Additional smoothness phenomena for analytic functions

The influence is studied of the geometric properties of a domian on the smoothness of Hölder class analytic functions defined on it. The case of the disc is covered by classical results of Hurdy and Littlewood. We consider a domian $G$ with an inward cusp boundary point $\xi$ (this means that, $\operatorname{meas}U_{\xi}\cap(\mathbb C\setminus G)/\operatorname{meas}U_{\xi}\to0$ as $\operatorname{meas}U_{\xi}\to0$, where the $U_{\xi}$ stands for a neighborhood of $\xi$). Three zones are distinguished near such a point: the outer zone, where high smoothness occur; the boundary zone, where the smoothness is “standard”, and the intermediate zone, where the smoothness decays steadily from high to standard. A sharp geometric description of these zones is given.