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

Solynin, Alexander

doi:10.1090/S0002-9939-08-09309-X

A Schwarz lemma for meromorphic functions and estimates for the hyperbolic metric
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by Alexander Yu. Solynin PDF

Proc. Amer. Math. Soc. 136 (2008), 3133-3143 Request permission

Abstract:

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

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