A schwarz lemma for meromorphic functions and estimates for the hyperbolic metric

We prove a generalization of the Schwarz lemma for meromorphic functions f mapping the unit disk D onto Riemann surfaces ℛ with bounded in mean radial distances from f(0) to the boundary of ℛ. A newvariant of the Schwarz lemma is also proved for the Carathèodory class of analytic functions having positive real part in D. Our results lead to several improved estimates for the hyperbolic metric.