  In this paper, we initiate the study of the geometric function theory for
slice starlike functions over quaternions and its subclasses. This allows us to
answer negatively some questions about the Bieberbach conjecture, the growth,
distortion, and covering theorems for slice regular functions. Precisely, we
find that the Bieberbach conjecture holds true for slice starlike functions in
contrast to the fact that the Bieberbach conjecture fails for biholomorphic
starlike mappings in higher dimensions. We also establish some sharp versions
of the growth, distortion, and covering theorems for quaternions.
