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Harmonic mappings in the exterior of the unit disk

Jarosław Widomski, Magdalena Gregorczyk (2010)

Annales UMCS, Mathematica

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.

Harmonic mappings onto parallel slit domains

Michael Dorff, Maria Nowak, Magdalena Wołoszkiewicz (2011)

Annales Polonici Mathematici

We consider typically real harmonic univalent functions in the unit disk 𝔻 whose range is the complex plane slit along infinite intervals on each of the lines x ± ib, b > 0. They are obtained via the shear construction of conformal mappings of 𝔻 onto the plane without two or four half-lines symmetric with respect to the real axis.

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