Harmonic Mappings onto Convex Domains
1987 ◽
Vol 39
(6)
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pp. 1489-1530
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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 equationwhere a is analytic and |a| < 1, such that f(U) ⊂ D and
1987 ◽
Vol 39
(1)
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pp. 54-73
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2007 ◽
Vol 2007
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pp. 1-11
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2017 ◽
Vol 95
(3)
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pp. 457-466
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2019 ◽
Vol 12
(02)
◽
pp. 1950017
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