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

Facultad de Ingeniería y Ciencias

Producción científica: Contribución a una revista › Artículo › revisión exhaustiva

8 Citas (Scopus)

Resumen

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.

Idioma original	Inglés
Páginas (desde-hasta)	1099-1122
Número de páginas	24
Publicación	Mathematische Annalen
Volumen	367
N.º	3-4
DOI	https://doi.org/10.1007/s00208-016-1418-x
Estado	Publicada - 1 abr. 2017

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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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Affine and linear invariant families of harmonic mappings

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