Some generalizations of Ahlfors Lemma

One of the most influential versions of the classical Schwarz Pick Lemma is probably that of Ahlfors. Pulling back a conformal semimetric on a Riemann surface under any holomorphic map from the open unit disk equipped with a Poincare metric, the curvature of which is assumed to bound from above the curvature of the Riemann surface, he successfully showed that a conformal semimetric to be compared with the Poincare metric is obtained. In the present paper, we give a comparison theorem between two conformal semimetrics of variable curvature in the same spirit. Our main theorem is a local one by its nature, but global results can be derived therefrom. (C) 2019 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.