Harmonic Mappings Convex in One or Every Direction

Abstract

Given a convex complex-valued analytic mapping on the open unit disk in ℂ, we construct a family of complex-valued harmonic mappings convex in the direction of the imaginary axis. We also show that adding the condition of direction convexity preserving to the analytic mapping is a necessary and sufficient condition for the harmonic mapping to be convex. Using analytic radial slit mappings, we provide a three parameter family of harmonic mappings convex in the direction of the imaginary axis and show in some cases that as one parameter varies continuously, the mappings vary from being convex in the direction of the imaginary axis to being convex. In so doing, we also provide information on whether or not some analytic mappings are direction convexity preserving. Lastly, we will provide coefficient conditions leading to harmonic mappings which are starlike or convex of order α.