Distortion theorems for polynomials on a circle
type: Item
instance of: scholarly article
title: Distortion theorems for polynomials on a circle
zbMATH Open document ID: 1010.30004
author: V. N. Dubinin
published in: Sbornik: Mathematics
publication date: 2001-08-28T00:00:00Z
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 30C85
zbMATH DE Number: 1638034
MaRDI profile type: Publication
full work available at URL: sm2000v191n12abeh000528
OpenAlex ID: W2051775014
DOI: 10.1070/SM2000V191N12ABEH000528
review text: The author considers polynomials NEWLINE\[NEWLINEP(z)= \sum^n_{k=0} c_kz^k, \quad c_n\neq 0,NEWLINE\]NEWLINE for points \(z=e^{i\varphi}\) on the unit circle. He proves new and interesting estimates for the derivatives NEWLINE\[NEWLINE{\partial\text{Re} P(e^{i \varphi}) \over\partial \varphi},\;{\partial\bigl|P(e^{i\varphi}) \bigr |^2 \over\partial\varphi} \text{ and } {\partial\arg \bigl(P(e^{i\varphi}) \bigr)\over\partial \varphi}NEWLINE\]NEWLINE using the moduli of \(c_0\) and \(c_n\) and the minimum and the maximum of the real part and of the modulus of \(P\) on the unit circle. Since most of these inequalities are to involved to be cited here, we give only two examples where the estimates are short:NEWLINENEWLINENEWLINE(1) If \(P\) is non-vanishing in the open unit disc, then NEWLINE\[NEWLINE{\partial\arg \bigl(P(e^{i\varphi}) \bigr)\over\partial \varphi}\leq{n-1\over 2}+\sqrt{{|c_n|\over 4|c_0|}}.NEWLINE\]NEWLINE (2) If all zeroes of \(P\) lie in the closed unit disc, then NEWLINE\[NEWLINE{\partial\arg \bigl(P(e^{i\varphi}) \bigr)\over\partial\varphi} \geq{n+1\over 2}-\sqrt{ {|c_0|\over 4|c_n|}}.NEWLINE\]
reviewed by: Karl-Joachim Wirths
description: scientific article; zbMATH DE number 1638034