Sufficient conditions for quasiconformality of harmonic mappings of the upper halfplane onto itself

Abstract

Abstract. In this paper we introduce a class of increasing homeomorphic self-mappings of R. We define a harmonic extension of such functions to the upper halfplane by means of the Poisson integral. Our main results give some sufficient conditions for quasiconformality of the extension.1. Introduction. Let F be a complex-valued sense-preserving diffeomorphism of the upper halfplane C + := {z ∈ C : Im z > 0} onto itself, where C stands for the complex plane. Then the Jacobianis positive on C + and so the functionis well defined. We recall that D F (z) is called the maximal dilatation of F at z ∈ C + . Here and in the sequel ∂ := (∂ x −i∂ y )/2 and∂ := (∂ x +i∂ y )/2 stands for the formal derivatives operators. From the analytical characterization of quasiconformal mappings (see [3]) it follows that for any K ≥ 1, F is 2000 Mathematics Subject Classification. 30C55, 30C62.