Rosette Harmonic Mappings

Abstract

A harmonic mapping is a univalent harmonic function of one complex variable. We define a family of harmonic mappings on the unit disk whose images are rotationally symmetric rosettes with $n$ cusps or n nodes, where $n \ge 3$. These mappings are analogous to the $n$-cusped hypocycloid, but are modified by Gauss hypergeometric factors, both in the analytic and co-analytic parts. Relative rotations by an angle $\beta$ of the analytic and anti-analytic parts lead to graphs that have cyclic, and in some cases dihedral symmetry of order $n$. While the graphs for different $\beta$ can be dissimilar, the cusps are aligned along axes that are independent of $\beta$. For certain isolated values of $\beta$, the boundary function is continuous with arcs of constancy, and has nodes of interior angle $\pi/2-\pi/n$.