Two New Criteria for Normal Families

Let a0, . . . , ak−1 be analytic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f 6= 0 and f (k) + ak−1f (k−1) + . . . + a0f 6= 0 on Ω, for all f ∈ F . Then {f ′/f : f ∈ F} is a normal family. Furthermore, let a0, . . . , ak−1 be meromorphic functions on a domain Ω. Let F be a family of meromorphic functions f on Ω such that f 6= 0, f ′ 6= 0 and f (k) + ak−1f (k−1) + . . .+ a0f 6= 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]. 1 Mathematics Subject Classification (1991): 32A19, 32A20, 32A22