A criterion of normality based on a single holomorphic function

Xiao Jun Liu, Shahar Nevo

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k - 1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ {pipe}f(k)(z){pipe} ≤ c, where c is a constant. Then F is normal on D.

Original languageEnglish
Pages (from-to)141-154
Number of pages14
JournalActa Mathematica Sinica, English Series
Volume27
Issue number1
DOIs
StatePublished - Jan 2011

Bibliographical note

Funding Information:
Received July 8, 2009, accepted January 18, 2010 The first author is supported by the Gelbart Research Institute for Mathematical Sciences and by National Natural Science Foundation of China (Grant No. 10671067); the second author is supported by the Israel Science Foundation (Grant No. 395107)

Keywords

  • Holomorphic functions
  • Normal family
  • Zero points

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