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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

Abstract

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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?

How to cite

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Daoud 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

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  1. [1] Ahlfors, L., Complex Analysis, Third Edition, McGraw-Hill, New York, 1979. Zbl0395.30001
  2. [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. [3] Duren, P., Harmonic Mappings in the Plane, Cambridge University Press, Cambridge, 2004. Zbl1055.31001
  4. [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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