Domain Bloch constants

Abstract

The classical Bloch constant B \mathcal {B} is defined for holomorphic functions f f defined on B = { z : | z | > 1 } {\mathbf {B}} = \{z:|z| > 1\} and normalized by | f ′ ( 0 ) | = 1 |f’(0)| = 1 . Let R f {R_f} denote the Riemann surface of f f and B f {B_f} the set of branch points. Then B \mathcal {B} can be regarded as a lower bound for the radius of the largest disk contained in R f ∖ B f {R_f}\backslash {B_f} . The metric on R f {R_f} used to measure the size of disks on R f {R_f} is obtained by lifting the euclidean metric from C {\mathbf {C}} to R f {R_f} . The surface R f {R_f} can also be regarded as spread over B {\mathbf {B}} and the hyperbolic metric lifted to R f {R_f} . One may then ask for the radius of the largest hyperbolic disk on R f ∖ B f {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 : B → X f:{\mathbf {B}} \to X , where X 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.