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(eit) = eiϕ(t), 0 ≤ t ≤ 2π, where ϕ is a continuously non-decreasing function that satisfies ϕ(2π)−ϕ(0) = 2Nπ, assume every value finitely many times in the disc?
@article{bwmeta1.element.doi-10_2478_v10062-012-0010-4,
author = {Daoud Bshouty and Abdallah Lyzzaik},
title = {On a question of T. Sheil-Small regarding valency of harmonic maps},
journal = {Annales UMCS, Mathematica},
volume = {66},
year = {2012},
pages = {25-29},
zbl = {1300.31002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0010-4}
}
Daoud Bshouty; Abdallah Lyzzaik. On a question of T. Sheil-Small regarding valency of harmonic maps. Annales UMCS, Mathematica, Tome 66 (2012) pp. 25-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0010-4/
[1] Ahlfors, L., Complex Analysis, Third Edition, McGraw-Hill, New York, 1979. | Zbl 0395.30001
[2] Bshouty, D., Hengartner, W., Lyzzaik, A. and Weitsman, A., Valency of harmonicmappings onto bounded convex domains, Comput. Methods Funct. Theory 1 (2001), 479-499. | Zbl 1017.30015
[3] Duren, P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004. | Zbl 1055.31001
[4] 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. | Zbl 0148.05201