Hiroshima Math. J.
33 (2003), 433–443
Radial growth of C 2 functions satisfying Bloch type condition
Dedicated to Professor Makoto Sakai on the occasion of
his sixtieth birthday
Toshihide Futamura and Yoshihiro Mizuta
(Received May 27, 2003)
Abstract. The aim of this paper is to give a simple proof of results by GonzálezKoskela concerning the radial growth of C 2 functions satisfying Bloch type condition.
Our results also give generalizations of their results.
1.
Introduction
Denote by B the Bloch space of all holomorphic functions f on the unit
disk U which satisfy
k f kB ¼ j f ð0Þj þ supð1 jzj 2 Þj f 0 ðzÞj < y:
zAU
The radial growth of Bloch functions was extensively discussed by ClunieMacGregor [2], Korenblum [4], Makarov [5] and Pommerenke [7]. The law of
the iterated logarithm of Makarov [5] states that if f A B, then
j f ðrzÞj
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a Ck f kB
1
1
r!1
log log log 1r
log 1r
ð1Þ
for almost every z A qU, where C is a universal constant. Pommerenke [7]
proved that this inequality is true for C ¼ 1 and this inequality is false for
C a 0:685. Recently, González and Koskela studied the radial growth of C 2
functions on the unit ball B n of R n which satisfy
j‘uðxÞj 2 þ juðxÞDuðxÞj a
ð1 jxjÞ
for all x A B n , where c > 0 and g a 1.
Theorem 1.2]).
2
c
log
2
1jxj
g
ð2Þ
They showed the following result ([3,
2000 Mathematics Subject Classification. Primary 31B25.
Key words and phrases. radial growth, Bloch condition, law of the iterated logarithm, Hausdor¤
measures.
434
Toshihide Futamura and Yoshihiro Mizuta
Theorem A. Let u be a C 2 function on B n satisfying (2).
almost all z, jzj ¼ 1,
Then, for
juðrzÞj
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a c1
r!1
1 1g
1
log 1r
log log 1r
if g < 1; and
lim sup
r!1
if g ¼ 1.
juðrzÞj
a c2
1
log log 1r
Here the constants c1 and c2 depend only on n; c; g.
We denote by Bðx; rÞ and Sðx; rÞ the open ball and the sphere of center
x and radius r, respectively. We set B n ¼ Bð0; 1Þ and S n1 ¼ Sð0; 1Þ. The
Hausdor¤ measure with a measure function h is written as Hh . In case
hðrÞ ¼ r a , we write Ha for Hh .
Our first aim in the present note is to extend Theorem A by GonzálezKoskela. For this purpose, let j be a positive, continuous and non-decreasing
function on the interval ½0; 1Þ satisfying
jð1 r=2Þ a Ajð1 rÞ
for every r A ð0; 1Þ
ð3Þ
with a constant A b 1 and
ð1
ð1 tÞjðtÞdt ¼ y:
ð4Þ
0
Set
FðrÞ ¼
ðr
ð1 tÞjðtÞdt:
0
Theorem 1.
Let u be a C 2 function on B n with uð0Þ ¼ 0 such that
Au ðxÞ ¼ j‘uðxÞj 2 þ juðxÞDuðxÞj a jðjxjÞ
for all x A B n :
ð5Þ
Then for Hn1 -a.e. z A S n1 ,
pffiffiffiffi
juðrzÞj
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a A:
1
r!1
FðrÞ log log 1r
Remark 1. If we take jðrÞ ¼ cð1 rÞ2 flogð2=ð1 rÞÞgg for c > 0 and
g a 1, then Theorem 1 gives Theorem A.
On the other hand, we have the lower limit result as follows:
Theorem 2.
If u is as in Theorem 1, then
Radial growth of C 2 functions
435
juðrzÞj
lim inf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2
r!1
FðrÞ log log FðrÞ
for Hn1 -a.e. z A S n1 .
By Theorems 1 and 2, we have the following corollary.
Corollary 1. Let u be a C 2 function on B n satisfying
1
1 g
1
1
1
Au ðxÞ a cð1 jxjÞ2 logð1Þ
. . . logðl1Þ
logðlÞ
;
1 jxj
1 jxj
1 jxj
where c > 0, g a 1 and logðkþ1Þ ðtÞ ¼ logðkÞ logð1Þ ðtÞ with logð1Þ ðtÞ ¼ logðe þ tÞ.
