CRITERIA FOR UNIVALENCE AND QUASICONFORMAL EXTENSION FOR HARMONIC MAPPINGS ON PLANAR DOMAINS

If 11 is a simply connected domain in C then, according to the Ahlfors-Gehring theorem, 11 is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in 11 in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to C and, under the additional assumption of quasiconformality in 11, they admit a quasiconformal extension to C. The Ahlfors-Gehring theorem has been extended to finitely connected domains 11 by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in 11 if and only if the components of a11 are either points or quasicircles. We generalize this theorem to harmonic mappings.