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

Daoud Bshouty; Abdallah Lyzzaik

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2012)

  • Volume: 66, Issue: 2
  • ISSN: 0365-1029

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 ( e i t ) = e i φ ( t ) , 0 ≤ t ≤ 2 π where φ is a continuously non-decreasing function that satisfies φ ( 2 π ) - φ ( 0 ) = 2 N π , 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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 66.2 (2012): null. <http://eudml.org/doc/289840>.

@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 $f(e^\{it\}) = e^\{i\phi (t)\}$, $0\le t \le 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?},
author = {Daoud Bshouty, Abdallah Lyzzaik},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Harmonic mapping; cluster set},
language = {eng},
number = {2},
pages = {null},
title = {On a question of T. Sheil-Small regarding valency of harmonic maps},
url = {http://eudml.org/doc/289840},
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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2012
VL - 66
IS - 2
SP - null
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 $f(e^{it}) = e^{i\phi (t)}$, $0\le t \le 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?
LA - eng
KW - Harmonic mapping; cluster set
UR - http://eudml.org/doc/289840
ER -

References

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

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