Then for Hn1 -a.e. z A S n1 ,
juðrzÞj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
lim sup r
a c1
1g
r!1
1
1
logðlÞ 1r
logð2Þ 1r
and
juðrzÞj
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi a c2
lim inf r
1g
r!1
1
1
logðlÞ 1r
logðlþ2Þ 1r
when g < 1;
juðrzÞj
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c3
1
1
r!1
logðlþ1Þ 1r
logð2Þ 1r
and
juðrzÞj
lim inf qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a c4
r!1
1
1
logðlþ3Þ 1r
logðlþ1Þ 1r
when g ¼ 1.
2.
Here c1 ; c2 ; c3 and c4 are constants depending only on c; g and l.
Exponential integral
In this section, we present an exponential estimate for C 2 functions
satisfying (5). For this we prepare the following lemma, which is a generalization of [3, Theorem 2.2].
Lemma 1.
Let j be a positive continuous function on ½0; 1Þ, and set
ðr
FðrÞ ¼ ð1 tÞjðtÞdt:
0
436
Toshihide Futamura and Yoshihiro Mizuta
Let u be a C 2 function in B n with uð0Þ ¼ 0 which satisfies condition (5).
ð
S
juðrzÞj 2k dSðzÞ a sn 4 k k!½FðrÞ k
n1
Then
ð6Þ
for all k A f0; 1; 2; . . .g and all r A ð0; 1Þ, where sn denotes the surface measure of
S n1 .
Proof. First we show that
d
dt
for each v A C 2 ðB n Þ.
d
dt
ð
vðtzÞdSðzÞ ¼ t 1n
S n1
ð
DvðwÞdw
ð7Þ
Bð0; tÞ
Using the divergence theorem, we have
ð
S n1
vðtzÞdSðzÞ ¼
ð
S n1
z ‘vðtzÞdSðzÞ
ð
w
‘vðwÞdSðwÞ
Sð0; tÞ t
ð
¼ t 1n
DvðwÞdw:
¼t
1n
Bð0; tÞ
Thus (7) holds.
We prove this lemma by induction on k. Clearly, (6) holds for k ¼ 0.
Suppose that (6) holds for k. Using (7) and the assumption on induction, we
obtain
d
dt
ð
S n1
juðtzÞj 2ðkþ1Þ dSðzÞ
¼ 2ðk þ 1Þt 1n
ð
juðwÞj 2k ðuðwÞDuðwÞ þ ð2k þ 1Þj‘uðwÞj 2 Þdw
Bð0; tÞ
a 4ðk þ 1Þ 2 t 1n
ð
juðwÞj 2k Au ðwÞdw
Bð0; tÞ
2 1n
a 4ðk þ 1Þ t
ðt
r
n1
ð
jðrÞ
S n1
0
a sn 4 kþ1 k!ðk þ 1Þ 2 t 1n
ðt
juðrzÞj dSðzÞ dr
2k
r n1 jðrÞ½FðrÞ k dr:
0
Integrating both sides from 0 to r and applying Fubini’s theorem, we have
Radial growth of C 2 functions
ð
S
n1
juðrzÞj 2ðkþ1Þ dSðzÞ a sn 4 kþ1 k!ðk þ 1Þ 2
ðr
t 1n
ðt
0
¼ sn 4 kþ1 k!ðk þ 1Þ 2
a sn 4 kþ1 ðk þ 1Þ!
r n1 jðrÞ½FðrÞ k drdt
0
ð r ð r
0
ðr
437
t 1n dt r n1 jðrÞ½FðrÞ k dr
r
ðk þ 1Þð1 rÞjðrÞ½FðrÞ k dr
0
¼ sn 4
kþ1
ðk þ 1Þ!
ðr
d
½FðrÞ kþ1 dr
dr
0
¼ sn 4 kþ1 ðk þ 1Þ!½FðrÞ kþ1 :
Hence (6) also holds for k þ 1.
The induction is completed.
Lemma 2. Let u be a function in B n satisfying condition (6).
c, 0 < c < 1=4, and for all r, 0 < r < 1,
!
ð
cjuðrzÞj 2
sn
:
exp
dSðzÞ a
n1
FðrÞ
1
4c
S
Then for all
ð8Þ
Proof. If k is a non-negative integer, then, by (6), we have
!k
ð
1
cjuðrzÞj 2
dSðzÞ a ð4cÞ k sn
k! S n1
FðrÞ
for c > 0.
