Application of Zalcman's lemma to Qm-normal families

A family F of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in F has a subsequence which converges locally χ-uniformly on D \ E, where E = E(S) is a subset of D having no accumulation points in D. The notion of quasi-normality generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of quasi-normal family further in an inductive fashion. According to Chuang, a family F of meromorphic functions on a plane domain D is Qm-normal (m = 0, 1, 2, . . . ) if each sequence S of functions in F has a subsequence which converges locally χ-uniformly on the domain D \ E, where E = E(S) ⊂ D satisfies E (m) D = ∅. (Here E (m) D is the m-th derived set of E in D.) In particular, a Q0-normal family is a normal family, and a Q1-normal family is a quasi-normal family. This paper gives generalizations of Zalcman's Lemma to Qm-normal families, together with applications of these generalizations – specifically, determining the degree of normality (m) of families of meromorphic functions obtained as linear combinations of functions taken from given families