citation count

  • 1

publication date

  • April 2016