I nternat. J. Math.
Mh. S ci.
369
Vol. 2 #3 (1979) 369-413
THE BASIC PROPERTIES OF BLOCH FUNCTIONS
JOSEPH A. CIMA Mathematics Department The University of North Carolina at Chapel Hill Chapel Hill, North Carolina 27514
U.S.A.
(Received February 2, 1979)
ABSTRACT.
A Bloch function f(z) is an analytic function on the unit dsc
])
whose
derivative grows no faster than a constant times the reciprocal of the dstance
We reprove here the basic analytic facts concerning Bloch functions.
from z to ]).
We establish the Banach space structure and collect facts concerning the geometry of the space.
We indicate dualty relationships, and known somorphc correspond-
ences are given.
We give a rather complete llst of references for further study
in the case of several variables.
KEY WORDS AND PHRASES. Bloch function, Schlicht discs, Normal funcion, is’omopi Bana’h spaces, Mobius invaiant subspac. 1980 MATHEMATICS SUBJECT CLASSIFICATION CES.
30A78, 46E15.
.
370
i.
A. CIMA
Introduction. The purpose of this article is to give a survey and some proofs of The basic idea goes back to
known results concerning Bloch functions. Andre Bloch
6 ].
on the unit disc
f
under
F
He considered the class
9,
with normalization
of functions holomorphlc
f’ (0)
Wf
is considered as a Riemann surface
(unramlfled) disc in
=
exists a domain
is an open disc
Wf 9
with
f
A
c
dr(Z).
Let
df(z)
be the supremum of
rf
such that there
one to one onto
mapping
denote the radius of the largest schllcht disc in
as
A schllcht
f((C)).
f
D
The image of
i.
4.
We
f
with center
z
varies over
as
f(z)
and set
b
Bloch showed that
b
inf
{rf
f
E
F}.
was posltive.
During the period from 1925 through 1968 Bloch’s result motivated
works of various nature.
One group of mathematicians considered the
generalizations of Bloch’s result to balls in
n.
and
mathematicians calculated upper and lower bounds for
b.
A group of
A third group
concentrated on the function theoretic implications for the case of the disc.
The Bloch theorem has been an ingredient in supplying a proof of
the Picard theorem which avoids the use of the modular function.
We will
not go into the generalizations for the n dimensional case but refer the interested reader to the papers of S. Bochner
and K. Sakaguchl
20 ].
and M.H. Helns
], S. Takahashi
25
We will also not discuss the best bounds but only
refer to the papers of L. V. Ahlfors 2
7
13 ].
i
], L. V. Ahlfors and H. Grunsky
PROPERTIES OF BLOCH FUNCTIONS
371
In the period from 1969 to the present a Banach space
B
of
holomorphic functions called the "Bloch functions" has been studied. Of course the requirement for membership in of the Bloch theorem.
B
is derived from the idea
Some progress has been made in studying the
B.
functional analytic properties of
The Banach space point of view has
allowed a somewhat broader viewpoint and consequently has given rise to a new set of questions concerning the Bloch
space.
This article will give a proof of the basic Bloch theorem.
will follow a theme developed by W. Seidel and J. Walsh [24 Ch. Pommerenke [17 ].
We
and by
We will supply proofs of the major results and
outline proofs of other ideas when they are not central to our interests.
We have borrowed freely from the text material available (especially
M. Heins [13 ]).
In many instances we have selected only partial results
from the journal articles quoted in the bibliography.
The reader should
consult the original article if he desires a more complete exposition.
Finally, I wish to point out a few other results which will not be included in this article but are extremely important to the overall
picture concerning Bloch functions.
First, L. A. Harris [12] has obtained
a strong form of the Bloch theorem for holomorphic mappings from the unit
of a Banach space X into X.
B,
ball
thesis of R. Timoney.
The second topic concerns the
He has made a definitive study of Bloch functions
on bounded symmetric domains in
n.
This work is quite expansive and
deep and would require material from areas which are not considered in the disc case.
2.
The Theorem of Bloch.
Let For
a
Aut (D) (C)
denote the group of holomorphic automorphisms of
we write
a(Z)
(z-a) (l-z)
-I
Aut(D).
D.
The inverse of
J.A. CIMA
372
a
is
Let
a
S-a" (0,I)
D + D
f
It can be shown that for
A
and denote A
a
A
a.
A
0 < x < x
if
B’(x 0)
0.
Thus
p
0. 0
B.
The remaining uniqueness part of the theorem is handled by noting that if
Of
p then
Im(O)[
1.
We proceed to a second necessary result. )
r
that
{lwl
0
< r}.
r
To each
f
sup {r
and
f
A
f
r
let
let a domain
maps
r > 0
For
nr --c
D
univalently onto
such
D
r
and s
inf
{f
f
A}.
J.A. CIMA
374 The number
Theorem 2.2.
is positive, equaling
s
2
Equality holds for some
,
for same constant
If(z)
(2.3)
(
in
A,
I1
->
0( 0
X.
p,
of
(C).
be a Moblus-lnvarlant semlnorm
p
If there exists a decent linear functional
with respect to
K
be a Mobius-invariant linear space of analytic
X
functions on the unit disc and let
on
K}
X
then
B
c
L
on
X
and there exists a constant
continuous
A > 0
such that
pB(f) for all
pB(f) =-
Sup
In this theorem
X.
f
< A p(f)
(l-lzlm) If’(z)l
pB(f)
so tht
zD
PB (f)
{l(fo)’(0)l
Sup
PB(C)
0
c
Aut(D)}
This motivates the following lemma.
Lemma [3.17].
For each
pn(f)
Sup
n
O.
By the Hahn-Banach Theorem L extends to a continuous linear functional (denoted again as on
D)
and the inequality for
known that (bN
+ 0)
L
L(f
e
)I
L
on
H() K
(all holomorphlc functions remains valid.
- -
holomorphic in
(z)
on
can be identified with a function
L(f)
Let
L)
z
[zl
>
r, r < i,
g(z)
It is well
[ n-Ne0
b Z -n n
in the following way
f(z) g(z) d__z z
be a rotation and change variables to deduce i
f
f(z) g(e
ilz )
dz
-
0
we conclude
by
f
Sup
f
IbNl IfN(0)
>
and taking the supremum we obtain
)N(0)I
{}(f
Aut(D)}
