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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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