NORMAL FAMILIES OF HOLOMORPHIC FUNCTIONS

摘要

Let F be a family of holomorphic functions in a domain D; let κ be a positive integer; let h be a positive number; and let a be a function holomorphic in D such that a(z) ≠ 0 for z ∈ D. For κ ≠ 2 we show that if, for every f ∈ F, all zeros of f have multiplicity at least κ, f(z) = 0 => f~((κ)) (z) = a(z), and f~((κ))(z) = a(z) => |f~((κ+1))(z)| ≤ h, then F is normal in D. For κ = 2 we prove the following result: Let s ≥ 4 be an even integer. If, for every f ∈F, all zeros of f have multiplicity at least 2, f(z) = 0 => f″(z) = a(z), and f″(z) = a(z) => |f~'"(z)| + |f~((s))(z)| ≤ h, then F is normal in D. This improves the well-known normality criterion of Miranda.