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
<
r)}
is a nonempty
0(z)’a(Z)"
B(z)
f
0, f’(0)
It contains the function
compact family.
For
the family of holomorphic functions
f(0)
with normalization
z
set
sup {r
Uf
is univalent in
f
Dr
(Izl
and
inf
P
A calculation shows that
Theorem (2.1).
f c A}.
{Uf
B’
The number
vanishes at a point of
is positive.
D,
The number
hence p < i.
Uf
if
and
only if
f(z)
where
Proof.
f
l
lB(1)
is a constant of modulus one.
Fix
f
A
and assume
Uf
<
I.
Recalling the normalization of
we see that equation
a (f.(z))=z zk(z)
yields a function
k,
holomorphic in D
and bounded by one.
Hence, by (2.1)
PROPERTIES OF BLOCH FUNCTIONS
Either
a
Iz
f(z)
(2.2)
z01 0f
z0,
zl
fails to be one to one on
f
f’ (z 0)
and
f(z 2)
f(zl)
0f
_<
or there exists
We will complete the proof by
0.
z
assuming that there exists distinct points
and
373
and
I
z 2,
The other case is similar.
c.
(c
m(z)--
-i
f(z))
zl
The function
-i
z2(Z)
(z)
has removable singularities and is bounded by one in D.
Im(0) l-<
Ici_<
implying
2
Of
z--z
Setting
z21 Of
Zll
(2.1)
in
I
Hence,
and using
the last inequality we have
_>
Uf
The functions
x
0
a
-I
0f
Thus
x
(I
a(X)
and
a
2
a
2
U
have a common value at
(0,I)
)
(I-’
_> a -I
-"’a
).
a(Z)
and
We note that
x
_>
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(<a).
there is
f(z*)
Let
0)
-
mlCz)
(r)
(0,a)
r
n
into
f
-i
D02)
02
and maps
z*l
maps the (simply connected) domain
B
observe that
0
2 <
s <
B
< 0
2
2 <
s.
_c D0.
Hence
D
univalently onto
2
(C)
2-
0-
s
0
2
B(0)
0
and
0
f
By definition
cannot be univalent on any domain containing
and so
Using
f(z)
Since
0
(0)
Since
If
maps the
f
B’ (0)
0
2
and
0
it follows by the argument principle that
has only one solution in
it follows that
onto
0
containing zero.
z*,
then there exists a
(lwl
r < i
and achieves a maximum value of
(2.3) again and the open mapping property one sees that
boundary of
Izl
a-r
(l--ar)
By inequality (2.3) this is absurd.
(C)02.
AB(Az)
f(z)
if and only if
s,
be the component of
(D \ D
z
f
i.
is a positive function on
at r
-/Y-a)] 2
inequality (2.2) implies that for
A
f
For any
Proof:
f
a-i
we
Thus
The remaining uniqueness results follow
in a routine way by a close examination of when equality holds in (2.2)
We are finally in a position to prove Bloch’s theorem.
is that used in the introduction.
Theorem (2.3).
The number
b
is positive.
The notation
PROPERTIES OF BLOCH FUNCTIONS
Elementary considerations show that for
Proof.
-i
we have
f(tz) t
g(z)
375
consider those
F
f
r
<
g
t
rf
-i
f
F,
0 < t < I
Thus it is sufficient to
which are analytic on the closed disc
lal
be so chosen and let
and
(C).
Let
f
be selected so that
<i
>1.
Now form the function
fo_ a (z)
g(z)
Note that
F and
g
(fo _a) (0)
is analytic on
9
with
I
<
{(fO_a)’(0) rf rf
g
Further,
If
-1./ zl
,y//
-<
lf’fa)
(1-
and so
(1Replacing
Ih(-){
Let
i
t
maps
is in
D
/
(0,I).
D,
S(0)
Hence,
t
=_ i
and normalize
h as follows
()
0
S
-
and integrating we deduce
h(tz)
St(z)
S
-<
g(0)
i+ Izl
1-Izl
og
(0,I)
be fixed in
t
-<
g’(z)
g(z)
h(z)
by
g
lzl 2)
[al 2
and
S ’(0)
Aa.
We apply Theorem (2.2) to
t
2ilL(t).
The number
St
L(t)
a
and note
376
J. A. ClMA
FL(t)
Since,
rf
> r
h
(),)-2-t
2)]
we conclude
t
If one checks the value of this expression (preferably with a calculator)
at
I
t
one concludes
r > 0.2.
The papers refered to in the introduction establish the bounds
_1 /--< b
4
We proceed to the paper of
the expression
f(D)
dr(Z)
concerning
dr(Z)
W. Seldel and J. Walsh [24].
d(z)
f(z).
with center
< 0 472
They introduced
as the radius of the largest schllcht disc in
In this paper they collect some known results
and they prove several theorems about its growth.
The
following two theorems are interesting and motivating.
Theorem (2.4).
Let
f
d(z) <
Proof.
Let
z
0
E
D
be holomorphic and univalent in
If’(-)l(l-lzl2)<
be given.
(C).
Then for
4 d(z).
We prove first the right inequality.
Form
the function
_zo(Z) f(Zo)
f
(z)
where
z E D.
omits the value
f’
(z0) (1-’Iz012
is a normalized univalent function and if
f(D)
then
PROPERTIES OF BLOCH FUNCTIONS
377
-f(z 0)
(f,
(z0)) (i_i z01 2)
suitably on
Applying the Koebe one-quarter theorem and choosing
f(D)
we conclude that
d(z O)
1
(l-{z0{2)f’(z 0)
To obtain the left inequality again let
a function
h
inverse
(z)
_z0(Z ).
f
h’ (0)
<
0
be fixed in
(0))
0
I 0 -I < d(z 0)
mapping the disc
The Schwarz lemma yields
f(z0)
If
z
9
into
(d(z0))-i
D
and form
then
with
has an
h( 0)
0.
and this completes the
proof.
