Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains

Univalence criteria for meromorphic functions and quasiconformal extensions

The aim of this paper is to obtain sufficient conditions for univalence of meromorphic functions in the U *. Also, we refine a quasiconformal extension criterion with the help of Becker's method. A number of univalence conditions would follow upon specializing the parameters involved in our main results. 1 Introduction We denote by U r = {z ∈ C : |z| < r} ( < r ≤ ) the disc of radius r and le...

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− ...

Locally Uniform Domains and Quasiconformal Mappings

We document various properties of the classes of locally uniform and weakly linearly locally connected domains. We describe the boundary behavior for quasiconformal ho-meomorphisms of these domains and exhibit certain metric conditions satissed by such maps. We characterize the quasiconformal homeomorphisms from locally uniform domains onto uniform domains. We furnish conditions which ensure th...

Harmonic Quasiconformal Mappings of Riemannian Manifolds

On Harmonic Quasiconformal Self-mappings of the Unit Ball

It is proved that any family of harmonic K-quasiconformal mappings {u = P [f ], u(0) = 0} of the unit ball onto itself is a uniformly Lipschitz family providing that f ∈ C. Moreover, the Lipschitz constant tends to 1 as K → 1.