Computational Methods and Function Theory 1 (2001), No. 2, 479--499
Copyright Heldermann Verlag 2001
Daoud Bshouty
daoud@techunix.technion.ac.il, Department of Mathematics, Technion, Haifa 32000, Israel.
Walter Hengartner
walheng@mat.ulaval.ca, Départment Des Mathématiques, Université Laval, Québec G1K 7P4, Canada.
Abdallah Lyzzaik
lyzzaik@aub.edu.lb, Department of Mathematics, American University of Beirut, Beirut, Lebanon.
Allen Weitsman
weits@math.purdue.edu, Purdue University, West Lafayette, IN 47907-1395, U.S.A.
We give two harmonic mappings of the open unit disc onto bounded convex domains that extend continuously to the unit circle as $2$-valent local homeomorphisms onto the boundary of the image domains such that the valency of one mapping is $6$ and of the other is $8$. This provides a negative answer to a conjecture of Sheil-Small.
Keywords: Harmonic mappings, p-valend functions.
MSC 2000: Primary 30C55; Secondary 31A05, 30F99.
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