Harmonic mappings in the exterior of the unit disk

Magdalena Gregorczyk, Jarosław Widomski


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
\(\sum_{n=1}^{\infty}n^{p}(|a_{n}|+|b_{n}|)\leq 1\). 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 mapping; meromorphic; quasiconformal extension; radius of convexity; radius of univalence

Full Text:



Hengartner W., Schober G., Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1-31.

Jahangiri, Jay M., Harmonic meromorphic starlike functions, Bull. Korean Math. Soc. 37 (2000), No. 2, 291-301.

Jahangiri, Jay M., Silverman H., Meromorphic univalent harmonic functions with

negative coefficients, Bull. Korean Math. Soc. 36 (1999), No. 4, 763-770.

Lehto O., Virtanen K. I., Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin-Heidelberg-New York, Second Edition, 1973.

Pommerenke Ch., Univalent Functions, Vandenhoeck & Ruprecht in Gottingen, 1975.

Sheil-Small T., Complex Polynomials, Cambridge University Press, 2002.

DOI: http://dx.doi.org/10.17951/a.2010.54.1.63-73
Data publikacji: 2016-07-29 22:06:16
Data złożenia artykułu: 2016-07-29 21:18:18


  • There are currently no refbacks.

Copyright (c) 2010 Magdalena Gregorczyk, Jarosław Widomski