TY - JOUR

T1 - Normal families of holomorphic functions

AU - Chang, Jianming

AU - Fang, Mingliang

AU - Zalcman, Lawrence

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=3042814330&partnerID=8YFLogxK

U2 - 10.1215/ijm/1258136186

DO - 10.1215/ijm/1258136186

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:3042814330

SN - 0019-2082

VL - 48

SP - 319

EP - 337

JO - Illinois Journal of Mathematics

JF - Illinois Journal of Mathematics

IS - 1

ER -