Quasiconformal Extension of Harmonic Mappings in the Plane

Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane. Introduction A well-known criterion due to Becker [5] states that if a locally univalent analytic function φ in the unit disk D satisfies (1) sup z∈D ∣∣∣∣φ′′(z) φ′(z) ∣∣∣∣ (1− |z|) ≤ 1, then φ is, indeed, univalent in D. Becker and Pommerenke [6] proved later that the constant 1 is sharp. The quotient Pφ = φ′′/φ′ is called the pre-Schwarzian derivative of φ, a function that is well-defined in the unit disk for every locally univalent function φ in D. In that paper [5], the author also proves that if (2) sup z∈D |Pφ(z)| (1− |z|) ≤ k < 1, then not only φ is univalent but it has a continuous extension φ̃ to D and φ̃(∂D) is a quasicircle. Indeed, using Löwner’s chains, Becker shows that φ has a Kquasiconformal extension to the whole complex plane C, where K = (1+ k)/(1− k). Moreover, Ahlfors [3] gives an explicit quasiconformal extension. Namely, the function Φ(z) = { φ̃(z), |z| ≤ 1, φ ( 1 z ) + u ( 1 z ) , |z| > 1, where, for z ∈ D\{0}, u(z) = φ′(z)(1−|z|2)/z. This mapping Φ is K-quasiconformal (with K as above) in the complex plane whenever (2) holds and coincides with φ in D. Let now f be a locally univalent harmonic mapping defined in the unit disk and f = h+g its canonical representation, where h and g are analytic in D and g(0) = 0. The (second) complex dilatation of f is ω = g′/h′. Since f is locally univalent, by doi:10.5186/aasfm.2013.3824 2010 Mathematics Subject Classification: Primary 30C45, 30C55.