Notice that the unlvalence of
df(z)
left inequality and hence
holomorphlc
If(z)
<
If’ (z)
(i
zl 2)
is valid for
Let
IlflI
f
M.
d(z)
Outline of proof.
be a bounded holomorphlc function on
<
If’(z)
1
(l-lz121
<
[SM d(zll
We have noted that the left inequality is valid.
z
func t ion
-z (z)
f
z)
(z)
0
and
D,
Then
idea of the proof in the right inequality is to fix
Then
any
A second result of interest in that paper is the following.
f.
Theorem (2.5).
<
is not used in the proof of the
f
l(z)
0
f(z 0)
f’(z0)(l-lz012)
2M
<
if, (z0)
(l_Iz012)
0
D
The
and form the
J.A. CIMA
378
and apply a variant of Theorem (2.2) to con-
One can then normalize
covers the disc
elude that
If’(Zo) l(l-lZo 12)
8M
This in turn implies that
unlvalently.
f (z
0)
2
If’(z 0)
f
covers the disc
(l-lz012)
8M
From this it follows that
univalently.
I
If’(z0)
(l-lz012)
<
[SM d(z 0) 2
The term
The next theorem is a statement of several equivalences.
The
Bloch function will be defined by any of these equivalent conditions.
24 ].
equivalence of (i) and (2) is in Seldel and Walsh
condition is due to Zygmund.
The last is in Anderson and Rubel
The conditions (3) and (4)were given by Pommerenke
few terms.
The fifth
f(z) (on D)
A holomorphlc function
17 ].
4
].
We recall a
is said to be finitely
normal if the family of functions
{f@(z)
f(@(0))
f(@(z))
@
E
Aut (D)}
,
is a normal family, where the constant infinity is not allowed as a limit.
A function
g
continuous on
zl
I
is in the class
condition holds
Ig(e i(8+h))
2g(e 18) +
g(ei(8-h))
0(h)
if the following
PROPERTIES OF BLOCH FUNCTIONS
uniformly for all
For
of
f
[0,2)
and
D,
holomorphic in
Aut())
under
f
e
n
.
{j=laj
379
h>O.
the absolute convex hull of the orbit
is defined as the family
(foj)(z)
N,
n
Aut(D),
a.3
n
with
aj
1
Theorem (2.6).
in
on
D
aj]
}.
-<
The following conditions on a function
f
holomorphic
are equivalent:
(C)}
+
(i)
sup {d(z)
(2)
sup
(3)
f
(4)
there exists a constant
z
<
{(l-lzl 2) If’(z)l
z e }} <
+
is finitely normal.
> 0
and a univalent holomorphic
g
such that
(C)
log g’(z)
f(z)
(5)
the primitive
g(z)
f(t) dt
0
is in the disc
(6)
Proof:
Form the
g(e ie) e
algebra and
,.
The absolute convex hull of
We know that (2) implies (i).
functions
and
@(z)
f
is a normal family on
Assume then
d(z)
g
f
-z
(z)
0
and
<
M
(C).
for
z e (C).
380
J. A. C4.A
* [z1+(1-I Zll )z]
(1-1Zll)’(z I)
where we have
function
z 0, z
E )
I
is analytic with
g
*E
Zli){*
that there exists
*
z
I + (1
Note that if
all
z
E )
and for the moment we assume
’(z I)
@)
0
g’(0)
such that
0.
The
It follows by Bloch’s theorem
i.
dg(*) > b.
Then with
the above inequality is still valid.
Hence, for
we have
i’(z) (1-1zl)
Normalize
’(z I)
M/b.
<
by considering
(z)
(z)
(0)
’(t) dr.
|
0
Fixing
t E
(0,i)
and noting that
[@(z)
if
Izl
< t,
M__ log (l-t)
b
<
we apply Theorem
(2.5) to the function
n(z)
@(tz)
to
conclude
t
2
If’(Zo)
2
The maximum of the function
log (l-t)
t/2(l-t)
df (z 0)
>
8M
( -lZoI 2 -<--(-log(l-t))
df(zo)
(t) (-t 2) (log(1-t))-I occurs
(0<t<l)
b (t 0)
which is approximately
If’ (z o) 12
t
o
when
072.
Hence,
(1-lZoi2) 2.
Assume that (2) holds and recall that a family
{f@}
defined and
PROPERTIES OF BLOCH FUNCTIONS
holomorphic on D
is normal if for
M(K)
positive number
381
every compact
K
c
D
there exists a
such that
< M (K)
f,
Aut(D)
Thus for
K
and
c
D
If (z)l)=
2)
i+
f,
sup
1
zK
(i-I zl 2
For the converse let
(z)
w
M(0)
positive number
compact
sup
/l_f’((z)) 2
zK k(+I f, (z)
D and choose (z)
Aut(D).
(l-lwl 2)
-i
There is a
such that
l+if,(0) 12
for all
(z+w)(l+wz)
<
M(0)
Thus
f’(w)
IV’(0) llf’(4(0))1
<
M(0).
The equivalence of (2) and (4) is dependent upon some results from the
theory of univalent functions.
log g’ (z),
that for such
where
g
Let
f(z)
is univalent and
have the representation
> 0.
