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

Daoud Bshouty; Abdallah Lyzzaik

Annales UMCS, Mathematica (2012)

- Volume: 66, Issue: 2, page 25-29
- ISSN: 2083-7402

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topDaoud Bshouty, and Abdallah Lyzzaik. "On a question of T. Sheil-Small regarding valency of harmonic maps." Annales UMCS, Mathematica 66.2 (2012): 25-29. <http://eudml.org/doc/267579>.

@article{DaoudBshouty2012,

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(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?},

author = {Daoud Bshouty, Abdallah Lyzzaik},

journal = {Annales UMCS, Mathematica},

keywords = {Harmonic mapping; cluster set.; harmonic functions in the disc; harmonic extension from the circle},

language = {eng},

number = {2},

pages = {25-29},

title = {On a question of T. Sheil-Small regarding valency of harmonic maps},

url = {http://eudml.org/doc/267579},

volume = {66},

year = {2012},

}

TY - JOUR

AU - Daoud Bshouty

AU - Abdallah Lyzzaik

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

JO - Annales UMCS, Mathematica

PY - 2012

VL - 66

IS - 2

SP - 25

EP - 29

AB - 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?

LA - eng

KW - Harmonic mapping; cluster set.; harmonic functions in the disc; harmonic extension from the circle

UR - http://eudml.org/doc/267579

ER -

## References

top- [1] Ahlfors, L., Complex Analysis, Third Edition, McGraw-Hill, New York, 1979. Zbl0395.30001
- [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. Zbl1017.30015
- [3] Duren, P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004. Zbl1055.31001
- [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. Zbl0148.05201

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