# Marty’s theorem

###### Theorem (Marty).

A set $\mathrm{F}$ of meromorphic functions is a normal family on a domain $G\mathrm{\subset}\mathrm{C}$ if and only if the spherical derivatives are uniformly bounded (uniformly over $\mathrm{F}$) on each compact subset of $G$.

Here normal convergence (convergence on compact subsets) is given using the
spherical metric and not the standard metric of the complex plane^{}. That is, if
$\sigma $ is the spherical metric then we will say ${f}_{n}\to f$ normally
if $\sigma ({f}_{n}(z),f(z))$ converges to 0 uniformly on compact subsets.

A related theorem can be stated.

###### Theorem.

If ${f}_{n}\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\to}f\mathit{}\mathrm{(}z\mathrm{)}$ uniformly in the spherical metric on compact subsets of $G\mathrm{\subset}\mathrm{C}$ then ${f}_{n}^{\mathrm{\u266f}}\mathit{}\mathrm{(}z\mathrm{)}\mathrm{\to}{f}^{\mathrm{\u266f}}\mathit{}\mathrm{(}z\mathrm{)}$ uniformly on compact subsets of $G$.

Here ${f}^{\mathrm{\u266f}}$ denotes the spherical derivative of $f$.

## References

- 1 Theodore B. Gamelin. . Springer-Verlag, New York, New York, 2001.

Title | Marty’s theorem |
---|---|

Canonical name | MartysTheorem |

Date of creation | 2013-03-22 14:18:39 |

Last modified on | 2013-03-22 14:18:39 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 30D30 |