A Non explicit counterexample to a problem of quasi-normality

Details A Non explicit counterexample to a problem of quasi-normality A Non explicit counterexample to a problem of quasi-normality

2011-11-03

arxiv.org/abs/1111.0841v1

Mathematics

Complex Variables

Scientific paper

In 1986, S.Y. Li and H.Xie proved the following theorem:Let k>=2 and let F be a family of functions meromorphic in some domain D, all of whose zeros are of multiplicity at least k. Then F is normal if and only if the family F_k={f^(k)/(1+|f^k+1|):f in F} is locally uniformly bounded in D. 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 F_2 does not imply even quasi-normality. In addition, we give a simpler proof for the Li-Xie Theorem that does not use Nevanlinna Theory which was used in the original proof.

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