A Non explicit counterexample to a problem of quasi-normality

TL;DR

This paper provides a counterexample showing that omitting zero multiplicity conditions invalidates a normality criterion for meromorphic functions, and offers a simpler proof of the original theorem without Nevanlinna Theory.

Contribution

It presents a counterexample for the case k=2 demonstrating the necessity of zero multiplicity conditions and simplifies the proof of the Li-Xie Theorem.

Findings

Counterexample shows local uniform boundedness does not imply quasi-normality without zero multiplicity conditions

Simplified proof of Li-Xie Theorem avoiding Nevanlinna Theory

Highlights importance of zero multiplicity in normality criteria

Abstract

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.