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Harmonic Mappings onto Convex Domains

Published online by Cambridge University Press:  20 November 2018

Yusuf Abu-Muhanna  and
Glenn Schober
Yusuf Abu-Muhanna
Affiliation:
University of Petroleum and Minerals, Dhahran, Saudi Arabia
Glenn Schober
Affiliation:
Indiana University, Bloomington, Indiana
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Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.

In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equation

where a is analytic and |a| < 1, such that f(U) ⊂ D and

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 6 , 01 December 1987 , pp. 1489 - 1530
Copyright
Copyright © Canadian Mathematical Society 1987

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