On the Bieberbach conjecture and holomorphic dynamics

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.