On the Hurwitz metric

Abstract

The Hurwitz metric was recently defined by Minda by considering a variational problem that involves holomorphic maps from the disc that are globally injective at the origin. In this note, sharp boundary estimates for this metric are obtained on $C^2$-smooth planar domains and as a consequence, it is shown that it is uniformly comparable with the Caratheodory and Kobayashi metrics on such domains. In addition, estimates for the generalized curvatures of this metric are also given.