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 \(f^nf^{(k)}-a\).
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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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Communicated by Lawrence Zalcman.
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Van Tan, T., Van Thin, N. On Lappan’s Five-Point Theorem. Comput. Methods Funct. Theory 17, 47–63 (2017). https://doi.org/10.1007/s40315-016-0168-9
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DOI: https://doi.org/10.1007/s40315-016-0168-9