Hence it follows that
!
!k
ð
y
X
cjuðrzÞj 2
1 cjuðrzÞj 2
exp
dSðzÞ
dSðzÞ ¼
FðrÞ
FðrÞ
S n1
S n1 k¼0 k!
ð
a sn
y
X
ð4cÞ k :
k¼0
The series on the right converges if 0 < c < 1=4 and thus our lemma is proved.
3.
Proof of Theorem 1
Let j and F be as in the Introduction, and let u be as in Theorem 1.
To prove Theorem 1, we need the following two lemmas.
Lemma 3.
For every 0 < r < 1,
Fð1 r=2Þ a
A
Fð1 rÞ þ Fð1=2Þ:
4
438
Toshihide Futamura and Yoshihiro Mizuta
Lemma 4. Let u be a C 2 function in B n such that j‘uðxÞj 2 a jðjxjÞ.
Then for every x A B n nBð0; 1=2Þ,
juð yÞ uðzÞj a A½FðjxjÞ 1=2
whenever y; z A Bðx; ð1 jxjÞ=2Þ.
Proof. We see that
juðyÞ uðzÞj a ð1 jxjÞ½jðð1 þ jxjÞ=2Þ 1=2 a A 1=2 ð1 jxjÞ½jðjxjÞ 1=2
for all y; z A Bðx; ð1 jxjÞ=2Þ. On the other hand, we have for 1=2 < t < 1,
ðt
ðt
FðtÞ b
ð1 sÞjðsÞds b jð2t 1Þ
ð1 sÞds b A1 ð1 tÞ 2 jðtÞ:
2t1
2t1
Thus Lemma 4 follows.
Proof of Theorem 1. From Lemma 2, we see that
!
1 d
ð
2
cjuðxÞj 2
1
ð1 jxjÞ
log
exp
dx < y
1 jxj
FðjxjÞ
Bn
for all c, 0 < c < 1=4, and all d > 0. Then there exists a set E H S n1 such
that Hn1 ðEÞ ¼ 0 and
!
ð1
2 1 d
cjuðrzÞj 2
1
ð1 rÞ
log
exp
dr < y;
1r
FðrÞ
0
for each z A S n1 nE, 0 < c < 1=4 and d > 0, which implies that
!
ð ð1þrÞ=2
2 1 d
cjuðtzÞj 2
1
lim
ð1 tÞ
log
exp
dt ¼ 0:
r!1 r
1t
FðtÞ
Fix z A S n1 nE. For 0 < r < 1, define Ir ¼ ½r; ð1 þ rÞ=2Þ.
obtain
!
1 1 d
c inf t A Ir juðtzÞj 2
lim log
exp
¼ 0;
r!1
1r
Fðð1 þ rÞ=2Þ
ð9Þ
From (9), we
which implies that
c inf t A Ir juðtzÞj 2
1
a ð1 þ dÞ log log
1
1r
4 AFðrÞ þ Fð1=2Þ
ð10Þ
for r near 1, by Lemma 3. Hence it follows from (10) and Lemma 4 that
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
juðrzÞj
Að1 þ dÞ
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a
:
4c
1
r!1
FðrÞ log log
1r
Radial growth of C 2 functions
439
Here, letting c ! 1=4 and d ! 0, we obtain
pffiffiffiffi
juðrzÞj
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a A;
1
r!1
FðrÞ log log 1r
which proves Theorem 1.
4.
Proof of Theorem 2
In this section we complete the proof of Theorem 2.
By Lemma 2, we see that
!
ð
cjuðxÞj 2
1
1d
ð1 jxjÞjðjxjÞFðjxjÞ ðlog FðjxjÞÞ
exp
dx < y
FðjxjÞ
B n nBð0; r0 Þ
for all c, 0 < c < 1=4, and d > 0, where r0 ¼ F1 ðeÞ. Consequently,
!
ð1
cjuðtzÞj 2
1
1d
lim ð1 tÞjðtÞFðtÞ ðlog FðtÞÞ
exp
dt ¼ 0
r!1 r
FðtÞ
for Hn1 -a.e. z A S n1 , 0 < c < 1=4 and d > 0. This implies that
!
ð1
cgr ðzÞ 2
1
1d
lim ð1 tÞjðtÞFðtÞ ðlog FðtÞÞ
exp
dt ¼ 0;
r!1 r
FðtÞ
ð11Þ
Since e dx b dx for x > 0, we have
!!
