Global C ^2+α-Estimates for Conformai Maps

Summary

For conformai maps g on the closed unit disc $\overline B$ we shall estimate $\left\| g \right\|\,_{C^{2 + a} \,(\bar B)}$ by the relevant geometric data from above. Here we bound the modulus of their derivatives $\left| {g'(w)} \right| > 0,w \in \,\overline B$ quantitatively from below. Only the maximum principle and Minding’s formula for the geodesic curvature are necessary in the proof. For instance, our estimates suffice to construct conformai maps of the class ${C^{{2 + \alpha }}}(\overline B )$ approximating C ^2+α-domains by real-analytic Jordan-domains.