Transfinite Extension to Q m-normality Theory

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The theory of Qm-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 ED α of E with respect to D by ED 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 Qα-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 ED 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 Qm-normality theory and extend Zalcman’s Lemma to Q α -normal families where α is an infinite countable (enumerable) ordinal number.

Original languageEnglish
Pages (from-to)141-156
Number of pages16
JournalResults in Mathematics
Issue number1-2
StatePublished - 1 Aug 2003

Bibliographical note

Publisher Copyright:
© 2003, Birkhäuser Verlag, Basel.


  • Q-normal family
  • ordinal number
  • transfinite induction


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