Preface

Miodrag Mateljević

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We give a quasiconformal version of the proof for the classical Lindelöf theorem: Let f map the unit disk D conformally onto the inner domain of a Jordan curve C. Then C is smooth if and only if arh f'(z) has a continuous extension to D. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.

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We study a dual analogue of the class Σ(κ) of hydrodynamically normalized schlicht conformal mappings g(z) of the exterior of the unit circle with a [...] -quasiconformal extension, namely now those (non-schlicht) mappings g(z) for which g(z) has such a quasiconformal extension.

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Quasiconformal mappings with Sobolev boundary values

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We consider quasiconformal mappings in the upper half space $ℝ_{+}^{n + 1}$ of $ℝ^{n + 1}$ , $n \geq 2$ , whose almost everywhere defined trace in $ℝ^{n}$ has distributional differential in $L^{n} (ℝ^{n})$ . We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space $H^{1}$ . More generally, we consider certain positive functions defined on $ℝ_{+}^{n + 1}$ , called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems...

A remark on the Ahlfors--Lehto univalence criterion.

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In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition [...] . We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.

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