# 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 $f\colon{\mathbb{D}}\to{\mathbb{C}}$ is a schlicht function  (univalent, $f(0)=0$ and $f^{\prime}(0)=1$) then $f$ has a power series  representation as

 $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots=z+\sum_{k=2}^{\infty}a_{k}z^{k}.$
###### Theorem (Bieberbach).

Suppose that $f$ is a schlicht function, then $\lvert a_{k}\rvert\leq k$ for all $k\geq 2$ and furthermore if there is some integer $k$ such that $\lvert a_{k}\rvert=k$, then $f$ is some rotation of the Koebe function.

In fact if $f$ is a rotation of the Koebe function then $\lvert a_{k}\rvert=k$ for all $k$.

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Bieberbach’s conjecture BieberbachsConjecture 2013-03-22 14:24:07 2013-03-22 14:24:07 jirka (4157) jirka (4157) 7 jirka (4157) Theorem msc 30C55 msc 30C45 Bieberbach conjecture