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.
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Communicated by Ngaiming Mok.
The authors are partially supported by grants Fondecyt 1110321, 1150115, and 1150284, Chile. The second author also thankfully acknowledges partial support from Faculty of Forestry and Sciences UEF, Finland (930349). The third author is supported by Academy of Finland grant 268009 and by Spanish MINECO Research Projects MTM2012-37436-C02-02 and MTM2015-65792-P.
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Chuaqui, M., Hernández, R. & Martín, M.J. Affine and linear invariant families of harmonic mappings. Math. Ann. 367, 1099–1122 (2017). https://doi.org/10.1007/s00208-016-1418-x
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DOI: https://doi.org/10.1007/s00208-016-1418-x