It is known
Ii
g
z D.
g’ (z)
The expression
define
> 0
(1-1zl 2) If’Cz)
by the equality
is then bounded by
6.
If (2) holds
382
J.A. CIMA
-i
and define
{(l-lzl2)If’(z)l
3 sup
m}
z
so that
g
g(z)
f().,
exp
The definitions of
g
and
d
s
0
-
yields
e
g’(z)
This is sufficient to conclude that
g
is univalent.
The proof that (2) and (5) are equivalent requires a lemma.
Lemma (2.7).
then
f
Let
f
is continuous on
f’
An integration of
f"
proves that
implies that
f
and has radial limits at each point of
81, 82
e
[0,2)
f"(z)
0((l-lzl)-l),
D.
The growth condition on
Proof.
D with
be analytic in
and
0<p<l
r.
{re
F3
{ e i8
F4
{e i8
3
81
_< 8 <
< @ _<
0(log(l-lzl)).
is a bounded holomorphic function
e i8 e D.
o_<r<l}
81
f’(z)
j
If
1,2
82
82 }
the Cauchy integral theorem implies
f(ei82)
f(eiSl)
1
f’ (6) d6
-F I u F 2 u F 3
Each of the integrals can be evaluated and there is a constant
c >
0
383
PROPERTIES OF BLOCH FUNCTIONS
(independent of
8
i)
such that
f(eiSl)
If(e i82)
102 -011 llog182 -8111
-< c
This completes the lemma.
Proof of the equivalence of (2) and (5).
implies that
f
is continuous on
Assume (2) holds.
Let
9.
l>h>0
Lemma (2.7)
be given and fixed.
It
is convenient to assign a symbol to the difference expression
f(e
i(8+h))
and to ignore the dependence of
f(e
i8)
A
on
A f(8)
A
h.
is linear and homogeneous.
We are required to show
A
8,
uniformly in
2
f(e
i8)
(as h/0).
f(e
18)
f(e
0(h)
i8)
l-h
t
Fix
f(te
i8
and write
+
f(teiS).
An integration by parts yields that
(l-r) f"(re
e
i8)
dr
t
(l-t) e
i8
f’(te
i8
i8)+ f(e
The integral in this equality is
g(t,8)
f(te i8)
0(h),
uniformly in
A(e i8 f’(te i8 ))
f’(te i(8+h)) A(e i8) + e i8 A(f (te i8))
0(hllog h I) +
e
i8
A(f’(te
))
8.
With
384
J. A. CIMA
we can compute
A(f’(te i8)
Hence,
A
2
I
it
h
f,,(te i(8+s)ds
0 (i).
0
g(t,8)
0(I)
f(e ie)
h A(g(t,8))
and
+ 0(h)
0(h).
We need only show that
A
2
f(te i8)
0(h)
Since,
A
2
f(te
i(8-h))
h
i( (e+s)
(_e i (e+s) -e le ) f’.te
i t
ds
0
h
+
i t
e
i8
(f’(te i (e+s)
f’(t e i(e-s) ))ds
0
+
_e
i t
i8
f’ (tei (e-s) )ds
e i(8-t)
0
and the integrands of the first and third integrals are dominated by
c s
flog
h
while that of the second is dominated by
we conclude that
2
A
f(te
1o)
0(h)
c s (h)
-I
uniformly in
Let us show that (5) implies (2). As usual
P(r,8)
is the Poisson kernel.
The partials
Po(r,0)
Pee (r,e)
-2r(l-r
2r(l-r
2)
[sine (l-2r cosO +
2)[2r(l+sin2o)
(l-2r cos 8
have the following properties.
of
8
and
+
cosO
r2) -2]
(l+r2)]
r2) 3
The second derivative
P
8e
is an even function
PROPERTIES OF BLOCH FUNCTIONS
I Pe
e (r,t)
0
For
Pee
(0,i),
fixed in
r
dt
385
0
is zero on
(0,H)
only at the point
s(e)
which satisfies
cos s
2r
l+sin2s
l+r
2
A computation shows that
Since
f
is in the disc algebra it has a representation
f(z)
where
E
re
z
P(r,e-t) f(e It) dt
-n
Taking the partials and changing variables yields
fee (z)
In
i
The hypothesis
P
2--
0
(r,t) {f(e i(e+t))
f
{f(e i(e+t)) + f(e i(8-t)) }dt
Pee (r,t)
2f(e ie) +
implies
If 88(z)
<
A
t
0
IP ee (r
We compute the integral on the right
I
s
t P
0
-2s
The expression
f(ei(e-t))
ee (r t)
Pe’(r’s)
dt
+
+ 2P(r,s)
t
s
P88(r,t)
P(r,O)
dt
P(r,n)
t)
dr.
dt
s
J.A. CIMA
386
-2s
P8
and so there is a constant
0 < -2s
4r(1-r)
4
(l-r)
(r,s)
so that
c
Ps(r,s)
_<
c(1-r)
0 ((-I zl
<
term.
Hence
)-)-
The Polsson integral representation for
Ifs(z)
-i
P(r,s)
A similar inequality is valid for the
(z)
3
c(l-r2) llfll
f implies
2rsin t dt
0
(l-2r cos t + r
2,
thus
0((l-lzl)-l).
fs(z)
An easy computation yields
f"(z)
r
-2
e
-2i8 {i
O((l-r)
fs(z)
f88(z)}
-I)
This completes the equivalence of (2) with (5).
We omit the proof for (6).
A perusal of these equivalences and a slight amount of study give
some insight into the size of the set of Bloch functions.
that the bounded holomorphic functions are Bloch functions.
