On a Lipschitz estimate for conformal maps in the plane

Abstract

By a C2 boundary I mean that aD is a not necessarily connected, compact, 1-dimensional C2 submanifold of C, or, equivalently, that aD consists of finitely many disjoint closed Jordan curves whose parametrizations in terms of arclength are twice continuously differentiable. Stronger results, with more complicated proofs, have been known for a long time; for a more recent proof, one may consult Warschawski [4], where one can also find references to the older literature. I thank Professor H. Grunsky for pointing out this reference to me. PROOF. Fix z e D and restrict f to the disc with center z and radius d(z, aD). By Koebe's Theorem [1, Chapter IV, Satz 59], the image of this disc contains a disc with center f(z) and radius p f'(z)l * d(z, aD), where p > 0 is independent of z. This implies