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.


30A99 General properties of functions of one complex variable
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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