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Removable singularities for analytic functions. (English) Zbl 0805.30001
Let $$f$$ be analytic in $$G \backslash E$$ where $$E$$ is a relatively closed subset of a plane domain $$G$$. The following condition, based on a bound for the growth of $$f'(z)$$, is introduced for the removability of $$E$$: Let $$0 \leq s<2$$, let $$\dim_ M(E) < 2- s$$, $$H^ 1 (\text{Re} E) = H^ 1(\text{Im} E) = 0$$, and suppose that for each $$z_ 0 \in E$$ there is $$r>0$$ and $$C$$ such that $$| f'(z) | \leq C$$ $$d(z,E)^{-s}$$ for all $$z \in B(z_ 0,r) \backslash E$$. Here $$H^ 1$$ is the one-dimensional Hausdorff measure and $$\dim_ M$$ refers to the Minkowski dimension, i.e. disks of equal radii are used for a covering of a set. The bound for the derivative and the bound for the Minkowski dimension are used to obtain the local integrability of $$f'(z)$$; the rest of the conditions are needed to employ Weyl’s lemma. It is shown that the result is rather sharp. Related results for quasiregular mappings are also discussed.

##### MSC:
 30A99 General properties of functions of one complex variable 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
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