• Corpus ID: 234741936

Ahlfors-Weill extensions for harmonic mappings

@inproceedings{Efraimidis2021AhlforsWeillEF,
  title={Ahlfors-Weill extensions for harmonic mappings},
  author={Iason Efraimidis and Rodrigo Hern'andez and Mar'ia J. Mart'in},
  year={2021}
}
We provide two new formulas for quasiconformal extension to C for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension for holomorphic functions. 

References

SHOWING 1-10 OF 25 REFERENCES
Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative
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 in the unit disk and can be
Quasiconformal extension for harmonic mappings on finitely connected domains
We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal
Hyperbolic derivatives and generalized Schwarz-Pick estimates
In this paper we use the beautiful formula of Faa di Brune for the nth derivative of composition of two functions to obtain the generalized Scliwarz-Pick estimates. By means of those estimates we
Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings
In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition
Quasiconformal Extensions of Harmonic Mappings
We derive a very general condition for a sense-preserving harmonic mapping with dilatation a square to be injective in the unit disk $${{\mathbb {D}}}$$ and to admit a quasiconformal extension to
Harmonic Mappings in the Plane
1. Preliminaries 2. Local properties of harmonic mappings 3. Harmonic mappings onto convex regions 4. Harmonic self-mappings of the disk 5. Harmonic univalent functions 6. Extremal problems 7.
Quasiconformal extensions to space of Weierstrass-Enneper lifts
The Ahlfors-Weill extension of a conformal mapping of the disk is generalized to the Weierstrass-Enneper lift of a harmonic mapping of the disk to a minimal surface, producing homeomorphic and
Affine and linear invariant families of harmonic mappings
We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk. By using the famous shear construction of Clunie and Sheil-Small, we construct a function to
The Schwarzian derivative for harmonic mappings
The Schwarzian derivative of an analytic function is a basic tool in complex analysis. It appeared as early as 1873, when H. A. Schwarz sought to generalize the Schwarz-Christoffel formula to
Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains
If $\Omega$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $\Omega$ is a quasidisk if and only if there exists a sufficient condition for the
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