A Riemannian Bieberbach estimate

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)| ≤ 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 → Rn, n ≥ 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.