On holomorphic functions satisfying 𝑓(𝑧)(1-𝑧²)≤1 in the unit disc
-
Published:1982
Issue:1
Volume:85
Page:19-23
-
ISSN:0002-9939
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Container-title:Proceedings of the American Mathematical Society
-
language:en
-
Short-container-title:Proc. Amer. Math. Soc.
Author:
Wirths Karl-Joachim
Abstract
Let
f
f
be holomorphic in
D
=
{
z
|
|
z
|
>
1
}
D = \{ \left . z \right |\left | z \right | > 1\}
,
|
f
(
z
)
|
(
1
−
|
z
|
2
)
⩽
1
\left | {f(z)} \right |(1 - {\left | z \right |^2}) \leqslant 1
in
D
D
,
lim
¯
|
z
|
→
1
|
f
(
z
)
|
(
1
−
|
z
|
2
)
>
1
{\overline {\lim } _{\left | z \right | \to 1}}\left | {f(z)} \right |(1 - {\left | z \right |^2}) > 1
and
L
(
f
)
:=
{
z
|
|
f
(
z
)
|
(
1
−
|
z
|
2
)
=
1
}
L(f): = \{ \left . z \right |\left | {f(z)} \right |(1 - {\left | z \right |^2}) = 1\}
. It is shown that the set
L
(
f
)
L(f)
consists of one simple closed curve
γ
\gamma
and a finite number of points in the bounded component of
C
∖
γ
{\mathbf {C}}\backslash \gamma
if
L
(
f
)
L(f)
is an infinite set.
Publisher
American Mathematical Society (AMS)
Subject
Applied Mathematics,General Mathematics
Reference2 articles.
1. Die Grundlehren der mathematischen Wissenschaften, Band 77;Behnke, Heinrich,1962
2. Extreme points of the unit ball of the Bloch space \cal𝐵₀;Cima, Joseph A.;Michigan Math. J.,1978
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