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On the Bieberbach conjecture and holomorphic dynamics


Author: Xavier Buff
Journal: Proc. Amer. Math. Soc. 131 (2003), 755-759
MSC (2000): Primary 37F10, 30C50
Published electronically: October 18, 2002
MathSciNet review: 1937413
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Abstract: In this note we prove that when $P$ is a polynomial of degree $d$ with connected Julia set and when $z_0$ belongs to the filled-in Julia set $K(P)$, then $\vert P'(z_0)\vert\leq d^2$. We also show that equality is achieved if and only if $K(P)$ is a segment of which one extremity is $z_0$. In that case, $P$ is conjugate to a Tchebycheff polynomial or its opposite. The main tool in our proof is the Bieberbach conjecture proved by de Branges in 1984.


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Additional Information

Xavier Buff
Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France
Email: buff@picard.ups-tlse.fr

DOI: https://doi.org/10.1090/S0002-9939-02-06864-8
Received by editor(s): June 25, 2001
Received by editor(s) in revised form: August 14, 2001
Published electronically: October 18, 2002
Communicated by: Linda Keen
Article copyright: © Copyright 2002 American Mathematical Society