In 1984, a simple and useful univalence criterion for harmonic functions was given by Clunie and Sheil-Small, which is usually called the shear construction. However, the application of this theorem is limited to the planar harmonic mappings convex in the horizontal direction. In this paper, a natural generalization of the shear construction is given. More precisely, our results are obtained under the hypothesis that the image of a harmonic mapping is a sum of two sets convex in the horizontal direction.

Let ϕ be a self-map of the unit disk and let C ϕ denote the composition operator acting on the standard Dirichlet space D. A necessary condition for compactness of a difference of two bounded composition operators acting on D, is given. As an application, a characterization of disk automorphisms ϕ and ψ for which the commutatorwhere the branch of the logarithm is chosen such thatBy a self-map of D we mean an analytic function ϕ such that ϕ(D) ⊂ D. We will also assume that a self-map ϕ is not a constant function. For a self-map of the unit disk ϕ, the composition operator C ϕ on the Dirichlet space D is defined by C ϕ f := f • ϕ. The composition operator C ϕ on Dirichlet space is not necessarily bounded for an arbitrary self-map of the unit disk. However, C ϕ is bounded on D if, for example, ϕ is a finitely Date: May 22, 2018. 2010 Mathematics Subject Classification. 47B33.

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

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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.