Publicacions
Ma emá iques,
Vol
35
(1991),
403-427
.
Abs ac
ON
BLOCH
FUNCTIONS
AND
GAP
SERIES
DANIELGIRELA
Kennedy
ob ained
sha p
es ima es
o
he
g ow h
o
he Ne anlinna
cha -
ac e is ic
o
he
de i a i e
o
a
unc ion
analy ic
and
wi h
bounded
cha ac e is ic
in
he
uni
disc
.
Ac ually,
Kennedy's
esul s
a e
sha p
e en
o
VMOA
unc ions
.
I is
well
known
ha
any
BMOA
unc ion
is
a Bloch
unc ion
and
any
VIVIOA
unc ion
belongs
o
he
li le
Bloch space
.
In
his
pape
we
s udy
he
possibili y
o
ex ending
Kennedy's
esul s
o
ce -
ain
classes
o
Bloch
unc ions
.
Also,
we
p o e
somemo e
gene al
esul s
ob aining
sha p compa ison
esul s
be ween
he
in eg al
means
M
p
( ,
)
wi h
T( , )
o
ce ain
classes
o
unc ions
analy ic
in
he
uni disc
.
The
Ne anlinna
class,
deno ed
by
N,
consis s o
hose
unc ions
analy ic
in
he
uni disc
U
o
which
sup
o<
,
«
T
( ,
)
<
oc,
whe e
T
( ,
)
deno es
he
Ne anlinna
cha ac e is ic
o
.
Kennedy
p o ed
ha
i
E
N
hen
i

1
(1)

(1-
)
exp(2T( ,
'»
d
<
oo
and
(2)

lim
l
Clog
1
-
o

-
T( , '))
=00
.
Bo h
esul s
a e
sha p
.
We
no e
ha
(2)
ollows
om
(1)
.
Le
B
deno e
he
space
o
Bloch
unc ions
.
Two
impo an
subspaces
o
B
a e
hose
deno ed
by
BO
and
B
l
.
The
space
BO
consis s
o
hose
E
B
such
ha
(1
-
Iz1)
j'(z)j
->
0,
as Iz1
-
1,
and
Bl
consis s
o
hose
E
B
such
ha
i
{z

,}
C
U
and
1 (z,,)1
-oo
hen
(1
-
1z

,1)1
'(z
n
)1
-
0
.
I is
well
known
ha
VMOAC
Bo
and
BMOAC
B
.
Kennedy's
es ima es
a e
ac ually
sha p
o
VMOA
unc ions
.
In his
pape
we
s udy
he
ques ion
o
whe he
o
no
(1)
and/o
(2)
emain
ue
o
a
unc ion
in
B,
B
l
o
BO
.
We
p o e
ha
(2)
need
no
be
ue
o
a
Bloch
unc ion
showing
ha
he
i ial
es íma e
T( ,
)
_<
log
i
1
+0(1)
is
he
bes ha
we
can say
in
gene al
.
Howe e
(2)
is
ue
o
any
E BO
e en
hough
i
may
no
sa is£y
(1)
.
We
do
no
know
whe he
o
no
(2)
is
ue
o
any
E
Bl
bu
we
can
p o e
ha
i
sa is ies
(2)
wi h
lim
sup
ins ead
o
lim
.
Also,
we
gene alize
hese
esul s
ob aining
sha p
compa ison
esul s
be ween
he
in eg al
means
M
p
( ,
)
wi h
T
( ,
)
o
ce ain
classes
o
unc ions
ana-
ly ic
in
U
.
40
4

D
.
GIRELA
Le
be
a
unc ion
analy ic
in
he
uni disc
U=
{z
E
C
:
Iz1
<
1}
.
Then,
he
Ne anlinna
cha ac e is ic T( ,
)
is
de ined
by
(1
.1)

T
( ,
)
=
27
l°g+
1
( e'%
d ,

0< <
1
.
The
Ne anlinna
.class,
deno ed
by N,
consis s
o
hose
analy ic
in
U
o
which
(1
.2)

sup

T
( ,
)
<
oo
.
o<T<I
I
is
well
known
ha
'
need
no
belong
o
N
e en
i
is
bounded
.
This
was
i s
p o ed
by
Ros man
[6]
who
showed
he
exis en e
o a
Blaschke
p oduc
whose
de i a i e
is
no
o
bounded
cha ac e is ic
.
Kennedy
de e mined
in
[14]
as
closely
as
possible
he
es ic ion
imposed
on
he
g ow h
o T( ,
')
by
(1
.2)
.
He
p o ed he
ollowing
wo
heo ems
.
(1
.3)

J
I(1
-
)
exp(2T( ,
'))
d
<
oo
.
0
Theo em
A
.
([14,
Th
.
I])
.
Le
E
N
.
Then
Theo em
B
.
([14,
Th
.
II])
.
Le
p
be
a posi i e
inc easing
unc ion
in (0,1)
such
ha
l
I
(üi)

(1
-
)
exp(2p( ))
d
<
oc
.
0
Then
he e
exis s
E
N
such
ha
o
all
su icien ly
close
o
1
.
1
.
In oduc ion
and
main
esul s
(1
-
)
exp
p( )
is
dec easing
.
1-
h( )
-
p(p)
-
oo
as