We have observed
In general
there are no containment relationships between the classical Hardy spaces
HP(D)
(0<p<=)
and the set of Bloch functions.
In fact one can construct
Bloch functions using gap series which are not of bounded characteristic.
PROPERTIES OF BLOCH FUNCTIONS
387
Although a BLoch function need not have angular (non-tangential) limits
D
almost everywhere on
={I z
one can prove using the Gross star
i}
theorem (from the theory of cluster sets) that every Bloch function has
D.
finite or infinite angular limits on an uncountably dense subset of
[.
f(z)
If
a
n=0
n
z
n
a Bloch function a straight forward estimate
is
yields
lanl
for
_< c
{(l-lzl 2) If’(z)
D}
z e
The following result will be useful in posing
i, 2, 3,
n
sup
some unanswered questions later in the paper.
Theorem (2.8).
Let
j
a. z
[
f(z)
limits almost everywhere on
Proof.
Integrating
2(C).
i
2n
j=l
r
- (i-
n
)
j
n
j-l
Then
a
n
O.
/
if (z)I 2
[. j21 ajl 2 r2j-2
Choosing
be a Bloch function with radial
J
j=l
.2
f’
(rei8)I 2
0
we can find a constant
ajl
<
0
c > 0
so that
2
(l-r)I
n f’(rneiS)l
A known property of normal functions (see Lehto and Virtanen
that if
f
then
has angular limit
f
lira f(rl)
r+l
is analytic in
A.
.
D with radial limit
A
at
With
A
15 ])
at a point
states
e
assume
e
The discs
Cn
lie in a fixed angle at
{Iz
rn I
<
i
and tend to
(i-
rn )}
as
n
/
Apply the Cauchy formula
J.A. CIMA
388
I
Cn
1
(-r)
Cf Cz)-A)
(Z-rn)
zcn
Since the
n-
(l-m) If’ (rneiS)
Integrand
dz
is uniformly bounded we may apply
the Lebesgue bounded convergence theorem to conclude.
n n
Thus
3.
a
/
n
0.
The Banach space structure.
B
Theorem (2.6) implies that the set of Bloch functions
vector space.
It is possible to equip
a Banach space.
f
If
E
B
B with a norm in which it becomes
we can define
sup{ If’ (z) l(-lz] 2)
The addition of the term
is a complex
If(O)
z
D)
is to account for the constant functions.
sup{
We prefer to work with the quantity
If’ (z) (l-[zl 2)}
limit ourselves to functions holomorphic on
B
D
for which
M(f)
f(0)
and so
0
and
M(f)
Henceforth when norm considerations for Bloch functions are involved we
always assume this normalization is in force.
of
B
denoted as
B0
and defined as
There is a natural subspace
PROPERTIES OF BLOCH FUNCTIONS
It is straightforward to check that
Also if
f
B
E
Since each
0
f
norms of
B.
is a closed subspace of
B0
B.
f (z)
f(pz) then f
as
tends f in B
0
P
can be uniformly approximated on D by polynomlals we
and
P
see that polynomlals are dense in
subspace of
389
B
B
0.
Thus
B
0
is a separable closed
is nonseparable as we can easily see by checking the
(log(1-ze ih)
log(l-z)
There is one result which
for h>0.
is very useful and should be noted.
MCf)
for every
MCf
)
Aut(D).
Although there is no natural inclusion relationship
2
p
between the Hardy spaces H
and B(i.e. g(z)
(log(l-z)) E Hp %c p<
yet
g
E
B)
there are some other interesting containment relations.
Let
us show for example that each holomorphlc function with finite Dirichlet
Integral is in
B.
Assume
f(z)
a z
n=l n
If(z) 12dc(z)
1
For each
than
> 0
we may choose
N
n
nl,l 2
and
<
(R).
so that the last square root is less
390
5. A. CIMA
Hnce
lira sup (1-r
rl
2) f’(z)[. <
fE B 0.
and
F
For a second and less obvious example we consider an
I(F)
of
I,
over an interval
F
be the average of
I
c
D,
III
E
L2(D).
Let
the measure
and
F(0) dO
I(F)
I
F
is said to have bounded mean oscillation if
I
in the Hardy space
H2(D)
this is equivalent
In the case of a function
f
to being able to write
as the sum of two holomorphlc functions
f2
for
fl
and
and the reader might wish to refer to the notes of Sarason
21
Which Re
equlvalece
f
and
fl
Im
f2
are bounded.
for a comprehensive discussion of this topic.
This is a nontrlvlal
But if
is decomposed as
f
above
f(z)
since
fl
function.
and
Thus
f2
fl(z)
+
f2(z)
are Bloch functions it follows that
BMOA
is a Bloch
(the space of analytic functions with bounded mean
oscillation) is a subspace of
B.
We establish now the basic duality relationships.
idea really begins
f
with the paper of Rubel and Shields
The fundamental
18
].
In that
PROPERTIES OF BLOCH FUNCTIONS
391
paper the following basic principle is confirmed.
Loosely stated it
shows that the second dual of a Banach space of holomorphic functions
satisfying a "little oh" growth condition is isomorphic to the Banach
space of analytic functions satisfying the corresponding "big oh"
relationship.
Theorem (3.1).
The inclusion mapping of
isomorphism of
B0**
onto
B0
B
into
extends to an
B.
We prove this theorem by a sequence of results.
introduce an auxiliary space
K,
on
g(O)
0
E
I
g
holomorphic
are the normalized holomorphic functions whose derivaties
One checks that
I
c
I
H (D)
and hence each
I has well deflmed boundary values
llm g (re
r+l
with
is the space of functions
drde<(R)
are in the Bergman one space
g
I
and
I
The functions in
I.
