On Lappan’s Five-Point Theorem

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 $f^nf^{(k)}-a$.