BOUNDARY BEHAVIOR OF UNIVALENT HARMONIC MAPPINGS ONTO BOUNDED CONVEX DOMAINS

GEBRESLASSIE ATABHA  WELDEGEBRIAL; HUNDUMA LEGESSE  GELETA

GEBRESLASSIE ATABHA WELDEGEBRIAL DEPARTMENT OF MATHEMATICS, COLLEGE OF NATURAL AND COMPUTATIONAL SCIENCE, ADDIS ABABA UNIVERSITY, ADDIS ABABA ,ETHIOPIA.
HUNDUMA LEGESSE GELETA DEPARTMENT OF MATHEMATICS, COLLEGE OF NATURAL AND COMPUTATIONAL SCIENCE, ADDIS ABABA UNIVERSITY, ADDIS ABABA ,ETHIOPIA

Keywords: Univalent harmonic function, Stolz angle, Angular limit, Dilatation, Boundary Behavior

Abstract

Many authors have examined various boundary behaviors of univalent 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.

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