It is necessary to
g (e
i8
)
Lemma (3.2).
E
L’ (D).
For
i8)
g(e i8)
a.e.
The next lemma is crucial to the development.
f(z)
and
g(z)
[
n= 1
Hadamard convolution
h(z)
f,g(z)
n=l
is in the disc algebra.
abz
n n
b z
n
n
the
3. A. CD
392
Proof.
z
re
iO
is analytic on
D.
We show that it is uniformly
and so can be extended to a continuous function on
D
continuous on
Let
h
The function
and
D,
/:
2)
(l-r
0
0
f’() (zg(z))’
e-i0d0dr
, anbnn-I
Also
]
-
g(rei0)
0
<
dOdr
0
g’ (tel0)
0
dtdrd0
0
0
We find using this result
Ih(=)l
-< c(sup
-<
Let points
g(z)
h(/:2)
g(z).
ze D)llg
llfllB’llgll x
be given in
i’ 2
[h( 1)
where
(X-lz[2) If’(E)l
< c
D
and observe
[I fll B
[[ g1
2
It is routine to check that as
lg-g:[[i
o
[1I
V.
PROPERTIES OF BLOCH FONCTIONS
Hence,
D.
is uniformly continuous on
h
Theorem (3.3).
The spaces
In the notation of Lemma (3.2) we have for
Proof.
I*<:)l
Hence each
eI*
f
g(z)-,
1
b
n
i<f,,>l -<
fh()l
c
can be identified with a
B
and let
is analytic in
(zn)
a
n > I.
0 < P < 1
Let
D.
for
to
.
g(z)
and
f e B
I
g e
II 11’11,11
in I*. Now assume
The function
and select
g
[
f(z)
a z
n
I
z"
anbnpn
But
The pairing is
are isomorphic.
where
f
given by
I*
and
B
393
tends to
g
in
I"
and hence the analytic function
It remains to prove that f
-2
z (l-z)
is in I for
<
f>
B.
is in
fixed in
.
(;,-’1:1 r)
("l-I l:12) -i
Thus
f’(’:)l
I<f,
%>1
< ,:
I1,11.
With
(:.-I:12)-1
z
re
18
s= j
0
0
_< c
corresponds
The kernel
s..,.,
’:
f
:l.-zi-2
d=
J.A. ClMA
394
and so
f
B.
is in
The next theorem will be the last result we need.
This result and
Theorem (3.3) will imedlately yield Theorem (3.1).
Theorem (3.4).
given by
/
1
The spaces
g,
f
Let
Proof.
(zn) -bn,
But for any
g
B 0.
$ e
B0*
n > i.
f e B,
g(z)
and form the holomorphic function
.
n=l
For
p
f
e B
(0,I)
and
f
bnzn
BO,
hence
0
>
I.
We have already observed that
on
The pairing is
<f,g>
SU
Thus
are isomorphic.
with
(f)
for each
B0*
and
induces a bounded linear functional
g
B 0.
We have already noted that for
f(z)
.
n=l
m
B/
defined by
re(f)
(an)
a z
n
B
the mapping
395
PROPERTIES OF BLOCH FUNCTIONS
is a one to one continuous linear mapping.
(nk)
It can also be shown if
is a strictly increasing sequence of positive integers with
nk+I/nk
>
2,
(ak)
and
then the function
E
[ akz
k--i
f(z)
B
is in
and
lfll
nk
If(an) If
-< 4
n
Hence, the
mapp.n,
/B
given by
E (a
k)
k=l
akz
2
k
is a (continuous) isomorphism onto its range.
P
B
/
B
In particular the map
given by
2
is a projection.
k
Although this is not sufficient to conclude that
linearly isomorphic to
Z
one might anticipate this result.
.
B
A denoue-
ment to the results of this nature was given by Shields and Williams
Their work coupled with a result of Lindenstrauss and Pelczynskl
will show that indeed
that
B
0
B
is isomorphic to
is isomorphic to
c
o (sequences
sup norm (Shields and Williams,
22, 23
establish a direct sum decomposition for
and a complemented subspace.
LI(D)
22 ].
16
They also obtain the result
tending to zero)
]).
is
with the usual
Shields and Williams
in terms of a
copy of
One can then apply the result of Lindenstrauss
and Pelczynski on bounded projects to prove that
Thus
I*--
B.
[
is isomorphic to
E
1.
J.A. CIMA
396
We prefer to use the notation of Shields and Williams and simply identify
To begin they choose positive continuous
otational connections necessary.
functions
$
and
[0,1)
on
$
$(r)
with
0
/
r
as
/
1
and
1
(r)dr <
(R).
0
A(D)
The space
on D.
f
consists of holomorphic functions
Three other
spaces are denoted by
A A() =-
where
o
{f
A(D)
sup
{If(z) (Iz[)
is normalized Lebesgue measure on
D.
D}
z
(R)}
We make the obvious
identifications and isomorphic correspondences
Letting
and
M(D)
bounded Borel measures on D
denote the Banach space of complex-valued,
(with variation norm) and
such that each
Banach space of continuous functions on
)
vanishes on
C0(D)
we define some isometrics
For
f
A, g
the
CO(D)
f
A
1
we
define
T=of
Then
T=
f,
Tlg
g
is an isometry of
and
A
Mg
to
and
g do
L’(D)
(T(R)IA0
A0
is an isometry of A
0
to C )
O
39?
