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 language | English |
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Pages (from-to) | 386-391 |
Number of pages | 6 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 406 |
Issue number | 2 |
DOIs | |
State | Published - 15 Oct 2013 |
Bibliographical note
Funding Information:The research was supported by the Israel Science Foundation , Grant No. 395/2007 .
Funding
The research was supported by the Israel Science Foundation , Grant No. 395/2007 .
Funders | Funder number |
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Israel Science Foundation | 395/2007 |
Keywords
- Differential inequality
- Interpolation theory
- Quasi-normal family
- Zalcman's lemma