ð1 þ dÞ1 cgr ðzÞ 2
exp
FðtÞ
where gr ðzÞ ¼ inf rar<1 juðrzÞj.
d
ðlog FðtÞÞ1d
dt
1
¼ ð1 þ dÞð1 tÞjðtÞFðtÞ ðlog FðtÞÞ
2d
ð1 þ dÞ1 cgr ðzÞ 2
exp
FðtÞ
þ ðlog FðtÞÞ1d ð1 tÞjðtÞFðtÞ1
!
!
ð1 þ dÞ1 cgr ðzÞ 2
ð1 þ dÞ1 cgr ðzÞ 2
exp
FðtÞ
FðtÞ
1
1
a ð1 þ d þ d Þð1 tÞjðtÞFðtÞ ðlog FðtÞÞ
for r0 < t < 1.
1d
cgr ðzÞ 2
exp
FðtÞ
From (11), we obtain
limðlog FðrÞÞ
r!1
1d
ð1 þ dÞ1 cgr ðzÞ 2
exp
FðrÞ
!
¼ 0;
!
!
440
Toshihide Futamura and Yoshihiro Mizuta
which implies that
ð1 þ dÞ1 cgr ðzÞ 2
a ð1 þ dÞ log log FðrÞ
FðrÞ
for r near 1.
By letting c ! 1=4 and d ! 0, we have
lim sup
r!1
gr ðzÞ 2
a 4;
FðrÞ log log FðrÞ
which completes the proof of Theorem 2.
Corollary 2. Let j and F be as in the Introduction.
harmonic function on B n satisfying
j‘uðxÞj 2 a jðjxjÞ
Let u be a
for all x A B n :
Then
juðrzÞj
lim sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2
FðrÞ log log FðrÞ
r!1
for Hn1 -a.e. z A S n1 .
Proof. Consider the radial maximal function of u defined by
for 0 < r < 1 and z A S n1 .
Ru ðr; zÞ ¼ max juðtzÞj
0atar
By the Hardy-Littlewood maximal theorem [1, Chapter 6] and Lemma 1,
ð
S n1
jRu ðr; zÞj 2k dSðzÞ a c1
ð
S n1
juðrzÞj 2k dSðzÞ a c1 sn 4 k k!½FðrÞ k
for all k and 0 < r < 1, where c1 is a constant depending only on n. As in the
proof of Theorem 1, we have
!
ð1
c2 jRu ðt; zÞj 2
1
1d
lim ð1 tÞjðtÞFðtÞ ðlog FðtÞÞ
exp
dt ¼ 0
r!1 r
FðtÞ
for Hn1 -a.e. z A S n1 , all c2 , 0 < c2 < 1=4, and d > 0.
proof of Theorem 2 that
Hence we see as in the
Ru ðr; zÞ
lim sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2
FðrÞ log log FðrÞ
r!1
holds for Hn1 -a.e. z A S n1 .
Since Ru ðr; zÞ b juðrzÞj, we have
Radial growth of C 2 functions
441
juðrzÞj
lim sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2
FðrÞ
log log FðrÞ
r!1
for Hn1 -a.e. z A S n1 , which yields the required conclusion.
Remark 2.
Let u be a harmonic function on B n satisfying
kuk ¼ sup ð1 jxjÞj‘uðxÞj < y:
x A Bn
Then Corollary 2 says that
juðrzÞj
lim sup qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2kuk
1
1
r!1
log 1r log log log 1r
for Hn1 -a.e. z A S n1 .
5.
Hausdor¤ measures and radial growth
Let j and F be as in the Introduction. Take a positive non-decreasing
function C on ½0; 1Þ satisfying
FðrÞ log logð1=ð1 rÞÞ
½CðrÞ 2
!0
as r ! 1:
For l > 0, consider the measure function
hl ðtÞ ¼ t
n1
!
½Cð1 tÞ 2
exp 4 A l
:
Fð1 tÞ
3
4 2
We finally establish the following result.
Theorem 3.
If l > 0 and u is as in Theorem 1, then
lim sup
r!1
for Hhl -a.e. z A S
n1
juðrzÞj
al
CðrÞ
.
Proof. In view of Lemma 2, we see that
!
