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The Carathéodory topology for multiply connected domains. II. (English) Zbl 1297.30045

The paper concerns the space \(X\) of pointed subdomains of \(\overline {\mathbb C}\) with the topology given by Carathéodory convergence. A pointed domain \((D,u)\) is non-degenerate if \(\mathbb C \setminus U\) has no \(1\)-point components. A family \(U\) of \(n\)-connected non-degenerate pointed domains is bounded if every sequence in \(U\) which converges in \(X\) has a limit which is \(n\)-connected and non-degenerate. A family \(U\) of pointed hyperbolic domains is non-degenerate if every sequence in \(U\) which converges in \(X\) has a limit which is a hyperbolic domain.
The paper establishes various conditions which ensure that a family of non-degenerate domains is bounded or that a family of hyperbolic domains is non-degenerate. This allows to extend the notions of convergence and equicontinuity to families of functions defined on varying domains. For Part I see [ibid. 11, No. 2, 322–340 (2013; Zbl 1282.30017)].

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C20 Conformal mappings of special domains
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)

Citations:

Zbl 1282.30017
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References:

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