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