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Domain Bloch constants


Author: C. David Minda
Journal: Trans. Amer. Math. Soc. 276 (1983), 645-655
MSC: Primary 30D45; Secondary 30F15
DOI: https://doi.org/10.1090/S0002-9947-1983-0688967-2
MathSciNet review: 688967
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Abstract: The classical Bloch constant $ \mathcal{B}$ is defined for holomorphic functions $ f$ defined on $ {\mathbf{B}} = \{z:\vert z\vert < 1\} $ and normalized by $ \vert f^{\prime}(0)\vert = 1$. Let $ {R_f}$ denote the Riemann surface of $ f$ and $ {B_f}$ the set of branch points. Then $ \mathcal{B}$ can be regarded as a lower bound for the radius of the largest disk contained in $ {R_f}\backslash {B_f}$. The metric on $ {R_f}$ used to measure the size of disks on $ {R_f}$ is obtained by lifting the euclidean metric from $ {\mathbf{C}}$ to $ {R_f}$. The surface $ {R_f}$ can also be regarded as spread over $ {\mathbf{B}}$ and the hyperbolic metric lifted to $ {R_f}$. One may then ask for the radius of the largest hyperbolic disk on $ {R_f}\backslash {B_f}$. A lower bound for this radius is called a domain Bloch constant. The determination of domain Bloch constants is nontrivial for nonconstant analytic functions $ f:{\mathbf{B}} \to X$, where $ X$ is a hyperbolic Riemann surface. Upper and lower bounds for domain Bloch constants are given. Also, domain Bloch constants are given an interpretation as a radius of local schlichtness.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0688967-2
Article copyright: © Copyright 1983 American Mathematical Society

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