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On a class of exceptional sets in the theory of conformal mappings. (Russian) Zbl 0688.30006
Let U be the unit disk. A set \(E\subset \partial U\) is called an L-set if there exists a univalent function in U which maps E to a set of zero linear measure. It is proved that L-sets E satisfy the following condition: for every compact \(F\subset E\) \[ \sum \phi (\ell_{\nu})=\infty, \] where \(\ell_{\nu}\) are the lengths of intervals complementary to F and \[ \phi (t)=t\sqrt{\log (1/t)\log \log \log (1/t)}. \] This \(\phi\) is the best possible.
Reviewer: A.E.Eremenko

MSC:
30C35 General theory of conformal mappings
30C99 Geometric function theory
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