Computational Methods and Function Theory 4 (2004), No. 1, 127--142
Copyright Heldermann Verlag 2004
Martin Chuaqui
mchuaqui@mat.puc.cl , Facultad de Matemáticas, P. Universidad Católica de Chile, Santiago, Chile.
Peter Duren
duren@umich.edu , Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109--1109, U.S.A.
Brad Osgood
osgood@ee.stanford.edu , Department of Electrical Engineering, Stanford University, Stanford, California 94305, U.S.A.
A geometric interpretation of the Schwarzian of a harmonic mapping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions. A formula for curvature of image arcs under harmonic mappings is applied to derive a known result on concavity of the boundary. It is also used to characterize fully convex mappings, which are related to fully starlike mappings through a harmonic analogue of Alexander's theorem.
Keywords: Harmonic mapping, Schwarzian derivative, geodesic curvature, minimal surface, convex, starlike.
MSC 2000: Primary 30C99; Secondary 31A05, 53A10.
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