1
-
T( , ')
>
M( )
Le
us
no ice
ha , since
T
( ,
)
is
an
inc easing
unc ion
o
,
Theo em
A
implies
ha
i
E
N
hen
1
(1
.5)

log
1

-T( ,
')
-
oo,
as
-->
1
.
ON
BLOCH
FUNCTIONS
ANDGAP
SERIES

405
Also,
since
he
unc ion
o
Theo em
B
is
inc easing,
(iii)
shows
ha
(i)
is
equi alen
o
(i )

log
1 1
-
p( )
T
oo,
as
T
1
.
The
au ho
has
ecen ly
ob ained
in
[9]
he
analogues
o
Kennedy's
esul s
o
analy ic
unc ions
wi h
ini e
Di ichle in eg al
in
U
.
The
unc ion
cons uc ed
by
Kennedy
o
p o e
Theo em
B
is
gi en
by a
powe
se ies
E
Ckz'
k
wi h
Hadama d
gaps such
ha F-
ICk 12
<
oo
.
Such
a
unc ion
belongs
o
HP,
0
<p<
oo,
and,
e en
mo e,
o
VMOA
.
This
ollows
om
Paley's mul iplie
heo em
[3,
p
.
104]
and
he
duali y
o
H
land
BMOA
[7,
p
.
270]
.
Hence
(1
.3)
and
(1
.5)
a e
sha p
(in
he
sense
o
Theo em
B)
o
VMOA
unc ions
.
The
ques ion
as
o
whe he
o
no
he e
exis s
a
unc ion
analy ic
and
bounded
in
U
wi h
sa is ying
he
conclusion
o
Theo em
B
emains
open
.
Kennedy
poin ed
ou
in
[14]
ha
in
dealing
wi h
his
p oblem
one
could exclude
unc ions
(z)
=1
:
akznk
ha ing
Hadama d
gaps
.
This
is
because
i
such
a
unc ion
is
bounded
in
U
hen
E
Iakl
<
oo
[20,
ol
.
I
p
.
149
and
247]
and
so
1
(1 .6)

expT( ,
')d
<
oo
0
a
s onge
inequali y
han
(1
.5)
.
Clunie
p o ed
in
[2]
ha
he e
exis s
a
unc ion
analy ic
and
bounded
in
U
no
sa is ying
(1
.6)
.
A
unc ion
analy ic
in
U
is
said o
be
a
Bloch
unc ion
i
II IIB
=
SUP
(1
-
IZI 2
)I
l
(z)I
+
I (
0
)j
<
oo
.
I=I<1
The
space
o
all
Bloch
unc ions
will
be
deno ed
by
B
.
Two
impo an
subspaces
o
B
a e
hose
deno ed
by
BO
and
B
I
.
The
space
BO
consis s o
hose
E
B
such
ha
(1-
IZI
2
)
I
'(z)I
,
0, as
Iz1
->
1
.
Al e na i ely,
Bo
can
be
cha ac e ized
as
he
closu e
o
he
polynomials
in
he
Bloch
no m
[1,
Th
.
2
.1]
.
The
space
Bl
consis s
o
hose
E
B
such ha
i
{z
n
}
C
U
and
I
(z)(
-
o0
hen
(1
-
Izn1
2
)I '(zn)I

>
0
.
Clea ly,
BO
C
BI
.
I
(z)
=
E°°
o
a,,zn
E
B
hen
sup
Ia

,I
<
oo
[1]
while
i
E
BO
hen
a
n
-->
0
.
Ac ually,
he
weake
condi ion
(1
.8)

12
( ,
')
=
o((1
-
)
-2
),

as
--->
1,
is
enough
o
conclude ha
an

>
0
[16,
p
.
693]
.
He e,
o
g
analy ic
in
U
(1
.9)

12
( ,
g)
=
27
,~
Ig( e")
12
d ,

0< <
1
.
40
6

D
.
GIRELA
The
space
Bl
was
in oduced
in
[10,
p
.
30]
and
[1,
p
.
36]
whe e
i
was
conjec u ed
ha
i
(z)
=
E
ñ=o
a

,zn
E
BI
hen
a

,
~
0
.
This
was
disp o ed
by
Fe nández
[4],
[5]
.
Fe nández
ga e
in
[4]
examples
o
unc ions
E
Bl
no
sa is ying
(1
.8)
.
I
D
is
a
B
1
-domain, Le
.
i
e e y
unc ion
g
analy ic
in
U
wi h
g(U)
C
D
is
in
B,,
and
is
he
uni e sal
co e ing
map
o
D
hen
Hayman,
Pa e son
and
Pomme enke
p o ed
in
[12]
ha
sa is ies
(1
.8)
bu
Fe nández
p o ed
in
[5]
ha
he e
exis s
a
unc ion
analy ic
in
U
whose
ange
lies
in
a
B
1
-domain
o
which
(1
.8)
is
no
ue
.
The
in eg al
means
and
adial
g ow h
o
B
1
- unc ions
we e
s udied
by
he
au ho
in
[8]
.
Impo an examples
o
Bloch
unc ions
a e
gi en
by
powe
se ies
wi h
Hada-
ma d
gaps,
Le
.
powe
se ies
(z)
=
Eñ=o
akznk
analy ic
in
U
wi h
nk+1
>
"
1
nk
o
some
cons an
A
>
1
.
Fo
such
an
we
ha e
[1,
p
.
19]
E
B
i
and
only
i
sup
jakI
<
oo
and
[4],
[16]
E
Bo
e-*
E
B
I
e*
ak
-+
0
.
I is
well
known
ha
VMOA
C
Bo
and
BMOA
C
B
.
The
main
objec
o
his
pape
is
s udying
whe he
o
no
(1
.3)
and
(1
.5),
which
a e
sha p
o
VMOA
unc ions,
emain
ue
o
unc ions
in
he spaces
B,
BI,
o
Bo
.
I
EB
hen
(1
.10)

T( ,
')
<
log
1 1
+O(1)
.
The
i s
esul
in
his
pape
asse s
ha
his
is
essen ially
he
bes ha
we
can
say,
showing
ha
(1
.5)
and,
hence,
(1
.3)
need
no
be
ue
o
a
Bloch
unc ion
.
Howe e ,
we
will
p o e
ha
(1
.5)
holds
o
any
E
Bo
e en
hough
i
may
no
sa is y
(1
.3)
.
Theo em
1
.
Fo
each
in ege q
>
5
le
Then
00
q(z)
=
E
z
qk
,

~
z1
<
1
.
k-o
Then
q
is
a
Bloch
unc ion
and
he e
exis s
a
cons an
C
q
such
ha
(1 .11)

T( , ')>log
1
+C
q
,
0< <1
.
-
1-
Theo em
2
.
(i)
Le
be
a
Bloch
unc ion
sa is ying
12
( , ~)
=
o((1
-
)as
log

1

-
T
( ,
')
--->
oo,

as
1-
(ii)
This
esul
is
bes
possible,
e en
in
BO,
in he
ollowing
sense
.
Le
<D
be
a
posi i e
unc ion
in
[T,
oo)
sa is ying
(1
.12)

-

lim
-¿(x)
=
0
.
x-oo
(1
.13)

<D
(2x)
-
~D
(x),

as
x
--->
oo
.
Le
0< <1
.
1-
Then
he e
exis s
E BO
such
ha
(1
.14)

T( ,
')
>
log
1 1
+
log
T
( )
o
all
su ícien ly
close
o
1
.
Le
us
no ice
.
ha
he
unc ion
T
o
Theo em
2
can
be
aken
o
be
and,
hence,
we
ob ain
.
ON
BLOCH
FUNCTIONS
ANDGAP
SERIES

407
Co olla y
1
.
The e
exis s
E
Bo
such
ha
o
all
suicien ly
close
o
1
and,
hence,
sa is ying
I
D
is
a
B1-domain
and
is
he
uni e sal
co e ing
map
o
D
hen
[12]
sa is ies
(1
.8)
and,
hence,
(1
.5)
.
I
is
known
ha
(1
.8)
may
no
be
ue
o
a
unc ion
in
B1
[4], [5]
.
We
do
no
know
whe he
o
no
(1
.5)
emains
ue
o
any
unc ion
E
Bl
.
We
can
p o e
he
ollowing
esul
.
Theo em
3
.
Le
E
B
l
.
Then
1/2
T
( )
=
(log
1
e
)

0
<
<
1,
-
T( ,
')
>
log

1

-
1
loglog

e
1-
2
1-
1
(1
-
)
exp(2T( ,
'))
d
=
oo
.
0
(1
.15)

limin (1
-
)
2
I2( ,
')
=
0
-1
and
(1
.16)

limsup
Clog

1

-
T( , ')
/
=
oo
.
-+1

1
-T
The
ea ly
s ages
o
his
wo k
bene i ed
om
con e sa ions
wi h
A
.
Bae ns ein
.
He
e en
old
me
ha
he
conclusion
o
Theo em
1
should
be
ue
a
leas
o
su icien ly
la ge
alues
o q
.
I is
a
pleasu e
o
exp ess
my
g a i ude
.

408

D
.
GIRELA
E en
hough
he
mo i a ion o
his
wo k
was
s udying
he
possibili y
o
ex ending
Kennedy's
esul s
o
Bloch
unc ions,
some
o
ou
esul s
a e
mo e
gene al
han
s a ed
and,
in
ac ,
could
be
s a ed
wi hou
making
any
e e ence
o
Bloch
unc ions
.
Fo
analy ic
in
U
and
0
<
<
1,
de ine
and
Le
us
no ice
ha ,
clea ly
2
.
P oo
o
he
main
esul s
Ip
( ,
)
=
27
,~

( e")¡'
d ,

0
<p<
oo,
MP( ,
)
=
Ip( ,
)
1/p

0
<
p
<
oo,
M
.( , )
=
maxI (z)j
.
Ixl=
Fo
s
>
0
and
0
<
p
<
oo, le
X
9
>p
deno e
he
space
o
hose
unc ions
analy ic
in
U
o
which
M
p
( ,
)
=
O((1
-
)
-
S),
as
--->
1,
and
le
Xó'p deno e
he
space
o
hose unc ions
analy ic
in
U
o
which
M
P
( ,
)
=
o((1
-
)
-S
),
as
-->
1
.
Since
M,
( ,
)
is
an
inc easing
unc ion
o
p,
we
ha e
(2
.1)

X'
,P
C
X'g
,
P
and
Xó
,
p
C
X''P,

0
<
p
<
p
<
oo
.
I
p
>
1
and
(z)
=
E°°
o
a
nz'
E
X'
,
P
( espec i ely
X¿,p)
hen
an
applica-
ion
o
Cauchy's
o mula
easily
gi es a
n
=
O(ns)
( espec i ely
a,
=
o(ns))
.
On
he
o he hand,
an
a gumen
simila
o
ha
used
in
[16,
Example
1,
p
.
694]
p o es
ha
i
(z)
_k
o
akznk
is
analy ic
in
U
and
has
Hadama d
gaps
hen
(2
.2)

E
XS,'
<~
:>
ak
=
O(n'),
as
k
(2
.3)

E
X
ó
`
<--¿
ak
=
o(nk),
as
k
-->
oo
.
EBg
'
E
X
1
''
and
EBp
~¿
,
EXó'
,
.
Hence
heo ems
1
and
2
will
be
co olla ies
o
he
mo e
gene al
esul s
ha
we
will
p o e
o
he
spaces
X'
,
P
and
Xos,p
.
ON
BLOCH
FUNCTIONS
AND
GAP
SERIES

409
I
p
<
p'
and
E
Xs
,
P
( espec i ely
X`)
hen
a
esul
o
Ha dy
and
Li lewood
(see
[3,
Th
.
5
.9])
shows
ha
E
X",P
( espec i ely
Xo~'P)
whe e
1 1
s'=s+--
;
.
p p
The
exponen
s'
is
bes
possible
.
Using
his
esul
and
a guing
as
in
.
[3,
Th
.
6
.4]
we
can
deduce
ha
i
0
<
p
<
1
and
(z)
=
ñ=o
a

