A non explicit counterexample to a problem of quasi-normality

Jürgen Grahl, Shahar Nevo, Xuecheng Pang

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4 Scopus citations

Abstract

In 1986, S.Y. Li and H. Xie proved the following theorem: let k≥ 2 and letFbe a family of functions meromorphic in some domainD, all of whose zeros are of multiplicity at leastk. ThenFis normal if and only if the familyFk={f(k)1+|f|k+1:f∈F}is locally uniformly bounded inD.Here we give, in the case k=2, a counterexample to show that if the condition on the multiplicities of the zeros is omitted, then the local uniform boundedness of F2 does not even imply quasi-normality. In addition, we give a simpler proof for the Li-Xie theorem (and an extension of it) that does not use Nevanlinna's Theory which was used in the original proof.

Original languageEnglish
Pages (from-to)386-391
Number of pages6
JournalJournal of Mathematical Analysis and Applications
Volume406
Issue number2
DOIs
StatePublished - 15 Oct 2013

Bibliographical note

Funding Information:
The research was supported by the Israel Science Foundation , Grant No. 395/2007 .

Keywords

  • Differential inequality
  • Interpolation theory
  • Quasi-normal family
  • Zalcman's lemma

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