# Koebe 1/4 theorem

###### Theorem (Koebe).

Suppose $f$ is a schlicht function^{} (univalent function^{} on the unit disc
such that $f\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}$ and ${f}^{\mathrm{\prime}}\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{1}$) and $\mathrm{D}\mathrm{\subset}\mathrm{C}$ is the unit disc in the complex plane^{}, then

$$ |

That is, if a univalent function on the unit disc maps 0 to 0 and has derivative 1 at 0, then the image of the unit disc contains the ball of radius $1/4$. So for any $w\notin f(\mathbb{D})$ we have that $|w|\ge 1/4$. Furthermore, if we look at the Koebe function, we can see that the constant $1/4$ is sharp and cannot be improved.

## References

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

Title | Koebe 1/4 theorem |
---|---|

Canonical name | Koebe14Theorem |

Date of creation | 2013-03-22 14:23:57 |

Last modified on | 2013-03-22 14:23:57 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 8 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30C45 |

Synonym | Köbe 1/4 theorem |

Synonym | Koebe one-fourth theorem |

Synonym | Köbe one-fourth theorem |

Related topic | SchlichtFunctions |