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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
$\lambda \geqslant 0$, 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.
Keywords:
unimodular polynomial, ultraflat polynomial, angular derivative.
UDC:
517.518.862
MSC: 41A17 Received: 08.01.2015 and 09.09.2015
DOI:
10.4213/sm8469
Fulltext:
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