Abstract
In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative S(f) of a locally univalent analytic function f in the unit disk was such that \(\limsup _{|z|\rightarrow 1} |S(f)(z)| (1-|z|^2)^2 < 2\), then there would exist a positive integer N such that f takes every value at most N times. Recently, Becker and Pommerenke have shown that the same result holds in those cases when the function f satisfies that \(\limsup _{|z|\rightarrow 1} |f''(z)/f'(z)|\, (1-|z|^2)< 1\). In this paper, we generalize these two criteria for bounded valence of analytic functions to the cases when f is only locally univalent and harmonic.
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We would like to thank the referees for their careful reading of the manuscript and for their useful suggestions.
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Communicated by Stephan Ruscheweyh.
This research is supported in part by Academy of Finland grant \(\#268009\). The first author is also supported by the Faculty of Science and Forestry of the University of Eastern Finland research project \(\#930349\). The second author thankfully acknowledges partial support from grants Fondecyt \(\#1150284\), Chile, Spanish MINECO/FEDER-EU research project MTM2015-65792-P, and by the Thematic Research Network MTM2015-69323-REDT, MINECO, Spain.
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Huusko, JM., Martín, M.J. Criteria for Bounded Valence of Harmonic Mappings. Comput. Methods Funct. Theory 17, 603–612 (2017). https://doi.org/10.1007/s40315-017-0197-z
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DOI: https://doi.org/10.1007/s40315-017-0197-z