Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T07:30:51.904Z Has data issue: false hasContentIssue false
  • English
  • Français

On the valence of some classes of harmonic maps

Published online by Cambridge University Press:  24 October 2008

Abdallah Lyzzaik
Abdallah Lyzzaik
Affiliation:
Department of Mathematics, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait
Get access Rights & Permissions [Opens in a new window]

Abstract

We give examples which (i) disprove a conjecture of Sheil-Small regarding the valence of harmonic mappings of the unit disc to bounded convex domains, and (ii) answer negatively a question of the author regarding the valence of harmonic mappings with polynomial analytic and co-analytic parts.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 2 , September 1991 , pp. 313 - 325
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abu-Muhanna, Y. and Lyzzaik, A.. A geometric criterion for decomposition and multivalence. Math. Proc. Cambridge Philos. Soc. 103 (1988), 487–495.Google Scholar
[2]Ahlfors, L. and Sario, L.. Riemann Surfaces (Princeton University Press, 1960).CrossRefGoogle Scholar
[3]Choquet, G.. Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques. Bull. Sci. Math. 69 (1945), 156–165.Google Scholar
[4]Farkas, H. and Kra, I.. Riemann Surfaces (Springer-Verlag, 1980).Google Scholar
[5]Kneser, H.. Lösung der Aufgabe 41. Jahresber. Deutsch. Math.-Verein. 35 (1926), 123–124.Google Scholar
[6]Lyzzaik, A.. The geometry of some classes of folding polynomials. Complex Variables Theory Appl. (to appear).Google Scholar
[7]Rado, . Zur Theorie der mehrdeutigen konformen Abbildungen. Acta Litt. ac. Sci. Univ. Hung. (Szeged) 1 (1922/1923), 55–69.Google Scholar
[8]Sheil-Small, T.. On the Fourier series of a finitely described curve and a conjecture of H. S. Shapiro. Math. Proc. Cambridge Philos. Soc. 98 (1985), 513–527.Google Scholar