  The Bieberbach estimate, a pivotal result in the classical theory of
univalent functions, states that any injective holomorphic function $f$ on the
open unit disc $D$ satisfies $|f"(0)|\leq 4 |f'(0)|$. We generalize the
Bieberbach estimate by proving a version of the inequality that applies to all
injective smooth conformal immersions $f : D\to \Bbb R^n, n\geq 2$. The new
estimate involves two correction terms. The first one is geometric, coming from
the second fundamental form of the image surface $f(D)$. The second term is of
a dynamical nature, and involves certain Riemannian quantities associated to
conformal attractors. Our results are partly motivated by a conjecture in the
theory of embedded minimal surfaces.
