### Abstract

An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere ℙ\{z_{1}, z_{2}, z_{3}} with singularities of order α_{j} ≤ 1 at z_{j} is obtained in all possible cases α_{1} + α_{2} + α_{3} > 2. The existence and uniqueness of such a metric was proved long time ago by Picard (J Reine Angew Math 130:243-258, 1905) and Heins (Nagoya Math J 21:1-60, 1962), while explicit formulas for the cases α_{1} = α_{2} = 1 were given earlier by Agard (Ann Acad Sci Fenn Ser A I 413, 1968) and recently by Anderson et al. (Math Z, to appear, doi:10. 1007/s00209-009-0560-5. We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel (J Lond Math Soc II Ser 20:435-445, 1979) and Minda (Complex Var 8:129-144, 1987). As applications, sharp versions of Landau- and Schottky-type theorems for meromorphic functions are obtained.

Original language | English |
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Pages (from-to) | 851-868 |

Number of pages | 18 |

Journal | Mathematische Zeitschrift |

Volume | 267 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Apr 1 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Mathematische Zeitschrift*,

*267*(3), 851-868. https://doi.org/10.1007/s00209-009-0649-x