QUASICONFORMAL EXTENSIONS OF STARLIKE HARMONIC MAPPINGS IN THE UNIT DISC

Abstract

Abstract. Let f be a harmonic mapping on the unit disc ∆ in C. Wegive some condition for f to be a quasiconformal homeomorphism on ∆and to have a quasiconformal extension to the whole plane C. We alsoobtain quasiconformal extension results for starlike harmonic mappingsof order α ∈(0,1). 1. IntroductionLet f be a complex-valued function of class C 1 on ∆ = {z∈ C;|z| 0 in ∆) or sense-reversing (if J f (z) <0 in ∆). A harmonic mappingof ∆ has the unique representation f= h+g, where hand gare analytic in ∆and g(0) = 0. Note that fis sense-preserving if and only if |g ′ (z)| <|h ′ (z)| forall z∈ ∆ (For univalent harmonic mappings, see [5]).Let f= h+ gbe a harmonic mapping of the form(1.1) h(z) = z+X ∞n=2 a n z n , g(z) =X