| n their 1984 paper, Clunie and Sheil-Small introduced harmonic mappings as generalizations of analytic functions. In this work, we attempt to continue the study of geometric function theory for harmonic mappings.;Analogous to the Schwarz-Christoffel formula, harmonic mappings can be constructed that map the unit disk onto the interior of a triangle. For mappings which map the boundary of the unit disk to a curve that bounds a convex domain, the extension to the unit disk by the Poisson integral is guaranteed to be univalent by the Rado-Kneser-Choquet Theorem. A harmonic univalent mapping ;The Noshiro-Warschawski Theorem, which gives a condition for the univalency of an analytic function, is generalized to apply to harmonic mappings. This leads to some results about harmonic close-to-convex mappings. Several methods for constructing harmonic univalent mappings from analytic functions that are convex or starlike of order... |
