## 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 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics