## Ellipses, near ellipses, and harmonic Möbius transformations.(English)Zbl 1068.30018

Analytic Möbius transformations take circles to circles. This is their most basic, most celebrated geometric property. In a previous paper [M. Chuaqui, P. Duren and B.  Osgood, J. Anal. Math. 91, 329–351 (2003; Zbl 1054.31003)], the authors introduced harmonic Möbius transformations as a generalization of Möbius transformations to harmonic mappings. Their basic geometric property is that they take circles to ellipses. In this paper, the authors consider the converse question. It is shown that a harmonic mapping that takes circles to ellipses must be a harmonic Möbius transformation. The authors also have some comments on the situation for analytic functions.

### MSC:

 30C99 Geometric function theory 31-06 Proceedings, conferences, collections, etc. pertaining to potential theory 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 58E20 Harmonic maps, etc. 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions

Zbl 1054.31003
Full Text: