Bieberbach’s conjecture

The following theorem is known as the Bieberbach conjecture, even though it hasnow been proven. Bieberbach proposed it in 1916 and it was finally proven in 1984 by Louis de Branges.

Firstly note that if $f:𝔻\to ℂ$ is a schlicht function (univalent, $f(0)=0$ and $f^{\prime }(0)=1$) then $f$ has a power series representationas

$$f(z)=z+a_2{z}^2+a_3{z}^3+\mathrm{\cdots }=z+\sum _{k=2}^{\mathrm{\infty }}{a}_k{z}^k.$$

Theorem (Bieberbach).

Suppose that $f$ is a schlicht function, then $\mathrm{|}{a}_k\mathrm{|}\mathrm{\le }k$ forall $k\mathrm{\ge }\mathrm{2}$ and furthermoreif there is some integer $k$ such that $\mathrm{|}{a}_k\mathrm{|}\mathrm{=}k$, then $f$ is some rotation of the Koebe function.

In fact if $f$ is a rotation of the Koebe function then $|a_k|=k$for all $k$.