VOL. 20 NO. 3 (1997) 503-510
SOME
PROPERTIES AND CHARACTERIZATIONS
OF A-NORMAL FUNCTIONS
JIE XIAO
Department
ofMathematicsPekingUniversity
Beijing, Beijing
100871,
China(Received
January
11, 1994 andinrevisedform October23,
1995)
ABSTRACT. Let
M
be the set ofall functions meromorphic onD
{z
E C[z[
< 1}.
For
a E(0, 1],
a functionf
e
M
is calleda-normal function of bounded(vanishing)
type orf
N
(N),
ifsupzeD(1 --[Z[)=f#(Z)
<
OC(limzl_.,(1 --[z[)=f#(z)
0).
In
this paper we not only show the discontinuity ofN
andN
relative to containment as a varies, which shows0<=<1
N=
CUBC’o,
but also give several characterizations ofN
andN
whicharereal extensions for characterizations ofN
andNo.
KEY
WORDS
AND PHRASES:
A-normal function,UBCo.
1991
AMS
SUBJECT
CLASSIFICATION CODES:
30C45, 31A20I.
INTRODUCTION.
Throughoutthis paper, let
D
{z
II
<
}
be the unit disk in C anddrn(z)
the two-dimensional Lebesgue measure on 0. Also let,(z)
(w- z)/(1
-z)
to be a canonical Mbbius map ofD
ontoD
determined by we D,
and letD(w,r)
{z
e D
[,(z)[ < r}
a pseudohyperbolic disk with center w E
D
and radius r[0, 1].
Suppose
thatg(z,
w)
-logldp(z)l
is theGreen
function ofD
with logarithmic singularity at we
D.
Alsoassume that a(0,
1]
andM
is the class offunctions meromorphic onD. For
f
M,
letf#(z)
[f’(z)[/(1
+
If(z)[2),
which isthespherical derivative off.
Furtherwe sayf
is an a-normal function of boundedtypeifIlfllN"
sup(1-
[zl)=f(z)
<
,
(1.1)
zD and
f
isana-normal functionofvanishing typeiflim
(I
-Izl)f(z)
0.(1.2)
The familiesofall a-normal functions of bounded and vanishing typearedenotedby
N
andN,
respectively.It
iseasy toobserve thatN’
CN
and that fora E(0, 1),
N
andN
are proper subsets ofN
andNo,
whicharethe classical setsofnormal andlittlenormal functions, namely,N
N
andNo
NJ,
respectively.of the families
N
andN
as avaries, andwefind that bothare discontinuous moreverthatN
andJ
/s<<1
N
0<<1/2
areproper subsetsofD#
andUBCo
respectively, whereD#
is the familyof functionsf
M
satisfying=/D
If#
(z)]s
din(z)
<
x,(1.3)
and
UBCo
isthefamily of functionsf
M
satisfyinglim
f
[f#(z)]Sg(z, w)din(z)
O.(1.4)
Here,
it isworth while mentioning thatD#
CUBCo
and thatUBCo
isanimportant meromor-phic counterpart of VMOA--the space of analytic functions with vanishingmeanoscillationonD
(see
[4,8]).
We
then characterize functionsinN
andN’
and obtainthreecriterionswhich areextensionsofcriteriaforN
andNo.
2.
CONTINUITY OF N
AND
N’.
In
thissection,wepayattention to the continuityofN
andN’.
Firstly, wesee the mono-tonicityofN
andN’.
More
preciselywehaveTI-IEOREM
2.1 Letal,as
(0,
1].
Ifal<
a2then(i).
Y
’
CN
2.(ii).
N2’
CN
’2.PROOF. It
sufficies to proveN
CN
’
fora
<
as.
Let
f
N
thenI[f][N
,
<
XandThis gives
f
N
’,
i.e.,N
1 C_N
2.As
tothestrict inculsion, wetake a functionf(z)
(1
z) -3,
a3(a,a2).
A
simple computation just givesf
N\N
’.In
fact,At
thesametime,(1
-Izl)-,
I1
zl
-I1,11-,
(
=)sup
D
l+[l--z[
Jim
(1
--Izl)fF(z)
(1
as)i!}m,
(1 -Izl)’ll
zl
-’
Izl"-’l
+
I1
zl
s’-’’3)Thus,
f,
N
’2\
N
’
So,
N’
CN
’.
Denote
byD
the class of functionsf
M
with=0.
II/11
spf*()<
.
(..1)
For
a(0, I),
it is easy to see thatD
CN.
Furthermore, Theorem 2.1, together withN
CN,
N
CN0
and[3,4]
suggest that wecoiderthe continuity ofN
andN;.
For
th ppo,weneedacorollarywhich canbe viewed anapplication of Theorem 2.1.COROLLARY
2.2Let
a, b(0, i].
Then(i).
U<
N
U<
N.
(n).
N<
N
N<
g,
PROOF.
(i).
