A note on the Bieberbach conjecture for some classes of slice regular functions

In this note we prove the Bieberbach conjecture for some classes of quaternionic functions, including quaternionic slice regular functions with specific geometric properties such as starlike and convex functions. At the same time, we investigate some interesting properties related to the concepts of starlikeness for these quaternionic slice regular functions. Furthermore, we introduce and obtain generalizations and results for the class of quaternionic Carath´eodory functions, such as a version of Rogosinski’s Theorem over the quaternions.