## Abstract

The theory of Q_{m}-normal families, m ∈ℕ, was developed by P. Montel for the cases m = 0 (normal families) [5] and m = 1 (quasinormal families) [4] and later generalized by C.T. Chuang [2] for any m ≥ 0. In this paper, we extend the definition to an arbitrary ordinal number α as follows. Given E ⊂D, define the α-th derived set E_{D} ^{α} of E with respect to D by E_{D} ^{α-1} (if α has an immediate predecessor and by if α is a limit ordinal. Then a family F of meromorphic functions on a plane domain D is Q_{α}-normal if each sequence S of functions in F has a subsequence which converges locally χ-uniformaly on the domain DE, where E = E(S) ⊂ D satisfies E_{D} ^{α} =0 Inparticular, a Q _{0} -normal family is a normal family, and a Q _{1} -normal family is a quasi- normal family. We also give analogues to some basic results in Q_{m}-normality theory and extend Zalcman’s Lemma to Q _{α} -normal families where α is an infinite countable (enumerable) ordinal number.

Original language | English |
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Pages (from-to) | 141-156 |

Number of pages | 16 |

Journal | Results in Mathematics |

Volume | 44 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Aug 2003 |

### Bibliographical note

Publisher Copyright:© 2003, Birkhäuser Verlag, Basel.

## Keywords

- Q-normal family
- ordinal number
- transfinite induction

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