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Mat. Sb., 2016, Volume 207, Number 4, Pages 123–142 (Mi msb8469)  

On the derivatives of unimodular polynomials

Paul Nevaia, Tamás Erdélyib

a Upper Arlington (Columbus), Ohio, USA
b Department of Mathematics, Texas A&M University, College Station, TX, USA

Abstract: Let $D$ be the open unit disk of the complex plane; its boundary, the unit circle of the complex plane, is denoted by $\partial D$. Let $\mathscr P_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. For $\lambda \ge 0$, let
$$ \mathscr K_n^\lambda \stackrel{\mathrm{def}}{=}\{P_n:P_n(z)=\sum_{k=0}^n{a_k k^\lambda z^k},   a_k \in\mathbb C, |a_k| = 1 \} \subset\mathscr P_n^c. $$
The class $\mathscr K_n^0$ is often called the collection of all (complex) unimodular polynomials of degree $n$. Given a sequence $(\varepsilon_n)$ of positive numbers tending to $0$, we say that a sequence $(P_n)$ of polynomials $P_n\in\mathscr K_n^\lambda$ is $\{\lambda, (\varepsilon_n)\}$-ultraflat if
$$ (1-\varepsilon_n)\frac{n^{\lambda+1/2}}{\sqrt{2\lambda+1}}\leq|P_n(z)|\leq(1+\varepsilon_n)\frac{n^{\lambda +1/2}}{\sqrt{2\lambda +1}}, \qquad z \in \partial D,\quad n\in\mathbb N_0. $$
Although we do not know, in general, whether or not $\{\lambda, (\varepsilon_n)\}$-ultraflat sequences of polynomials $P_n\in\mathscr K_n^\lambda$ exist for each fixed $\lambda>0$, we make an effort to prove various interesting properties of them. These allow us to conclude that there are no sequences $(P_n)$ of either conjugate, or plain, or skew reciprocal unimodular polynomials $P_n\in\mathscr K_n^0$ such that $(Q_n)$ with $Q_n(z)\stackrel{\mathrm{def}}{=} zP_n'(z)+1$ is a $\{1,(\varepsilon_n)\}$-ultraflat sequence of polynomials.
Bibliography: 18 titles.

Keywords: unimodular polynomial, ultraflat polynomial, angular derivative.
Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8469

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English version:
Sbornik: Mathematics, 2016, 207:4, 590–609

Bibliographic databases:

UDC: 517.518.862
MSC: 41A17
Received: 08.01.2015 and 09.09.2015

Citation: Paul Nevai, Tamás Erdélyi, “On the derivatives of unimodular polynomials”, Mat. Sb., 207:4 (2016), 123–142; Sb. Math., 207:4 (2016), 590–609

Citation in format AMSBIB
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