Affine and linear invariant families of harmonic mappings

Martin Chuaqui; Rodrigo Hernández; María J. Martín

doi:10.1007/s00208-016-1418-x

Affine and linear invariant families of harmonic mappings

Martin Chuaqui, Rodrigo Hernández, María J. Martín

Faculty of Engineering and Science

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

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 determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class S_H of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of S_H and the Schwarzian norm of the harmonic Koebe function.

Original language	English
Pages (from-to)	1099-1122
Number of pages	24
Journal	Mathematische Annalen
Volume	367
Issue number	3-4
DOIs	https://doi.org/10.1007/s00208-016-1418-x
State	Published - 1 Apr 2017

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30C55

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AU - Hernández, Rodrigo

AU - Martín, María J.

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N2 - 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 determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class SH of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of SH and the Schwarzian norm of the harmonic Koebe function.

AB - 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 determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class SH of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of SH and the Schwarzian norm of the harmonic Koebe function.

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JO - Mathematische Annalen

JF - Mathematische Annalen

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Affine and linear invariant families of harmonic mappings

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