Some Note on Exceptional Values of Meromorphic Functions

Abstract

1. Let E be a totally-disconnected compact set in the z-plane and let Ω be its complement with respect to the extended z-plane. Then Ω is a domain and we can consider a single-valued meromorphic function w = f(z) on Ω which has a transcendental singularity at each point of E. Suppose that E is a nullset of the class W in the sense of Kametani [4] ( -the class N<$ in the sense of Ahlfors and Beurling [1]). Then the cluster set of f(z) at each transcendental singularity is the whole w-plane, and hence f(z) has an essential singularity at each point of E. We shall say that a value w is exceptional for f{z) at an essential singularity CE£ if there exists a neighborhood of C where the function f(z) does not take this value w.In our previous paper [6], we showed that, even if E is of capacity 1 ' zero, the set of all exceptional values of f(z) at a point C of E may be non-countable.