On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik

Abstract


The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form \(f(e^{it}) = e^{i\phi(t)}\), \(0\leq t \leq 2\pi\) where \(\phi\) is a continuously non-decreasing function that satisfies \(\phi(2\pi)-\phi(0) = 2N\pi\), assume every value finitely many times in the disc?

Keywords


Harmonic mapping; cluster set

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References


Ahlfors, L., Complex Analysis, Third Edition, McGraw-Hill, New York, 1979.

Bshouty, D., Hengartner, W., Lyzzaik, A. and Weitsman, A., Valency of harmonic mappings onto bounded convex domains, Comput. Methods Funct. Theory 1 (2001), 479-499.

Duren, P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004.

Markushevich, A. I., Theory of functions of a complex variable. vol. III, English edition translated and edited by Richard A. Silverman, Prentice-Hall Inc., N. J., 1967.




DOI: http://dx.doi.org/10.17951/a.2012.66.2.25-29
Data publikacji: 2016-07-25 12:22:15
Data złożenia artykułu: 2016-07-24 22:29:03

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