Characterization of univalent harmonic mappings with integer or half-integer coefficients

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Research Article

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Analysis (Germany)


Let δHdenote the usual class of all normalized functions f = h + 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 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 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 Hiranuma and Sugawa. One of the examples generated in this way helps to disprove a conjecture of Bharanedhar and Ponnusamy.

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