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