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ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS

  • ÁLVARO FERRADA-SALAS (a1), RODRIGO HERNÁNDEZ (a2) and MARÍA J. MARTÍN (a3)

Abstract

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.

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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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References

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ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS

  • ÁLVARO FERRADA-SALAS (a1), RODRIGO HERNÁNDEZ (a2) and MARÍA J. MARTÍN (a3)

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