Univalent harmonic mappings with integer or half-integer coefficients

TL;DR

This paper classifies univalent harmonic mappings with integer or half-integer coefficients, exploring their convexity properties and extending known results about univalent analytic functions with such coefficients.

Contribution

It extends the classification of univalent functions with integer or half-integer coefficients to harmonic mappings, analyzing their convexity in specific directions.

Findings

Classified univalent harmonic mappings with integer or half-integer coefficients.

Analyzed convexity properties in real and imaginary directions.

Extended known results from analytic to harmonic mappings.

Abstract

Let ${\mathcal S}$ denote the set of all univalent analytic functions $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ on the unit disk $|z|<1$. In 1946 B. Friedman found that the set $\mathcal S$ 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.