## 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

### Keywords:

meromorphic functions; spherical derivative

### Citations:

Zbl 0821.30021; Zbl 0584.30018
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