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Mathematische Annalen

, Volume 367, Issue 3–4, pp 1099–1122 | Cite as

Affine and linear invariant families of harmonic mappings

  • Martin Chuaqui
  • Rodrigo Hernández
  • María J. Martín
Article

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 \(\mathcal {S}_H\) 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 \(\mathcal {S}_H\) and the Schwarzian norm of the harmonic Koebe function.

Mathematics Subject Classification

30C55 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Martin Chuaqui
    • 1
  • Rodrigo Hernández
    • 2
  • María J. Martín
    • 3
  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Facultad de Ingeniería y CienciasUniversidad Adolfo IbáñezViña del MarChile
  3. 3.Department of Physics and MathematicsUniversity of Eastern FinlandJoensuuFinland

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