Sharp Lower Bounds for the Hyperbolic Metric of the Complement of a Closed Subset of the Unit Circle and Theorems of Schwarz–Pick-, Schottky- and Landau-type for Analytic Functions

• Published in: Constructive approximation Vol. 43; no. 1; pp. 47 - 69
• Main Authors: Kraus, Daniela, Roth, Oliver
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-02-2016
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Numerical Analysis; Hypergeometric functions; 35J65; Hyperbolic metric; 53A30; Primary 30E25; Covering properties of holomorphic functions
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-015-9313-3

Let
E
be a closed subset of the unit circle. We prove a sharp lower bound for the Poincaré metric
λ
C
\
E
(
z
)
|
d
z
|
of the domain
C
\
E
. This lower bound depends only on the minimal value of
λ
C
\
E
on the unit circle. For the case when
E
is the set
S
n
of
n
-th roots of unity, we explicitly find the minimal value in terms of hypergeometric functions and show that it is strictly increasing with respect to
n
. As a consequence, we obtain sharp Schwarz-, Schwarz–Pick-, Landau-, and Schottky-type theorems for analytic functions in the unit disk omitting the set
E
with precise numerical bounds for the special case
E
=
S
n
.