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Characterization of univalent harmonic mappings with integer or half-integer coefficients. (English) Zbl 1360.31003

Summary: Let \({\mathcal{S}_{H}}\) denote the usual class of all normalized functions \({f=h+\overline{g}}\) harmonic and sense-preserving univalent on the unit disk \({|z|<1}\). In this article we show that the set, consisting of those mappings \(f\) from \({\mathcal{S}_{H}}\) for which all Taylor coefficients of the analytic and co-analytic parts of \(f\) are integers, consists of only nine functions. The second aim is to discuss the set \({\mathcal{S}_{H}}\) of those functions which have half-integer coefficients. More precisely, we determine the set of univalent harmonic mappings with half-integer coefficients which are convex in real direction or convex in imaginary direction. This work generalizes the recent paper of N. Hiranuma and T. Sugawa [Comput. Methods Funct. Theory 13, No. 1, 133–151 (2013; Zbl 1278.30023)]. One of the examples generated in this way helps to disprove a conjecture of S. V. Bharanedhar and S. Ponnusamy [Rocky Mt. J. Math. 44, No. 3, 753–777 (2014; Zbl 1298.30001)].

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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