Rosette Harmonic Mappings

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