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On the behavior of meromorphic functions around some nonisolated singularities. II. (English) Zbl 0851.30014

Let \(G\) be a plane domain and \(E \subset G\) a relatively closed set. If \(f\) is meromorphic in \(G \backslash E\) with at least one essential singularity in \(E\), then the author showed [P. Järvi, Ann. Acad. Sci. Fenn., Ser. A I, Math. 19, No. 2, 367-374 (1994; Zbl 0821.30021)] that (i) \(\lim \sup f^* (z) d(z, E)^\beta = \infty\) as \(z \to E\) for all \(\beta < 1 - \alpha/2\) provided that the \(\alpha\)-dimensional Minkowski content of \(E\) is finite and \(E\) is of class \(N_D\). Here \(f^*\) is the spherical derivative of \(f\). The author improves this estimate if \(E\) has a regular distribution. For example if \(E\) lies on a quasicircle and if the Hausdorff dimension of \(E\) satisfies \(\dim_H E \leq \alpha\), then (i) holds for all \(\beta < \min (1,2 - \alpha)\). The proof analyzes the local lipschitz properties of \(f\) [F. Gehring and O. Martio, Ann. Acad. Sci. Fenn., Ser. A I 10, 203-219 (1985; Zbl 0584.30018)].

MSC:

30D30 Meromorphic functions of one complex variable (general theory)
30D40 Cluster sets, prime ends, boundary behavior
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