On the derivatives of unimodular polynomials

Paul Nevai ; Tamás Erdélyi

Sbornik. Mathematics, Tome 207 (2016) no. 4, pp. 590-609

Résumé

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 $\lambda \geqslant 0$, let $$ \mathscr K_n^\lambda \stackrel{\mathrm{def}}{=}\biggl\{P_n:P_n(z)=\sum_{k=0}^n{a_k k^\lambda z^k}, \, a_k \in\mathbb C,\,|a_k| = 1 \biggr\} \subset\mathscr P_n^c. $$ 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 $$ (1-\varepsilon_n)\frac{n^{\lambda+1/2}}{\sqrt{2\lambda+1}}\leqslant|P_n(z)|\leqslant(1+\varepsilon_n)\frac{n^{\lambda +1/2}}{\sqrt{2\lambda +1}}, \qquad z \in \partial D,\quad n\in\mathbb N_0. $$ 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.

Détail

Keywords: unimodular polynomial, angular derivative.
Mots-clés : ultraflat polynomial