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Sharp lower bounds for the hyperbolic metric of the complement of a closed subset of the unit circle and theorems of Schwarz-Pick-, Schottky- and Landau-type for analytic functions. (English) Zbl 1341.30040

A sharp lower bound is proved for the Poincaré metric \(\lambda_{\mathbb{C}}\setminus E(x)|dz|\) of the domain \(\mathbb{C}\setminus E\), where \(E\) is a closed subset of the unit circle. This lower bound depends only on the minimal value of \(\lambda_{\mathbb{C}}\setminus E\) on the unit circle. The minimal value in terms of hypergeometric functions for the case when \(E\) is the set \(S_n\) of \(n\)-th roots of unity is explicitly found, and it is shown that it is strictly increasing with respect to \(n\). As a consequence, sharp Schwarz-, Schwarz-Pick-, Landau-, and Schottky-type theorems are obtained for analytic functions in the unit disk omitting the set \(E\) with precise numerical bounds for the special case \(E=S_n\).

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
33C05 Classical hypergeometric functions, \({}_2F_1\)
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