Transfinite Extension to Q _m-normality Theory

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.