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 is defined for holomorphic functions defined on and normalized by . Let denote the Riemann surface of and the set of branch points. Then can be regarded as a lower bound for the radius of the largest disk contained in . The metric on used to measure the size of disks on is obtained by lifting the euclidean metric from to . The surface can also be regarded as spread over and the hyperbolic metric lifted to . One may then ask for the radius of the largest hyperbolic disk on . A lower bound for this radius is called a domain Bloch constant. The determination of domain Bloch constants is nontrivial for nonconstant analytic functions , where 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