,z
-
E
Xs
,
P
hen
The
unc ion
(z)
=
(1
-
z)
-
(s+
1/P)
o
which
a,,,
-
(s
+
Pl
ns
-1
+
1
/P
shows
ha
his
es ima e
is
sha p
.
Now,
i
p'
<
pand
E
Xs
,
P
hen
i
is
easy
o see
ha
he
i ial
esul
E
XS,P
is
he
bes ha
we
can
say
in
gene al
.
In
ac ,
he e
exis s
E
X','
such
ha
o
e e y
p
E
(0,
oo]
he e
exis s
a
cons an
B
P
,
s
>0
such
ha
(2
.4)

M
P
(T,
)
>
BP
.s
(1
-
T)
-s
,

1
< <1,

0
<p<
oo
.
_

2-
Indeed,
le
q
>
2
be
an
in ege
and
00
(z)
=
E
gkszqk,

Iz1
<
1
.
k=0
Then,
since
has
Hadama d
gaps,
(2
.2)
shows
ha
E
Xs
,
w
.
Now,
i is
a
simple
exe cise
o
show
ha
he e
exis s
a
cons an
,0s
=
,6
.,,q
>
0
such
ha
(2
.5)

M2
( ,
)
>_ i
.
(1
-
)-3,
1
<
<
1
.
2
_
This
implies
(2
.4)
o
2
<_
p
<
oo
wi h
B
P
,
s
=
J6
s
.
Using
Theo em
8
.20
o
[20,
ol
.
I,
p
.
215]
we
deduce
ha
o
each
p
E
(0,
2)
he e
exis s
a
cons an
A
P
=
A
P
,
q
>
0
such
ha
(2
.6)

M,
( , )>APM2( , ),

0< <1,

0<p<2,
which,
wi h
(2
.5)
implies
(2
.4)
o
0
<
p
<
2
wi h
(2
.7)

B
P
,
s
=
QsAP
.
Since
(2
.8)

logM
P ( ,
)
1
2~

log
( e")¡
d ,
as
p
10,
410

D
.
GIRELA
(2
.4)
shows
ha ,
o
2
<
<
1,
(2
.9)

T
( ,
)
>
27
J-

log
I
( e
2
%
d
>
s
log
1 1
+
ys
whe e
An
examina ion
o
he
p oo
o
Theo em
8
.20
in [20,
ol
.
I,
p
.
215]
shows
ha
he
cons an
A
P
gi en he e
is
o
he
o m
o
some
5
q
>
1
.
This
and
(2
.7)
shows
ha
7s
=
-oo
and
hence
(2
.9)
gi es
no
in o ma ion
a
all
.
Howe e ,
we
will
p o e
in
Theo em
4
ha
he e
exis s
E
X'
,1
sa is ying
(2
.9)
wi h
a
cons an
C
3
in
he
place
o
-ys
and,
also,
sa is ying
(2
.4)
wi h
a
cons an
B
3
>
0
independen
o
p
in
he
place
o
B
P
,
3
.
Theo em
4
.
(i)
Le
s
>
0,
0
<
p
<
oo
and
E
X3,P,
hen
(2
.10)

EX
s,
P
'
,

0<p'<p,
and
(2
.11)

T( ,
)
<
slog
1 1
+0(1)
.
(ii)
This
esul
is bes
possible
in
he
ollowing
sense
.
The e
exis s
E
X
-,
'
and
wo
cons an s
C
s
E
R
and
B
9
>
0 such
ha
(2 .13)

M,
( ,
)
>
B
3
(1
-
)
-s
,

0< <
1,

0
<P
:5
oo,
and
-ys
=
lim
in
log
B
P
,
9
.
P
-0
A
=
6(P-2)/p
p
q
(2
.14)

T
( ,
)
>
2

log
1
( e")
¡
d
>
s
log
1 1
+
Q,

0
<
<
1
.
Fo
s
=
1,
he
conclusion
o
(ii)
holds
wi h
=
é o
any
in ege
q
>
5
.
Theo em
5 gi es
he
analogous
esul s
o
he spaces
XOS,
.
Theo em
5
.
(i)
Le
s
>
0,
0
<p<
oo
and
E
Xoq'P
.
Then
(2
.15)

E
Xó'P
,

0
<
p'
<
p,
and
ON
BLOCH
FUNCTIONS
AND
GAP
SERIES

411
(2
.16)

s
log

1
1

-T
( ,
)
----+
oo,

as
-
(ii)
This
esul
is
bes
possible in
he
ollowing
sense
.
Le
D
be
a
posi i e
unc ion
in
[1,
oo)
sa is ying
(1
.12)
and
(1
.13)
and
le
Then
he e
exis s
E
Xó`
such
ha
(2
.17)
and
~Y( )=~~
i
~,
0< <1
.
1-
(
)
)
M
.
( ,
)
--,
oo,
as
--->
1,
o
e e y
p E
(0,
o0],
(2
.18)

T
( ,
)
>
-

log
1
( e
z )1
d
>
s
log
1
1
+
log
T
( )
27
,

-
o
all
suicien ly
Glose
o
1
.
P oo
o
Theoo ms
4(i)
and
5(i)
:
We
ha e
al eady
p o ed
(2
.10)
and
(2
.15)
.
Also,
(2
.11)
and
(2
.16)
a e
ob ious
o
p
=
oo
.
Now,
le
be a
unc ion
analy ic
in
U
and
0
<p<
oo
.
Using
he
a i hme ic-
geome ic
inequali y,
we
ob ain
T( , )
=
-,
log+
1 ( e")1
d
<
p
-
J_,
log(I ( e")¡P+
1)
d
Hence
<
p
log
C2~
~~(I ( e
i )I
P
+
1~
d
=p
log(I
P
( ,
)
+
1)
.
1

1

1
slog
1-
-T( , )>
plog(1- )sP(IP( , )+1)
.
Then
i is
clea
ha
(2
.11)
( espec i ely (2
.16))
holds
i
E
X"
( espec i ely
i
E
Xó'P)
.
P oo
o
Theo em
4(ií)
:
Le
s
>
0
.
Le
q
>
2
be
an
in ege o
be
de e mined
la e
and
(z)
=
k=0
k
ks
z
e
-i

l
z
l
<
1
.
41
8
Le
Then
(2
.38)
implies
,
Z
I
<
L
,(z)
12 dxdy
=
,
L
l
(z)
I
2
dxdy
+
F
<
+
m2)2
~

I ,(z)I2
(1

JIZI<
(1+
I (z)1
2
)
2
which
implies
D
.
GIRELA
F
=
{z
:
I
z1
<
and
i
(z)
l
<
M},
G
=
{z
:
I
z1
<
and
l
(z)
j
>
M}
.
d
IZI<
z
dxd
lim
in
d

d
I
/1

)1
2

y
=
0
-1
d
1-
L
,
(z)
12 dx
dy
2
dxdy
+

y

2
dxdy
~
IZI<
(1
-
IZI)
<
(1
+
M
2
)
2
7 S( ,
)
+
27 77
2
1-
Hence,
using
(2
.37),
we
ob ain
lim
in
(1
-
)

I
'
(z)
12
dxdy
<
27 12
.
il
.l<
-
Since
17
>
0
is
a bi a y,
we
ha e
lmin (1- )
L 1(z)12dxdy=0
,ZI<
and
his
is
equi alen
o
(1
.15)
.
Now,
an
a gumen
simila o
ha
used
in
he
p oo
o
Theo em
5(i)
shows
ha
(1 .15)
implies
(1 .16)
.
This
inishes
he
p oo
o
Theo em
3
.
3
.
Some
u he
esul s
and
inal
ema ks
a)
The
esul s
ha
we
ha e
p o ed
a e
compa ison
esul s
be ween
M
p ( ,
)
wi h
M
p
,
( ,
) and
T( ,
)
o
in
some
o
he
spaces
Xs
,
p o
XO'p
.
I is
well
known
(see
e
.g
.
[3,
Th
.
5
.10])
ha
he e
exis
unc ions
analy ic
in
U
wi h
M,,
.
( ,
)
g owing
o
in ini y
a bi a ily
slowly
which
a e
no
o
bounded
cha ac e is ic
.
This
leads
one
o
ask he
ollowing
ques ion
:
Le
p( )
be
a posi i e
inc easing
unc ion
on
0
<_
<
1
wi h
y,(0)
=
1
and
u( )
-
oo,
as
-
1,
and
le
be
a
unc ion
analy ic
in
U
sa is ying
M
p
( ,
)
=
0(p( )),
as
~
1
.
Wha
can
be
said
abou
he
g ow h
o
M
p
,
( ,
)
and
T
( ,
)?
In pa icula ,
i
seems
na u al
o
ask
whe he
o
no
he
analogue

ON
BLOCH
FUNCTIONS
AND
GAP
SERIES

419
o
Theo em
4(ii)
is
ue
in his
se ing,
Le
.
does
he e
exis
a
unc ion
analy ic
in
U
wi h
and
a
cons an
C
such ha
We
do
no
know
he
answe
o
his
ques ion
.
Howe e ,
we
do
belie e
ha
he
me hods
o
his
pape
a e no
enough
o
cons uc
such
an
.
b) Fi s
o
all le
us
ema k
ha
some
o
he
esul s
ha
we
a e
going
o s a e
below
(Theo em
6,
Co olla y
2,
and
Theo em
7)
could
be
s a ed
in
he
gene al
amewo k
o
he
spaces
XS,P
and
XÓ'P
.
Howe e ,
o
he
sake
o
simplici y,
we
will
s a e
hem
in
he
se ing
o
Bloch
unc ions
.
I
seems
na u al
o
conjec u e
ha
he conclusion
o
Theo em
1
emains
ue
o
q
=
2,
3,
and 4
.
Howe e ,
ou
a gumen
does
no
p o e
his
since,
wi h
he
no a ion
used
in
he
p oo
o
Theo em
1,
we
ha e
A4
<
0
.
A
mo e
gene al
ques ion
would
be
cha ac e izing
hose
Bloch
unc ions
gi en
by a powe
se ies
wi h
Hadama d
gaps
o
which
(1
.5)
o
a
leas
(1 .16)
is
ue
.
The
ollowing
heo em
gi es
a
pa ial
answe
o
his
ques ion
.
Theo em
6
.
Le
be
a
Bloch
unc ion
gi en
by
a
powe
se ies
(3
.1)
Then
(z)
=
M
.
( ,
)
=
0(p( )), as
--
1,
T( , )>logp( )+C,
0< <1?
lim
Su )
Clog
1
1

-
T
( ,
')
/
=
oo
.
-,1
-
Fu he mo e,
i
limsuplajj
>
0
hen
j_oo
lim
in
~log
1 1
-
T
( ,
')~
<
oo
.
Using
Theo em
2,
we
ob ain
as
an
easy
consequence
o
Theo em
6
he
ol-
lowing
esul
.
Co olla y
2
.
Le
be
a
Bloch
une ion
gi en
by
a
powe
se ies
zn
j
wi h
w
j+1
-->
oo,
as j
-
oo
.
nj
(z)
=
j=1
Then
E
BO
i
and
only
i
1
z"'',
wi h
n,
+

--~
oo,
as
j
-
oo
.
nj
lim
1
(log
11
-

-
T( ,
')
/
=
oo
.
---~
The
p oo
o
Theo em
6
depends
on
he
ollowing
wo
elemen a y
lemmas
whose
p oo s
will
be
omi ed
.
42
0
Lemma
3
.
[14,
p
.
339]
Le
{Sk}
be
a
sequence
o
posi i e
numbe s
such
ha
Sk+11Sk
->
oo
as
k
-
oo
.
Then,
as
k
--->
oo,
Sk+ll k
-
oo
and
k/Sk
-+
oo,
as
k
--->
oo
.
Then, as k
->
oo,
P oo
o
Theo em
6
:
Le
be
a
Bloch
unc ion
gi en
by
(3
.1)
.
Since
E
B,
he e
exis s
K
>
0
such
ha
(3
.2)

laj
j
<
K,

j
=
1, 2,
3,
. . .
Le
{mj}
be
an
inc easing
sequence o
posi i e
numbe s
such
ha
(3
.3)
k-1

