Normality Properties of the Family f (nz): n∈ N

A family Ƒ of meromorphic functions on a plane domain D is called quasi-normal on D if each sequence S of functions in Ƒ 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 was introduced by Montel. It generalizes the concept of normal family, which corresponds to E = ∅. Chi-Tai Chuang extended the notion of a quasi-normal family further in an inductive fashion. According to Chuang, a family Ƒ of meromorphic functions on a plane domain D is Q m-normal (m = 0,1, 2,…) if each sequence S of functions in Ƒ has a subsequence which converges locally χ-uniformly on the domain D E, where E = E(S) ⊂ D satisfies E D (m) = ∅. m is said to be the degree of Ƒ. (Here E D (m) is the m-th derived set of E in D.) In particular, a Q 0-normal family is a normal family, and a Q 1-normal family is a quasi-normal family. This paper is devoted to the study of non-normal families generated by a single function. Let f be a nonconstant meromorphic function on ℂ, and write f n(z) = f(nz). Then the family Π(f) = f n: n ∈ ℕ is not normal on the unit disk. We examine the degree of non-normality of this family. We show that if f is a rational function, then Π(f) is quasi-normal of exact order 1. In the opposite direction, if f is not rational, and there exist a, b ∈ Ĉ such that the equations f(z) = a and f(z) = b have only a finite number of solutions in ℂ, then Π(f) fails to be Q m-normal for any value of m.