Univalent harmonic mappings with integer or half-integer coefficients

arXiv1207.3768MaRDI QIDQ6234417

S. Ponnusamy, J. Qiao

Publication date: 12 July 2012

Abstract: Let $mathcalS$ denote the set of all univalent analytic functions $f(z)=z+su{m}_{n=2}^{infty}{a}_n{z}^n$ on the unit disk $|z|<1$. In 1946 B. Friedman found that the set $mathcalS$ of those functions which have integer coefficients consists of only nine functions. In a recent paper Hiranuma and Sugawa proved that the similar set obtained for the functions with half-integer coefficients consists of twelve functions in addition to the nine. In this paper, the main aim is to discuss the class of all sense-preserving univalent harmonic mappings $f$ on the unit disk with integer or half-integer coefficients for the analytic and co-analytic parts of $f$. Secondly, we consider the class of univalent harmonic mappings with integer coefficients, and consider the convexity in real direction and convexity in imaginary direction of these mappings. Thirdly, we determine the set of univalent harmonic mappings with half-integer coefficients which are convex in real direction or convex in imaginary direction.

Mathematics Subject Classification ID

Quasiconformal mappings in (mathbb{R}^n), other generalizations (30C65) Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) (30C45) Conformal mappings of special domains (30C20)

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