Normal families of holomorphic functions

Let ℱ be a family of holomorphic functions in a domain D; let k 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 k ≠ 2 we show that if, for every f ∈ ℱ, all zeros of f have multiplicity at least k, f(z) = 0 ⇒ f^(k)(z) = a(z), and f^(k)(z) = a(z) ⇒ |f^(k+1)(z)| ≤ h, then ℱ is normal in D. For k = 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.