Harmonic Mappings onto Convex Domains

Abstract

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