Bieberbach’s conjecture

The following theorem is known as the Bieberbach conjecture, even though it has now 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 representation as

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

Theorem (Bieberbach).

Suppose that $f$ is a schlicht function, then $\mathrm{|}{a}_k\mathrm{|}\mathrm{\le }k$ for all $k\mathrm{\ge }\mathrm{2}$ and furthermore if 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 $k$.

Title	Bieberbach’s conjecture
Canonical name	BieberbachsConjecture
Date of creation	2013-03-22 14:24:07
Last modified on	2013-03-22 14:24:07
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Theorem
Classification	msc 30C55
Classification	msc 30C45
Synonym	Bieberbach conjecture