On a question of T. Sheil-Small regarding valency of harmonic maps
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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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/10.2478/v10062-012-0010-4
Date of publication: 2016-07-25 12:22:15
Date of submission: 2016-07-24 22:29:03
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Copyright (c) 2012 Daoud Bshouty, Abdallah Lyzzaik