PROPERTIES O BLOCH FUNCTIONS
T
AI to L I
is an isometry of
I
M
and
is an isumetry to
M()).
The
following netational pairing is used
zl
I
I f(z)g()(l-izl 2)
() (I
(z)
(’g)
)
(I --I )
do (--)
du(z)
D
with
f
A.,
AI.
g
n
f(z)
a z
n
For example
{(z)
and
[
and
f
if
are polynomials
g
n
b z
n
ab
(n+l) (n+2)
The pair
{,#}
Lemma (3.5).
is a normal pair with
Let
z,
w
D
Kw (z)
-=
e
I-,2
k
AI
and
3
and
a
i.
2(1-wz) -(3)
Then
(i)
g(w)
(Kw,g)
(2)
f(,)
(,K) a
(3)
span
{K
span
{Kw;
(4)
Proof.
case of
all
AI
g
A(R)
w e D}
w
w
is dense in
D}
is weak star dense in
A
0
A.
This is just a computation and we limit ourselves to proving the
A1
in
to show that if
(3) and part (4).
h
L (D)
and if
Kw(z)
D
By the Hahn-Banach theorem it suffices
h(z) do(z)= 0
5. A. CIMA
398
for all
w
D,
AI
I.
T A
A calculation
annihilates all polynomials.
The polynomials
h anihilates all of
then
T
I
yields
0
for all
w
are dense in
2n
D.
n
znh()
(n+l) (n+2)w
do (z)
0
Hence,
I
A
h
and thus span
is dense in
{Kw}
For part (4) we recall that if
o n (f)
denotes the
means of the partial sums of the Taylor series for
sup
00<2
u
I.
A
f
nth Cesaro
then
sup fCreiO)
ncf) cre 1o)l 0<0<2
Hence,
If
f
A
sup
sup
0_<r<l
0
lonCf)(reie)l
(l-r2)]
we apply the aSove inequality to conclude that
(l-r2) Un (f) (reiS)
converges pointwlse and 5oundedly to
(l-r
2) f(reiS).
By the Lebesgue dom#inate convergence theorem we have
f(z) h() d(z)
for all
h
L’(D).
Thus each
sequence of polynomials.
f
in
A=
is the weak star limit of a
PROPERTIES OF BLOCH FUNCTIONS
The
L I(D)
L(D)
A
functional on
duality implies that each weak star continuous
is given by
Ce-[ z
h
where
L I (D).
12)
fCz) h(l) (--)
But as above if
I
Kw(Z)
h() (i-
z
12 )
(z)
annihilates polynomials and thus span
h
then
399
o
{Kw}
is weak star dense in
A
The next lemma is well known and the second follows by a calculation.
Lemma [3.6].
For
0 < r < i,
2Ell-felt[-2
dt
0 ((l-r) -I)
0
Lemma [3.7].
For
m _> 2
i
and
0 _< 0 < i
(1- Or)-m dr
0 ((1-0
2
)l-m).
0
Theorem 3.8.
The transformation
defined by
d. (z)
e
with
M(D),
w
The transformation
A I.
The operator
subspace
Proof.
TA
D
is a bounded operator mapping
M(D)
onto
I.
A
i -= Q]LI (D) is a bounded operator mapplnz LI(D)
T1 i is a bounded projection of LI(D) onto the
I.
We consider only
i
the proof for
is analogous.
For
onto
J. A. CIMA
400
1
f L (D)
I ( I d(w))
Kw()
(l-’z.2)’f(z)’d(z)
D
where we have used Lemmas 3.6 and 3.7.
Hence,
i
i
{g
A
is in the kernel of
w e D.
(which
Theorem 3.10.
I(D)
onto
I.
A
I
O,
A}
We show that the null space of
0
in
g,
(f,g)
L’(m)
all f
for all
if
We have the following direct sum decomposition
(TA)
g e LI(D)
L
is a bounded projection from
Corollary 3.9.
Proof.
If
1
("-I zl 2
I(-TII)
A1
(TmA) I.
A
if and only if
()
zCz) d(z)
The finite linear combinations of
s
is
weak star closed) and hence
g
is topologically isomorphic to
Kw are weak star dense
(TA) I.
i.
PROPERTIES OF BLOCH FUNCTIONS
with
{r
Let
Proof.
lim r
I -< J.
be a strictly increasing sequence of positive numbers
n
I.
n
A.3
Let
denote the annulus
R.3 LI(D) LI(AJ )’
Let
401
J
/
(restriction), with respect
L
R
1,2,...
be the natural projection
to Lebesgue measuz=_.
I(8)
+
(L
10% 1 )
L I(A 2)
The operator
L
(R)
I(4 3
defined by
(Rlf R2f
Rf
is an onto isometry.
R.I.I
A normal families argument shows that
Hence, if
S
then
Rj
is a totally bounded
S
LI(Aj).
set in
Consider
E
measure on E...
O(E)
I,
A
is the unit ball in
is a compact operator.
i.
Lebesgue measurable set and
a
c
-
d
Lebesgue
Further assume with out loss of generality that
For each
let
n,
n
measurable sets
n
E 1,...,E2n
Pn
E
be a partition of
1
B(E
with
into disjoint
for each
i.
Assume that
2
Pn+l
refines
Pn
Pn
f + f
P
f
n
X
in
EL
n.
Now for
i
p f
n
Of course
for each
fd
(E)
i=l
n
E
i
f
E
L I(E,),
Xgn
i
is the characteristic of the set
I
n
E
i-
norm, uniformly on compact subsets of
is of the form
2
n
[
i= I
C
i
X
E
=P
n
f-
define
One can prove that
L
I(E,).
