# 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