On Erdős-Lax theorem
type: Item
instance of: scholarly article
title: On Erdős-Lax theorem
zbMATH Open document ID: 0551.30002
DOI: 10.1007/BF02876763
author: N. E. Zubov
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1983-01-01T00:00:00Z
review text: Let \(P(z)=\sum^{n}_{\nu =0}c_{\nu}z^{\nu}\) be a complex polynomial and put \(\| P\|:=\max_{| z| =1}| P(z)|.\) The authors prove Theorem 1. If \[ P(z)=c_ 0+\sum^{n}_{\nu =\mu}c_{\nu}z^{\nu}\neq 0\quad for\quad | z| <K,\quad K\geq 1, \] then \(\| P'\| \leq n(1+K^{\mu})^{- 1}\| P\|\) holds. Here, equality holds for \(P(z)=(z^{\mu}+K^{\mu})^{n/\mu},\) where \(\mu\) divides n. Corollary. If \[ P(z)=\sum^{n-\mu}_{\nu =0}c_{\nu}z^{\nu}+c_ nz^ n\neq 0\quad for\quad | z| >k,\quad k\leq 1, \] then \(\| P'\| \geq n(1+k^{\mu})^{-1}\| P\|\) holds. Here, equality holds for \(P(z)=(z^{\mu}+k^{\mu})^{n/\mu},\) where \(\mu\) divides n. Theorem 2. If \(c_ n=1,P(z)\neq 0\) for \(| z| <1\) and \(\| P\| =2\), then \(P(z)=z^ n+c_ 0\) with \(| c_ 0| =1\).
Mathematics Subject Classification ID: 30A10
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 3877480
zbMATH Keywords: Erdős-Lax theorem
zbMATH Keywords: maximum modulus principle
MaRDI profile type: Publication
cites work: Q5784996
cites work: On the Derivative of a Polynomial
cites work: On the derivative of a polynomial
cites work: Proof of a conjecture of P. Erdös on the derivative of a polynomial
cites work: Q5774835
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description: scientific article; zbMATH DE number 3877480