00
Si
=
O( k)
and

E

S
.,-
1
=
O(Skl)
.
j=1

j=k+1
=
O( k)
and

E
S~
1
=
O( kl)
.
j=1

j=k+1
D
.
GIRELA
Lemma
4
.
Le
{Sk}
and
{ k} be
wo
sequences
o
posi i e
numbe s
such
ha
m'
,
oo and
nj+l
-
oo,
as
j
-->
oo
.
nj
m
i
Fo
example,
we
can
ake
mj
=
(njnj+l)1/2
.
Le
Iz1
=
1
-
mk1
.
Then
0o

k
(3
.4)

Iz '(z)j
=
E
njzn-
1
<
K
E
nj
+
K
E
nj(1
_
M
k
l)n,
.
j=1 j=1
j=k+1
Using
(3
.3)
and
Lemma
4,
we
ob ain
(3
.5)
=
O(mk),
as
k
,
oo
.
Now,
(2 .21)
wi h
m
=
3,
(3
.3)
and
Lemma
4
imply
(3
.6)

nj
(
1
-
mk
1)n'

(3e-1)3mk

n,-2
j=k+1

j=k+1
_
(3e-1)3mko(mk2)
=
O(mk),
as k
-,
oo
.
Then
(3
.4),
(3
.5)
and
(3
.6)
show
ha
and
hence
This
implies
ha
l '(z)l
>
Iz '(z)I
Lemma
3
implies
ha
ON
BLOCH
FUNCTIONS
ANDGAP
SERIES

421
sup

¡z '(z)1
=
o(mk),
as k
-
oo,
IzI-I-Mk
1
logmk
-T(1-
mk

oo,
as
k
-
oo
.
-~I
1_
Asume
now
ha
is
gi en
by
(3
.1),
sa is ies
(3
.2)
and
limsuplaki
>
0
.
Then
k-+oo
he e
exis s
M
>
0
such
ha
he
se
T
=
{k
:
laki
>
M}
is
in ini e
.
Take k E
T
and
le
k
=
1
-
nk
I
.
Then,
o
Iz
1
=
k
k-I
00
>Mnk(1-7Lk
I )
nk
-KEn~-K
E
nj(1-nk1)--,
=I_II_III
.
j-I
j=k+I
Using
(2
.18),
we
see
ha
he e
exis s
a
cons an
C
>
0
such
ha
I
>
Cnk
.
Finally,
(2
.21)
and
Lemma
3
show
ha
Consequen ly,
we
ob ain
ha
and,
hence
Izln
k
I '(z)j
?
Cnk-
o(nk),
as k
-
a
oo(k
E
T),
T( k,
')
>
log(nk
-
o(nk))
+
O(1),
as k
->
oo(k
E T)
.
This
easily
implies
ha
-+1

1
-
inishing
he
p oo
o
Theo em
6
.
lim
in
(log

-
T( j»
<
00
c)
So
a
we
ha e
p o ed
in
Theo em
2
ha
i
a
Bloch
unc ion
sa is ies
(1
.8)
hen
i
sa is ies
(1
.5)
.
Fu he mo e,
he
unc ions
conside ed
in
heo ems
3
and
6
sa is y
no
only
(1
.16)
bu
also
(1
.15)
.
These
ac s
migh
load
one
o
ask
whe he
o
no
he
con e se
o
Theo em
2(i)
is
ue
.
Theo em
7
shows
ha
he
answe
o
his
ques ion
is
nega i e
in
a
e y
s ong
sense
.
422

D
.
GIRELA
Theo em
7
.
Le 0
<
H
<
1
.
Then
he e
exis s
a
Bloch
unc ion
such
ha
(3
.7)

l min (1- )
2
I2( ,
')
>
0
and
(3
.8)

lim

T
( ,
~
)

=
H
.
li
1
log
11
1
P oo
.
We
will
use
he
ollowing
esul
due
o Spech
[18,
Th
.
III]
(see also
[13,
Lem
.
1
and
2])
on
he
con o mal
mapping
o
ce ain
nea ly
ci cula
egions
.
The e
exis s
a simply connec ed
domain
D
in
he
plane
wi h
(3
.9)

U
U
{ei
:
7 H
<
I
i
<
7 }
C
D
and
such
ha
i
w
deno es
he
con o mal
mapping
om
D
on o
U
wi h
w(0)
=
0
and
w'(0)
>
0,
hen
(3
.10)

lw'(z)
-
11
<
2,

z E
D,
and
(3
.11)

2
(1
-
)
<
1
-
jw( e'
)j
<
3(1
-
),

0
<
<
1,

¡ i
<
7 H
.
Le q
>
12 be
an
in ege
and
de ine
Le
.
(z)
=
e(w(z))
.
Since
he
Bloch
space
is
p ese ed
unde
subo dina ion
(see e
.g
.
[17,
p
.
35]),
i
ollows ha
is
a
Bloch
unc ion
.
Then
he e
exis s
a
cons an
C
such
ha ,
o
e e y
,
(3
.12)

log
+
J
'( e
i
%
<
log
1
+
C,

0< <
1
.
1-
Now,
(3
.9)
implies
ha
00
(Z)
=
W(Z)q",
n=1
log
+
i
'
( e
i
)
1
z
E
U,
lim

