# 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 $f:\mathbb{D}\to \u2102$ 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 $k$.

## References

- 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.

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 |