Each
402
Hence the map
J. A. ClMA
Cl
--+
Cix
Pn
is an isometry of the range of
2
P
P
and hence
Pn
n
I
<Aj)
ej
> 0
into itself is given so that
k.
13
some
for a suitable integer
for every
S.
f
kj
a projection
IPjll
< i,
Pj
pj
L
I
pj(ej) of
<Aj) is isometric
and
It is now clear that the subspace
(PI L1 (AI)
X
One can check that
i
is a projection.
n
There is for every given
L
2n
onto
X,
P2 L1 (A2) *
of
L
I(D)
is isometric to
I (A I)
$
LI
(A2)
i"
The operator
T(RI
L
f’
R2
liT
Rf
T
f’
R AI
/
X
(PI
defined by
RI
f’
P2
f’
satisfies
for
f e S.
1
for
f
AI.
Thus, if
e > 0 is given we can choose the
ej
> 0
so that
to
PROPERTIES OF BLOCH FUNCTIONS
is complemented in R L
R
Since
T R A
I
sufficiently small
403
we claim that for
I
is complemented in
RL
> 0
and
so in
I
us consider the general situation of a Banach space U with Y
U.
plemented subspace of
on
U
P equal
with range
IlYll
i,
For if not
y
Y
E
vn
E
z
y
U
Then
Let
a corn-
P
and there is a projection
We claim there is
Y.
X.
I
>n>
0
such that if
then
z
Y,
n
e Z
with
Thus
lPll/n
->
IP(yn
Now let
and this is absurd.
Zn) ll
T
i
be an operator on
Y
U
into
which
satisfies
We claim that
T(Y)
is a complemented subspace of
invertlble as an operator on
Y
T(Y).
/
Next define
U.
First,
T
T
on all of
is
U
follows,
Computations show
iS a
Y0
Y’
T(y)
T(y)
y e Y
T(z)
z
z
T
flY011
is linear on
i
and
Ty 0
Z.
U
and is one to-one.
Z
0
E
Z,
then
For if there
as
J.A. CIMA
404
Next if
fly
2
<
ITY 0 Y011
" z ll
<
1
llzll
lYll
Similarly,
< M
if
+ IIP(y
flY zll
z)ll
I.
is isomorphic to
T
and since
IIPII
+
< M
R L I.
is complemented in
complemented subspace of
it is known that an inflnite-dlmensional
i
i
<
Hence,
T R A
I
This establishes the claim that
is isomorphic to
<
then
flY zll
<
Y011
iz 0
But
1
is an isomorphism we find that
AI
i"
There are other results of interest which are known in regard to the
Banach space structure.
the multipliers from
B
B
p
into
in the following sense.
(A,B)
If
A
n
(e,8)
{k
n
2
{%nan
{an}
every
I
A and
to
B,
are defined as
(A,B) ={%= {%n
by
have characterized
are two vector spaces of sequences the multipliers from
denoted
For
5
Anderson and Shields
n+i
k < 2
n
the set of sequences
E
E
B
A}
0,i,2,...
{ak}
for
and
(k a I)
n=0
and
,8
kl
n=0
(o:(R))
i
,
for which
8
denote
PROPERTIES OF BLOCH FUNCTIONS
405
These are Banach spaces when they are given the obvious norm.
Anderson
and Shields prove that
/
t
(B’P)
(22_-p, p)
1p2
(’P)
2<p_
Questions of rotundity of the unit ball have also been investigated.
Examples of extreme points in the unit ball of
B0
nz
are
ha2,
and
other powers of Blaschke factors, suitably normed e.g.
(3.z)
Isl<l
No other examples of extreme points are known.
extreme points of the unit ball of
We are considering functions
{z
Lf
Theorem 3.11.
if and only
Let
Lf
f
D
f
in
B
A characterization of the
is given in Cima and Wogen
0
B0
lfll-
with
8 ].
Let
i.
Jf’(z) ( _izl ’)
be in the unit ball of
B
0.
Then
f
is an extreme point
is an infinite set.
A second result is also noteworthy in this regard.
Theorem 3.12.
If
f
is an extreme point of the unit ball of
there are simple closed pairwise disjoint analytic curves
with
k > i
and points
wI, w
2,
wj
with
J
k
Lf
(
yi
U
{Wl, w2,
wj}
>_ 0
B0, then
YI’ 72’’’ "’Yk
so that
406
J.A. CIMA
There are other examples of extreme points of the unit ball in
The function
log
(l+z)
B.
is an example and scalar multiplies of singular
inner functions corresponding to a point mass on
D are also examples.
The observation that
M(f)
sup
{If’(z) (l-lzl 2
M (fo)
for every
a
Aut(D)
leads one to consider the isometrics of
B
and
0
B.
In particular the extreme points of the unit ball are permuted under an
onto
isometry.
The process of composing
is a composition operator on the space.
B
C#
+
0
(or B
B0
a
(Aut(D)
with
f
(
B
or
We write
+
B)
and this is defined as
C(f)
In
9
(z)
Cima and Wogen have proven the following results.
Theorem 3.13.
If
B
S
0
/
(Sf) (z)
where
(z)
f
%
(
D and
Corollary 3.14.
Theorem 3.15.
B0
is an isometry,
lf((z))
%f((0))
$ (Aut(D).
Every isometry of
If
then
B
S
morphic automorphism
$
/
B
and
(Sf) (z)
B
0
is onto.
is an onto isometry, then there is a holo-
a
(
D such that
A(f((z)))
%f((0)).
