BOUNDARY BEHAVIOR OF UNIVALENT HARMONICMAPPINGS ONTO BOUNDED CONVEX DOMAINS

Abstract

 Many authors have examined various boundary behaviors of uni
valent harmonic mappings in the open unit disk. Building on the work of
Laugesen, Bshouty and others, this paper extends earlier results on the boundary behavior of univalent harmonic mappings under different conditions. We determine the angular limits of the arguments and logarithms of the analytic and co-analytic parts of univalent harmonic mappings in terms of the derivative of the boundary function and the dilatation. Explicit formulas are obtained when this derivative is finite. We also show that the dilatation possesses a finite number of zeros within any Stolz angle provided the derivative of the boundary function tends to infinity. For mappings onto bounded convex domains, the complex derivative has no interior zeros in any Stolz angle. These results
explore and complement earlier work and clarify the geometric role of the di
latation near the boundary.