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ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  07 August 2017

ÁLVARO FERRADA-SALAS ,
RODRIGO HERNÁNDEZ  and
MARÍA J. MARTÍN
ÁLVARO FERRADA-SALAS
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile email [email protected]
RODRIGO HERNÁNDEZ
Affiliation:
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile email [email protected]
MARÍA J. MARTÍN*
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, PO Box 111, FI-80101 Joensuu, Finland email [email protected]
*
✉[email protected]
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Abstract

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The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying

$$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$
for some $\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$, along with their rotations, play an important role among those functions that are harmonic and orientation-preserving and map the unit disc onto a convex domain. The main theorem in this paper generalises results in recent literature by showing that convex combinations of functions in ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ are convex.

Keywords

convex harmonic mappingsconvex combinations

MSC classification

Primary: 31A05: Harmonic, subharmonic, superharmonic functions
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 256 - 262
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The second and third authors are supported by Fondecyt, Chile, grant 1150284. The third author is also partially supported by grant MTM2015-65792-P from MINECO/FEDER, the Thematic Research Network MTM2015-69323-REDT, MINECO, Spain, and the Academy of Finland, grant 268009.

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