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On the Bieberbach Conjecture and Holomorphic Dynamics
Proceedings of the American Mathematical Society
Vol. 131, No. 3 (Mar., 2003), pp. 755-759
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/1194477
Page Count: 5
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In this note we prove that when P is a polynomial of degree d with connected Julia set and when z0 belongs to the filled-in Julia set K(P), then |P′(z0)|≤ d2. We also show that equality is achieved if and only if K(P) is a segment of which one extremity is z0. 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.
Proceedings of the American Mathematical Society © 2003 American Mathematical Society