Two New Criteria for Normal Families

Abstract

Let a 0, …, a k−1 be analytic functions on a domain Ω. Let F be a family of meromorphic functions f defined on Ω such that f ≠ 0 and f (k) + a k−1 f (k−1) + … + a 0 f ≠ 0 on Ω, for all f ∊ F. Then f′ / f: f ∊ F is a normal family. Furthermore, let a 0,…, a k−1 be meromorphic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f≠ 0, f′ ≠ 0 and f (k) + a k− f (k−1) + … + a 0 f ≠ 0 on Ω, for all f ∊ F. Then f′ / f: f ∊ F is a normal family. These two new criteria for normal families extend a recent result of Bergweiler and Langley, [1, Corollary 1.1].