Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative

Abstract

Abstract. We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping f in the unit disk is small enough, then f is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.

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

“…The proof of the univalence criterion in Theorem 1 follows closely the reasoning in [12]: We show that if a harmonic mapping f = h + g has small Schwarzian derivative then so does its analytic part h, and therefore h is univalent by Theorem A. The same can then be said about h + ag, the analytic part of the affine transformation f + af , a ∈ D. Finally, Hurwitz' theorem shows that h + ag is univalent for every a ∈ D and by an elementary rotational argument we get that f is injective.…”

“…This shows that if σ H (Ω) > 0 and Ω is finitely connected then, in view of Theorem B, every boundary component of Ω is either a point or a quasicircle. For Ω = D it was shown in [12] that σ H (D) > 0. We prove that the harmonic inner radius is positive for all quasidisks.…”

Criteria for Bounded Valence of Harmonic Mappings

“…The argument of the proof is analogous to the one used to prove Theorem 1. Condition (11) implies that there exist a real number ρ with 1/2 < ρ < 1 and ε > 0 such that (12) |S…”

The Schwarzian derivatives of harmonic functions and univalence conditions

“…In particular, the authors of [5] observed a deep connection of S f with lifts of harmonic functions onto minimal surfaces. In [15,16] some estimations of S f in some subclasses of univalent harmonic functions were obtained and many properties of the Schwarzian were established. Norms of the Pre-Schrarzian and Schwarzian derivative S f were estimated in [14] for the linear-and affine-invariant families of harmonic functions in terms of order of the family; so, analogues of the Krauss and Nehari theorem about the upper bounds of |S f | were obtained.…”

“…In [16] R. Hernández and M. J. Martín proved an analogue of Theorem A for the S f in the following form: they proved the existence of constant C such that for f = h + g the inequality…”