On a Class of Conformal Metrics

Last year when I was preparing for course lectures the work of Ahlfors 1 which establishes that the Bloch constant is at least as large as it appeared to me that the resources of the theory of metrics of negative curvature offered rich possibilities from a function-theoretic point of view. The parallelism between certain properties of subharmonic functions and those of the metrics introduced by Ahlfors 1 is so striking that we are led to ask whether one can introduce a class of metrics including the metrics of Ahlfors for which not only does a Schwarz-Pick-Ahlfors lemma hold, but also requirements of differentiability disappear, as in the modern theory of subharmonic functions.