Computational Methods and Function Theory

, Volume 17, Issue 1, pp 47–63 | Cite as

On Lappan’s Five-Point Theorem

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Abstract

A well-known result of Lappan states that a meromorphic function f in the unit disc \(\mathbb {D}\) is normal if and only if there is a subset \(E\subset \widehat{\mathbb {C}}\) consisting of five points such that \(\sup \{(1-|z|^2) f^{\#}(z): z \in f^{-1}(E)\} < \infty ,\) where \(f^{\#}(z)\) is the spherical derivative of f at z. An analogous result for normal families is due to Hinkkanen and Lappan: a family \(\mathcal F\) of meromorphic functions in a domain \(D\subset \mathbb {C}\) is normal if and only if for each compact subset \(K\subset D,\) there are a subset \(E\subset \widehat{\mathbb {C}}\) consisting of five points and a positive constant M such that \(\sup \{f^{\#}(z): f\in \mathcal F, z \in f^{-1}(E)\}<M.\) In this paper, we extend the above-mentioned results to the case where the set E contains fewer points. In particular, the Pang–Zalcman’s theorem on normality of a family \(\mathcal F\) of holomorphic functions f in a domain D,  \(f^nf^{(k)}(z)\ne a\) (for some given constant a), is also extended to the case where the spherical derivative of \(f^nf^{(k)}\) is bounded on the zero set of \(f^nf^{(k)}-a\).

Keywords

Normal family Normal function Nevanlinna theory 

Mathematics Subject Classification

Primary 30D35 30D45 
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Notes

Acknowledgments

The authors would like to thank the referee for a very careful reading of the manuscript, and for pointing out misprints. Tran Van Tan was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2013.13, and the Vietnam Institute for Advanced Studies in Mathematics. He is currently Regular Associate Member of ICTP, Trieste, Italy.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Department of MathematicsThai Nguyen University of EducationThai Nguyen cityVietnam

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