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 $|Pâ(z_0)|\leq d^2$. 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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.