On
theonehand,
the relation:U<
N$
U<
g
is clear.On
the otherhand, if
f
U<
N,
thenf
mu
be in someN
,
saying,N
,
where a E(0, b). However,
for
a’
(a,b)
we havef
N’
by theproofof Theorem 2.1.So
f
U<N,
and henceU<N =U<N
.
(ii).
This partcanbeprovedsimilarly.THEOREM
2.3Let
a, bE(0,
1].
Then(i).
U,<,,
gc
No
.
(ii).
F’I,<,,
N0
DN
.
PROOF.
OwingtoTheorem 2.1 andCorollary 2.2,weonly requke provhg(1).
U<
N{
N
and(2).
n<
Y #
g-First, let us consider
(1).
]ff(z)
=,
:
:(_,) thenwe get that
f
is bounded onD.
Since lim_ (-) for b
>
a,f2
<Nd
from[7,
Theorem1].
But
it follows that 6N
again from[7,
Theorem1].
Note
thatwehavehere usedafact:f#
equivalenttolf’[
once
f
isbounded and analyticonD.
The above facts tellusthat<
N
#
N
istrue.It
clear thatf
isSecond,
let us consider(2).
For
this, we pickfa(z)
=
bounded on
D. Moreover
f
N
by using[7,Theorem 1].
However,
lim_ 0 as b>
a, and thene
<
N
.
That istosay,<
N
#
N
.
Ts
completestheproof.Finally, we discussaspecial ce of Theorem 2.3. Theorem 2.3 implies that
0<<]
N
CN0.
Notingthe inclusion:UBCo
CNo
[8],
wewillnaturally ask what connection betweenN
isa proper subset of0<<
N
andUBCo
It
is alittle bitsurpring tousthat0<<
UBCo.
Thisresult shows that thereis abig gap from0<<]
N"
or0<<]
N
toN
orN.
Exactly speaking,weobt
THEOREM
2.4y
(i)
U0<.<,/
c
O,.
(iii).
0<<
N
DN
.
Indeed, PROOf.Firstofall, settg
(z)
log(1
z)
wecheck thatwehve
(2.2)
and
(l-t)
-1IIAIIo
>
li_m,
+
Zo:(_ t)
oofor anyaE
(0, 1).
Now
weturntotheproofsof(i), (ii)
and(iii).
N
then there isan a 6(0,
1/2)
such thatf
N
and thus(i).
Iff
e
Uo<<,/
(2.3)
II111
=,
_<
II711,o
fo
(1-
llzl)
=
din(z).
(2.4)
(2.3)
and(2.4)imply
that(i)is
true.(ii).
From
Theorem2.1,(2.1),
(2.2)
and(2.3)
it isseenthatweonly have needtodemonstrate(1).
U1/=_<<1
N"
D#
and(2).
U/=s=<
N
c
UBCo.
For
(1),
we takea functionf(z)
"="
It
is clearthatA
isbounded and also inN
=D#
ae
[1/2,1)
from[7,
Theorem=0
"
By
noting that-Zoll
_<
8(1 -Il)
foI1
>-
,
wealsoget-Zogll
Z
I1)
\(o,I)
(- I1)1
-1
(’-)rim(z) <_
8
I
2
z-i
din(z)
and/D
--log]z]
fO
1/4 dt(o,1/4)
(1- I1)=1-
1(,-=)
a.()
_<
)-=.
tZog
7
Furthermore,
wecanobtain aconstantC
>
0sothat[Y#
()]=g(, 0) am()
_<
CllIIl=
(1
Iwl)
:(1-)(2.5)
Here
we have usedLemma
4.2.2 in[101.
The above(2.5)
givesf
e
UBCo,
otherwords,(2)
holds.
(iii).
We
pickfs(z)
=
By
[7,Theorem 1]
it follows thatfs e
0<<, N2.
However,
it isve
ey to observe thatfs
D.
So,
0<<,
N
D.
3.
CHACTERIZATIONS OF N AND
N.
In
thissection,wechartere functionsinN
dN
fora(0,
1]
in terms of theweighted average, thepudhyperbolic dk and theGreen
function,respectively.We
use[EI
todenote themeureof thesetE
D
relativetodin(z),
i.e.,IEI
f
din(z).
THEOREM
3.1Let
f
M,a
(0,1]
and p(1,).
Then the following statementsare equivalent"(i).
I
e
g.
(ii).
Thereisan r0e (0,
1)
such that for anyre (0,
r0],
sup
Ire(z)]
din(z) <
.
(iii).
Thereis anr0e (0,
1)
such that for anyre (0,
r0],
[
[l(z)l( -Izl)
(-)din(z) <
.
supwedJD(w,r)
(iv).
f
]
[f#(z)](1
Izl)(=-’)g,(,
)din(z) <
x. supwEDJD
PROOF. We
prove thistheorem inaccordance with the order(i)
(ii)
(iii)
(i)
(iv)
(iii).
