Transfinite Extension to Q _m-normality Theory

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 $E^{(\alpha)}_D$ of E with respect to D by $(E^{(\alpha-1)}_D)^{(1)}_D$ if α has an immediate predecessor and by ${\mathop \bigcap\limits_{\beta<\alpha}} E^{(\beta)}_D$ if α is a limit ordinal. Then a family ${\cal F}$ of meromorphic functions on a plane domain D is Q_α-normal if each sequence S of functions in ${\cal F}$ has a subsequence which converges locally χ-uniformaly on the domain DE, where E = E(S) ⊂ D satisfies $E^{(\alpha)}_{D}=\emptyset$. 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.