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SMALE’S PROBLEM FOR CRITICAL POINTS ON CERTAIN TWO RAYS

Part of: Geometric function theory

Published online by Cambridge University Press:  25 March 2010

AIMO HINKKANEN  and
ILGIZ KAYUMOV
AIMO HINKKANEN*
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA (email: aimo@illinois.edu)
ILGIZ KAYUMOV
Affiliation:
Institute of Mathematics and Mechanics, Kazan State University, Kremlevskaya 18, 420 008 Kazan, Russia (email: Ilgis.Kayumov@ksu.ru)
*
✉For correspondence; e-mail: aimo@illinois.edu
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Abstract

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Let f be a polynomial of degree n≥2 with f(0)=0 and f′(0)=1. We prove that there is a critical point ζ of f with ∣f(ζ)/ζ∣≤1/2 provided that the critical points of f lie in the sector {reiθ:r>0,∣θ∣≤π/6}, and ∣f(ζ)/ζ∣<2/3 if they lie in the union of the two rays {1+re±iθ:r≥0}, where 0<θ≤π/2.

Keywords

polynomialscritical pointsSmale’s problem

MSC classification

Secondary: 30C10: Polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 2 , April 2010 , pp. 183 - 191
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This material is based upon work supported by the National Science Foundation under grant no. 0758226. The second author was supported by RFBR (grant nos 08-01-00381, 09-01-12188 ofi-m) and by the Russian Federal Agency of Education, grant no. P 944.

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