1

=
0,

7 H
<
¡ i
<
7 ,
--~1

log
1=
and
hence,
by
he
domina ed
con e gen e heo em,
lim
1

log
+
j
i
(
i
e
i
)
I
d
=
0
-~1
27
.RH<, ,<7,

log
1
1
which,
wi h
(3
.12),
easily
implies
which
implies
ha
Thenwe
ha e
ON
BLOCH
FUNCTIONS
AND
GAP
SERIES
(3
.13)

lim
sup
T( ,
')
<_
H
.
-+1
log
1-T
Le
n
=
1
-
q
-
n
.
Then,
o
z
=

,e
i
wi h
¡ i
<
7 H,
we
ha e
00
(3
.14)

w
(z)
l
(z)
I
=
gnw(z)en
k=i n-1
00
>
gnjw(z)I9

-
1
:
q
k
-

1
:

gklw(z)I9k
=
I
-
II
-
III
.
k=1
k=n+l
Using
(3
.11)
.
we
ob ain
jw( e'
)1
>
3
-
2,

¡ i
<
7 H,
I
=
gnp(z)I9

>
qn
(1
-
3q-n)9"
.
Hence,
since
(1
-
3j-1)j
>
e
-4
(j
>-
12),
we
ha e
1
(3
.15)

I
>
e-4qn
=
e_4

.
1
-
n
Now
ake
q
so
big
ha
q
1 1
<
4e
-4 and
(3e
-1
) 3
82
11
'<
4e_
n_1
(3
.16)

11
=

qk
<

qn
=

1

1

<
1e-4

1
k=1
q-1
g-11-
n 4 1
-
n
.
and,
by
Schwa z's
lemma
and
Lemma
1,
(3
.17)

III
=

gklW(Z)I9k
<-

E
q
k(1
_
q-n),"
k=n+1
k=n+1
<
(3e
-1
) 3

1

qn
=
(3e
-1
)
3

1

1

<
1e-4

1
q
2
-1

q
2
-11
-
n
4
1
-
n
Hence
(3
.14),
(3
.15),
(3
.16)
and
(3
.17)
show
ha
(3
.18)

I
w
()
l
(z)
I
>
2e-41

1
n'

z
=
ne
i

I l
<
7 H
.
423

424

D
.
GIRELA
No ice
ha
Iw(z)I
<
1
and
ha
(3
.10)
implies
ha
Iw'(z)I
>
1/2
.
Then
(3
.18)
easily
implies
(3
.19)

I '(z)I
?
4e-41
l n
,

z
=
neQe

¡ i
<
7 H
.
I
is
clea
ha
(3
.19)
shows
ha
he e
exis s
a
cons an
C
>
0
such
ha
C
Consequen ly,
12 (
.,
'
)
?
(1
-
n)2
Since
12
( ,
')
is
an
inc easing
unc ion
o
,
i
ollows
ha ,
o
n
<
<
,,
+
1,
12 12

_
C C
1
( ,
'
)
?

( n,
')

(1
-
C
n)
2
q
2
( 1
-
n+1)2
.>
q2
(1
-
)2'
This
p o es
(3
.7)
.
Finally,
(3
.19)
shows
ha
o
n
big
enough,
T(
n
, ~)
>
1

1
log
+
I
'( neZc)
I
d
>
H
log
21
iel<, x
4(1
-
n)
lim
in
T
( ,
)
>
H
-1
lo
g
I
-T
which,
wi h
(3
.13),
implies
(3
.8)
.
We
should
ema k
ha
he condi ion
H
>
0
is
needed
in
Theo em
7
.
In
ac ,
i is
a
simple
exe cise
o
p o e
ha
i
E
B
and
T
( ,
')
=
o
(log
1
1
),
as
->
1,
hen
12
( ,
)
=
o((1
-
)
-2
),
as

.
1
.
Pomme enke
p o ed
in
[16,
Th
.
2]
ha
i
a
Bloch
unc ion
has
adial
limi s
almos
e e ywhe e
on
Iz1
=
1
hen
i
sa is ies
(1
.8)
.
No ice
ha
he
unc ion
cons uc ed
o
p o e
Theo em
7
is
in
ac
analy ic
on
he
se
{e"
:
7 H
<
¡ i
<
7 }
and
consequen ly
i
has
adial
limi s
on
a
se
o posi i e
measu e
.
The
nex
esul
asse s
ha
i
a
Bloch
unc ion
sa is ies
his
las
condi ion
hen
i
sa is ies
(1
.5)
.
Theo em8
.
Le
be
a Bloch
unc ion
ha ing
adial limi s
on
a
se
o
posi i e
mensu e
.
Then
lim
1
(log
1 1
-
T( , ~)
/
=oo
.
P oo
.
Since
a
Bloch
unc ion
is
no mal
[16,
p
.
689],
[1,
p
.
12]
he
concep s
o adial
limi s
and
angula
limi s
a e
equi alen
o
[15]
(see also [17,
Th
.
ON
BLOCH
FUNCTIONS
ANDGAP
SERIES

42
5
9
.3])
.
By
P i alo 's
heo em
[19,
p
.
320]
hese
angula
limi s
a e
ini e
almos
e e ywhe e
.
Hence,
i
we
se
hen
¡El
>
0
.
Now,
E
=
{
E
[-7 ,
7 ]
:
has
a
ini e
angula
limi
a e
i }
,
(3
.20)

log 1 1
-
T
( ,
')
=
(LE¡
II
log
1
1
+
(
27
2~JEI
log
1
1
-
27
,

log+
I
'
(
ei
)
1
d
)
=
I
+
II
.
[_i ,7 1_E
I
¡El
=
27 ,
hen
II
=
0
.
O he wise
-

log
+
I
'( e'
)
1
d
27

E
I
II
=
27
-
¡El

log

1

-

1


log+(I '( e' )I
d
27

(

1
-

27
-
IEI

~_,~
, j_E
>
27
-
¡El
Clog

1

-
1

log(I '( e'
)I
+
1)
d
l
-

27

1
-

27
-
IEI

[,,,1
_
E
27
_,
, .
1
]
_
E
log
(1
-
)(L '( e
i
)I
+
1)
d
Hence,
since
E
B,
he e
exis s
a
cons an
C
such
ha
(3
.21)

