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The Bloch constant of bounded analytic functions on a multiply connected domain


Author: Flavia Colonna
Journal: Proc. Amer. Math. Soc. 112 (1991), 1055-1066
MSC: Primary 30D50; Secondary 30F99
DOI: https://doi.org/10.1090/S0002-9939-1991-1039529-9
MathSciNet review: 1039529
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Abstract: Let $ F$ be an analytic function on the bounded domain $ R$, a multiply-connected region of the complex plane. The Bloch constant of $ F$ is defined by

$\displaystyle \beta_F = \sup_{\vert z\vert < 1} (1 - \vert z\vert^2)\vert(F \circ p)'(z)\vert,$

where $ p$ is a conformal universal cover of $ R$ with domain $ \Delta $, the open unit disk. If $ F$ is bounded, then $ {\beta _F} \leq \vert\vert F\vert{\vert _\infty }$, the sup-norm of $ F$. In this paper we characterize those functions $ F$ for which $ {\beta _F} = \vert\vert F\vert{\vert _\infty }$ in terms of the zeros of $ F$ when the boundary of $ R$ is the union of finitely many curves. We conclude this paper by showing the existence of extremal functions, and generalizing the results to bounded harmonic mappings on these domains.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1039529-9
Keywords: Bloch functions, Blaschke products
Article copyright: © Copyright 1991 American Mathematical Society

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