We establish the sharp growth and distortion theorems for slice
monogenic extensions of univalent functions on the unit disc
in the
setting of Clifford algebras, based on a new convex combination identity. The
analogous results are also valid in the quaternionic setting for slice regular functions
and we can even prove a Koebe type one-quarter theorem in this case. Our growth
and distortion theorems for slice regular (slice monogenic) extensions to higher
dimensions of univalent holomorphic functions hold without extra geometric
assumptions, in contrast to the setting of several complex variables in which the
growth and distortion theorems fail in general and hold only for some subclasses with
the starlike or convex assumption.
