On Lappan's Five-Point Theorem

A well-known result of Lappan states that a meromorphic function f in the unit disc D is normal if and only if there is a subset E subset of (C) over cap consisting of five points such that sup{(1 - | z|(2)) f (#)(z) : z is an element of f(-1)(E)} < infinity, 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 F of meromorphic functions in a domain D subset of C is normal if and only if for each compact subset K subset of D, there are a subset E subset of <(C)over cap> consisting of five points and a positive constant M such that sup{f (#)(z) : f is an element of F, z is an element of 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 F of holomorphic functions f in a domain D, f(n) f((k))(z) not equal a (for some given constant a), is also extended to the case where the spherical derivative of fn f(n) f((k)) is bounded on the zero set of f (n) f ((k)) - a.