A Counterexample to Uniqueness in the Riemann Mapping Theorem for Univalent Harmonic Mappings |
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Authors: | Weitsman Allen |
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Institution: | Department of Mathematics, Purdue University West Lafayette, IN 47907, USA |
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Abstract: | Let f be an orientation-preserving univalent harmonic mappingof the unit disk U. Then , where h and g are analytic in U. Furthermore, f satisfies theequation
in U, where a(z)=g'(z)/h'(z), and |a(z)| < 1 in U. The functiona(z) is the analytic dilatation of f. In 2], Hengartner and Schober proved the following versionof the Riemann mapping theorem for univalent harmonic mappings.1991 Mathematics Subject Classification 31A05, 31A20. |
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