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A COUNTEREXAMPLE TO UNIQUENESS IN THE RIEMANN MAPPING THEOREM FOR UNIVALENT HARMONIC MAPPINGS
Published online by Cambridge University Press: 01 January 1999
Abstract
Let f be an orientation-preserving univalent harmonic mapping of the unit disk U. Then f=h+[gmacron], where h and g are analytic in U. Furthermore, f satisfies the equation
formula here
in U, where a(z)=g′(z)/h′(z), and [mid ]a(z)[mid ]<1 in U. The function a(z) is the analytic dilatation of f.
In [2], Hengartner and Schober proved the following version of the Riemann mapping theorem for univalent harmonic mappings.
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