2
ð
2
c1 juðxÞj 2
1
ð1 jxjÞ
log
exp
dx < y
1 jxj
FðjxjÞ
Bn
for all c1 , 0 < c1 < 1=4. By the covering theorem, there exists a set F H S n1
such that Hhl ðF Þ ¼ 0 and
442
Toshihide Futamura and Yoshihiro Mizuta
1
lim½hl ð5tÞ
t!0
ð
ð1 jxjÞ
log
1
Bðz; tÞVB n
2
1 jxj
2
!
c1 juðxÞj 2
exp
dx ¼ 0
FðjxjÞ
ð12Þ
for z A S n1 nF and 0 < c1 < 1=4 (cf. [6, Lemma 5.8.2]).
Fix z A S n1 nF . For 0 < t < 1, write Dt ¼ Bðð1 tÞz þ 41 tz; 41 tÞ.
Since Dt H Bðz; tÞ V B n , we have
!
2
ð
2
c1 juðxÞj 2
1
1
dx
½hl ð5tÞ
ð1 jxjÞ
log
exp
1 jxj
FðjxjÞ
Bðz; tÞVB n
!
2
ð
2
c1 juðxÞj 2
1
1
b ½hl ð5tÞ
ð1 jxjÞ
log
exp
dx
1 jxj
FðjxjÞ
Dt
!
4 2
c1 inf x A Dt juðxÞj 2
1
1
log
exp
b ½hl ð5tÞ jDt jt
t
Fð1 t=2Þ
!
2
2
½Cð1
5tÞ
c
inf
juðxÞj
1
x
A
D
2
t
2 log logð1=tÞ þ
b c2 exp 4 3 A4 l
;
Fð1 5tÞ
Fð1 t=2Þ
where c2 is a positive constant.
From (12), we obtain
c1 inf x A Dt juðxÞj 2
½Cð1 5tÞ 2
a 4 3 A4 l 2
þ 2 log logð1=tÞ
Fð1 t=2Þ
Fð1 5tÞ
for su‰ciently small t > 0.
By Lemma 3, we have
2
c1 inf x A Dt juðxÞj 2
1 2 Cð1 tÞ
a
A
l
þ 2 log logð1=tÞ
41 AFð1 tÞ þ Fð1=2Þ
Fð1 tÞ c3
where c3 ¼ Fð1=2ÞððA=4Þ 3 þ ðA=4Þ 2 þ ðA=4ÞÞ, which implies that
lim sup
inf x A Dt juðxÞj 2
t!0
Cð1 tÞ
2
a
1 2
l :
4c1
Hence it follows from Lemma 4 that
lim sup
t!0
juðð1 tÞzÞj
l
a pffiffiffiffiffiffiffi :
Cð1 tÞ
4c1
Letting c1 ! 1=4, we obtain
lim sup
t!0
which proves Theorem 3.
juðð1 tÞzÞj
a l;
Cð1 tÞ
Radial growth of C 2 functions
443
Remark 3. If we take jðrÞ ¼ ð1 rÞ2 flogð2=ð1 rÞÞgg , g < 1, then the
conclusion of Theorem 3 says that
lim sup
r!1
for Hh -a.e. z A S
n1
juðrzÞj
al
flogð1=ð1 rÞÞg a
, where 2a > 1 g and
hðtÞ ¼ t n1 exp½4 3 A4 l 2 flogð1=tÞg 2aþg1 :
Thus Theorem 3 can not cover [3, Theorem 1.3] by González-Koskela.
References
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[ 2 ] J. G. Clunie and T. H. MacGregor, Radial growth of the derivative of univalent functions,
Commentari Math. Helv. 59 (1984), 362–375.
[ 3 ] M. J. González and P. Koskela, Radial growth of solutions to the Poisson Equation,
Complex Variables, 46 (2001), 59–72.
[ 4 ] B. Korenblum, BMO estimates and radial growth of Bloch functions, Bull. Amer. Math.
Soc. 12 (1985), 99–102.
[ 5 ] N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc.
London Math. Soc. 51 (1985), 369–384.
[ 6 ] Y. Mizuta, Potential theory in Euclidean spaces, Gakkōtosyo, Tokyo, 1996.
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Toshihide Futamura
Department of Mathematics
Graduate School of Science
Hiroshima University
Higashi-Hiroshima 739-8526, Japan
E-mail address: [email protected]
Yoshihiro Mizuta
The Division of Mathematical and Information Sciences
Faculty of Integrated Arts and Sciences
Hiroshima University
Higashi-Hiroshima 739-8521, Japan
E-mail address: [email protected]