II>C,
0< <1
.
A guing
as
in
he
p oo
o
[16,
Th
.
2]
we
ob ain
ha
(1-
)
2

I
'( ei )
I2
d
-
0,
as
-
1,
E
and
hen
an
a gumen
like
ha
used
in
he
p oo
o
Theo em
5(i)
p o es ha
I
-->
oo,
as
->
1
.
This,
wi h
(3
.21),
shows
ha
log

1
1

-T
( ,
')
-+
oo,
as
--~
1
.
-
d)
The e
a e
o he
ques ions
ha
we
could
ask
in
his
con ex
.
Fo
ins ance,
i
seems
na u al
o
ask
whe he
o
no
(1
.5) is
ue
i
belongs
o
he
closu e
o
H°°
in
B
.
The
answe
o
his
ques ion
is
a i ma i e
.
Ac ually,
i
is
easy
o see
ha
(1
.8)
is
p ese ed
unde
con e gence
in
he
Bloch
no m
and
hence
Theo em
2
and
[16,
Th
.
2]
show
he
ollowing
:
I
E
CLB(B
n
N)
( he closu e o
B
n
N
in
B)
hen
sa is ies
(1
.5)
.
Acknowledgmen s
.
I
wish
o
hank
he
e e ee
o his
help ul
commen s,
specially o his
ema ks
abou
he
gene ali y
o
ou
esul s
.
O iginally
we
jus
s a ed
ou
esul s
in
he
se ing
o
Bloch
unc ions
and
no
in
he
mo e
gene al
amewo k
o
heoem
4
and
5
.
42
6

D
.
GIRELA
Re e ences
1
.

J
.M
.
ANDERSON,
J.G
.
CLUNIE
AND
CH
.
POMMERENKE,
On
Bloch
unc-
ions
and
no mal
unc ions,
J
.
Reine
Angew
.
Ma h
.
270
(1974),
12-37
.
2
.

J.G
.
CLUNIE,
On
he
de i a i e
o
a
bounded
unc ion,
P oc
.
London
Ma h
.
Soc
.
(3)
14A
(1965),
58-68
.
3
.

P
.
L
.
DUREN,
"Theo y
o
HP
spaces,"
Academic
P ess,
New
Yo k,
1970
.
4
.

J
.L
.
FERNÁNDEZ,
On
he
coe icien s
o
Bloch
unc ions,
J
.
London
Ma h
.
Soc
.
(2)
29
(1984),
94-102
.
5
.

J
.L
.
FERNÁNDEZ,
On
he
g ow h
and
coe icien s
o
analy ic
unc ions,
Ann
.
Ma h
.
120
(1984),
505-516
.
6
.

O
.
FROSTMAN,
Su
les
p odui s
de
Blaschke,
Kungl
.
Fysiog
.
Süllsk
.
i
Lund
Fó h
.
12,
15
(1942),
169-182
.
7
.

J
.B
.
GARNETT,
"Bounded
Analy ic
Func ions,"
Academic
P ess,
New
Yo k,
1981
.
8
.

D
.
GIRELA,
In eg al
means
and
adial
g ow h
o
Bloch
unc ions,
Ma h
.
Z
.
195
(1987),
37-50
.
9
.

D
.
GIRELA,
On
analy ic
unc ions
wi h
ini e
Di ichle
in eg al,
Complex
Va iables
Theo y
Appl
.
12
(1989),
9-15
.
10
.
W
.
K
.
HAYMAN,
"Resea chp oblems
in
unc ion
heo y,"
London
Uni e -
si y
P ess,
London,
1967
.
11
.
W
.
K
.
HAYMAN,
"Me omo phic
unc ions,"
Ox o d
Uni e si y
P ess,
Lon-
don,
1975
.
12
.
W
.K
.
HAYMAN,
S
.J
.
PATTERSON
AND
CH
.
POMMERENKE,
On
he
co-
e icien s
o
ce ain
au omo phic
unc ions,
Ma h
.
P oc
.
Camb idge
Phil
.
Soc
.
82
(1977),
357-367
.
13
.

P
.
B
.
KENNEDY,
A
p ope y
o
bounded
egula unc ions,
P oc
.
Roy
.
I ish
Acad
.
60,
sec
.
A
(1959),
7-14
.
14
.

P
.
B
.
KENNEDY,
On
he
de i a i e
o
a
unc ion
o
bounded
cha ac e is ic,
Qua
.
J
.
Ma h
.
Ox o d (2) 15
(1964),
337-341
.
15
.
O
.
LEHTO
AND
K
.J
.
VIRTANEN,
Bounda y
beha iou
and
no mal me o-
mo phic
unc ions,
Ac a
Ma h
.
97
(1957),
47-65
.
16
.
CH
.
POMMERENKE,
On
Bloch
unc ions,
J
.
London
Ma h
.
Soc
.
(2),
2
(1970),
689-695
.
17
.
CH
.
POMMERENKE,
"Uni alen
unc ions,"
Vandenhoeck
und
Rup ech ,
Gó ingen,
1975
.
ON
BLOCH
FUNCTIONS
AND
GAP
SERIES

42
7
18
.
E
.J
.
SPECHT,
Es ima es
on
he
mapping
unc ion
and
i s
de i a i es
in
con o mal
mapping
o
nea ly
ci cula
egions,
T ans
.
Ame
.
Ma h
.
Soc
.
7
1
(1951),183-196
.
19
.

M
.
TSU,Ii,
"Po encial
Theo y
in
modem
unc ion
heo y,"
Chelsea
Pu
.
Co
.,
New
Yo k,
1975
.
20
.
A
.
ZYGMUND,
"T igonome ic
Se ies,"
Camb idge
Uni e si y
P ess,
Cam-
b idge,
1959
.
Análisis
Ma emá ico
Facul ad
de
Ciencias
Uni e sidad
de
Málaga
29071
Málaga
SPAIN
P ime a
e sió
ebuda
el
15 de
Feb e
de
1990,
da e a
ue si,ó
ebuda
el
8
de
Mai,g
de
1990