B
0
PROPERTIES OF BLOCH FUNCTIONS
407
We conclude with the following interesting result of Rubel and Timoney
19 ].
and every Mobius transformation
X
f
invariant if
A real valued function
X
semi-norm on
#
of
D
X
p
and every
p(f)
fo#
#,
X.
(
is said to be a Mobius-invarlant
for every Mobius transformation
A non-zero linear functional
X.
f
[0,)
/
#)
p(f
if
of analytic functions is said to be Mobius-
X
A linear space
X
L on
is
said to be decent if
{If(z)
sup
for all
for some
X,
f
Theorem [3.16].
Let
z
and same compact subset
M > 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 <
I,
the expression
{I (f)n(0)l
.
is the seminorm given by
Aut(D)}
We observe that
408
J.A. ClMA
defined on analytic functions, is a semlnorm equivalent to the Bloch
Proof.
We follow Rubel and Timoney and consider the case
case requires more bookkeeping but is similar.
-a(Z)
(z+a) (l+z) -I
2) 2
I2I
1
Choose
2
a (fo b_a
The general
2) 2
a
Let
f(3)(a) (1-1zl 2)
- -
6f’ (a)(l-[zl
3.
and observe that
_a)(3)(0)
(f
n
(21- )
a
)(3)
e
tt3
9
(0) d(}
eff
9
6 f" (a) (1-1al
and observe that
12lI
/.
1
f’ (a) de
0
f’ (0)
Taking absolute values we find
If’(0)
If
is in
Sup
Hence,
Aut((C))
-<
() P3 (f)
we apply thls last inequallty to obtaln
{[(foe)’(0)
PB (f)
<
Aut(D)}
() p3(f)
_<
P3 (f)"
For the converse we apply the Cauchy integral theorem
f"(z)
1
[J 2n f,()
0
2
d0
An easy computation shows that if
2
l(fo+)"
(-z)
(0)
< c
< c
(PB
< c
PB
PB (f)
(f) + sup
(f)
+
If"(z)
Cl-lz12) 2
f’ ()
(-ll 2))
(0)
z
/
PROPERTIES OF BLOCH FUNCTIONS
A similar estimate will show that
Hence, there
inequality.
P3
f"’ (z)
is a constant
(f) -< c
PB
409
also satisfies such an
c
such that
(f)
Proof of the Theorem.
(X, p)
Let
satisfy the hypothesis of the theorem and let
L
be a
linear functional satisfying
L(f)
_< p
(f)
and
Sup
for all
X,
f
{[f(z)l
z
K
cc
D}
A > 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
-<p (f)
An application of Fubini’s theorem produces the following equality
410
J.A. CIMA
e
2
i
iN 1
dz
g(eiz) -7
f(z)
b fN (0).
This implies
p(f)
and replacing
pN(f)
If
N > 0
we conclude
by
f
Sup
f
IbNl IfN(0)
>
and taking the supremum we obtain
)N(0)I
{}(f
Aut(D)}
<
I.blN
p(f)
we apply the lemma (3.17) to produce the result.
X
and that
c
The sup norm dominates
p
and
PB
dominates the supremum norm
Hc
B.
N
If
0
II II
This conclude the proof of the
theorem.
4.
Open questions.
Of course the older question of getting a sharp estimate for
Let us define subspaces of
viable.
L
{f
M
{f(z)
B
B
is
as follows
has radial limits
f
b
a.e. on
and
[.
a
We know that if
(Theorem 2.8).
f
E
L
Hence,
is a Cauehy sequence and
fan(f)
E
n
i=n
B
lira a
n
0}.
then the Taylor coefficients of
f
L
For if
c
M.
fk,
But M
f
n(fk)
a
in
< c
is closed in
B
B.
we recall that
lf-fkllB
tend to zero
{fk}
c
M
PROPERTIES OF BLOCH FUNCTIONS
411
Hence,
=
lanCf)l
and
lira
a
(f)
n
H c__
0.
lf-fkl
+
B
lanCfk)l
The following inclusions have been pointed out
p
BMOA_c B n (i H ) =_ M c__ L
It would be a useful result to describe the Bloch closure of
or
BMOA
L.
in
the following.
H
A question raised by D. Campbell amd the author is
Produce a function
f
B
n(l Hp)
which is not in
BMOA.
In answer to a question posed by the author R. Timoney has produced
a function
but
B
f
such that
f
has radial limit values in
L (D)
p.
UH
Are there any natural conditions which can be placed on a
0<p
Bloch function f so that if f has radial limits in LP(D) then
f
f
HP?
Several questions remain concerning the structure of the unit
ball in
B
O.
Namely, is each extreme point of the unit ball of
the form (3.1)?
B0
of
What is the closed convex hull of the set of functions
given by (3.1)?
As a final question, we consider an oral communication of D. Sarason.
There are singular functions
S*(z)
in
B
0.
This is not an obvious
result but follows by some work of H. Shapiro and an application of the
Zgymund criteria for
B 0.
Hence, for a
D \ K,
where
K
has capacity
zero, the functions
s*(z)
l-a s*(z)
are Blaschke products in B
0 (Frostman).
for B
by designating its zeros
@
Give an explicit construction
412
J.A. CIl
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ERRATA
On pages 15 and 16:
Replace
f
by
g
On page 15, line 3(b)
Replace
g
by
s
on page 16,
Replace
g
by
s
Replace
A 2 f(e i8
lines 3, 4:
On page 16, line 4:
A2[f(e i8)
On page 33, line 6:
f(te
by
i8)]
The direct sum is taken in the
norm.
i
An___n.