Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 32, 2007, 301-315

QUASICONFORMAL AND QUASIREGULAR HARMONIC ANALOGUES OF KOEBE'S THEOREM AND APPLICATIONS

Miodrag Mateljevic

University of Belgrade, Faculty of Mathematics
Studentski Trg 16, Belgrade, Yugoslavia; miodrag 'at' matf.bg.ac.yu

Abstract. We show two versions of the Koebe theorem: one for quasiregular harmonic functions and another for quasiconformal functions. We also give an elementary proof of a version of the Koebe one-quarter theorem for holomorphic functions. As an application, we show the harmonic analogue of the Koebe one-quarter theorem and that holomorphic functions (more generally, quasiregular harmonic functions) and their modulus have similar behaviour in a certain sense.

2000 Mathematics Subject Classification: Primary 30C80, 30C62; Secondary 30C55, 30H05.

Key words: Grötzsch annulus, extremal length, the harmonic analogue of Koebe's one-quarter theorem, Lipschitz-type spaces.

Reference to this article: M. Mateljevic: Quasiconformal and quasiregular harmonic analogues of Koebe's theorem and applications. Ann. Acad. Sci. Fenn. Math. 32 (2007), 301-315.

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