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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the Bieberbach conjecture and holomorphic dynamics
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by Xavier Buff PDF
Proc. Amer. Math. Soc. 131 (2003), 755-759 Request permission

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 $|P’(z_0)|\leq d^2$. 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
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 755-759
  • MSC (2000): Primary 37F10, 30C50
  • DOI: https://doi.org/10.1090/S0002-9939-02-06864-8
  • MathSciNet review: 1937413