Domain Bloch constants

Minda, C. David

doi:10.1090/S0002-9947-1983-0688967-2

Domain Bloch constants
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Trans. Amer. Math. Soc. 276 (1983), 645-655 Request permission

Abstract:

The classical Bloch constant $\mathcal {B}$ is defined for holomorphic functions $f$ defined on ${\mathbf {B}} = \{z:|z| < 1\}$ and normalized by $|f’(0)| = 1$. 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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On Marden’s universal constant of Fuchsian groups