Step
1.(i)
(ii).
Letf
EN
,
thenIlfllo
<
o.For
wED
andr(0,
1)
wehave(i
-I1)
(3.1)
ID(,r)l--(1
11);
and
7Fr
din(z) <
(3.2)
Purther, wereadilyget
j
I*()]:’()
I,(w,r)
[D(w,r)[,_
....
r(4r)
(1
rP
II/11"
(3.3)
From
(3.3)
it follows that there exists anro
(0, i)
for whichsup,e
l,(w,
r) <
for any(0,
0],
i..,(ii)
hogan.Sep
2.(ii)
(iii).
For
wD
andre (0, 1),
by(3.1)
wehave(3.4)
Once
assuming(ii),
we canchoosero e (0,
1)
such thatsup
,e
Im(w,
[f#(z)]
din(z) <
zr for anyr(0,
ro].
Further,whenr(0,
to],
wehaver-(4r)’-sup
I(w,
r)
<
(3.5)
(1
r)(’-)"
Thus,there exists anr,
(0,
ro]
such that sup,heDI2(w, r)
<
r for anyr(0,
r,],
and hence(iii)
holds.Step
3.(iii)
=
(i).
If(iii)
holds,then thereexists anro
(0,
1)
satisfyingCo
=sup/D
wed 7r (,,r0)
Consequently,for allw
D,
[/#(z)]2(1 --Izl2)
(-’)din(z) <
1.(3.6)
S(ro,
f
w)=
_I/
7r
[f#(z)]
din(z)<_
Co
1.
Dufresnoy’slemma
[5,Lemma
I],p.216]
thenyields that(3.7)
Also
(1
--Iwl=)==[f*(w)]
_<
S(ro,
f,
w,)(1
I,1=)
-’)
rg(1
S(ro,
/,
w,))
(3.8)
From
(3.7),
(3.8)
and(3.9)
it derivesthat for allwe D,
(
-Il)f()
<
[o(
Co)(
o)
’-o]1/2’
namely,
f
N
.
Step
4.(i)
=),(iv).
Underf
N
,
wehaveI3(w)
JD
[f#(z)]2(1
IZl2)2(-’)
g’(z’
W) din(z)
<
Cllfll
Nwhere
C’
27rf0
t(1
t)-21ogP
dt<
0for p(1,
x)).
So,
(iv)
follows.Step
5.(iv)
(iii).
If(iv)
holds, then,forwD
andr(0, 1),
wehave(3.11)
[f(z)]=(1 -Izl=)
-’
din(z) <
I3(w)
(3.12)
(,) logr
Thatisto say,we can choosean
ro
(0,1)
sothatsup,eDI2(w,r
<
r foranyr(0,
r0].
Thiscompletestheproof.
For
N’
wehaveasimilarresult.THEOREM
3.2Let
f
M,
a(0,
1]
and p(1,
oo).
Then thefollowingstatementsare equivalent:(i).
f
N.
(ii).
Thereisan r0(0,
1)
such that for anyr(0,
r0],
lim
,,,-
ID(w, r)l
-
[f# (z)]
am(z)
O.(iii).
Thereisan r0(0,
1)
such that for anyr(0, r0],
[f#
(z)](1
Iz[)
z(-)din(z)
O.(iv).
lim
/[f*(z)](
Izl)(-’)g(z,
w)din(z)
O.JD
PROOF. We
show this theoremaccordingtotheorder(i)
(ii)
(iii)
(i)
(iv)
(iii).
Step
1.(i)
(ii). Suppose
thatf
N’.
Thenfor anycompactsubsetE
CD
and all zEE,
suchafunction
f
satisfies,lm(1
-I,,,(z)12)f#(,(z))
O. Thus, for anye>
0,thereexists ap(0, 1)
suchthat forIwl
>
,
As
itsconsequence,If#
(z)]
din(z) <
e/o
I
zl
’(-)
(0,>
lu)T (=L’-
iS
"
izl
)=
din(z).
(3.13)
I,(w,r)
<
(-_
)),:,o,
(3.4)
PROPERTIES OF A-NORMAL FUNCTIONS 509
Step
2.(ii)
=.
(iii).
Thisfollowsreadilyfrom(3.4).
Step
3.(iii) =. (i).
Assumingthatf
EM
isof(iii),
for anyeE(0, 1),
wecanfind p(0, 1)
tomake
I(w,
ro)
<
zre,andconsequentlyS(ro,
f,w,)
_
e<
forallwD\D(O,p).
Combining(3.8)
and(3.9)
weget(1-
[w[2)f#
(w)
<_
[(1--ro)(
-4)1
(3.15)
for allw
e
D
O(0,
p).
Thereforef
N.
Step
4.(i)
(iv).
Provided that(i)
is true. SinceC
<
,
for any>
0 there is anr
(0,
1)
such thatlog
(3.16)
(o,
(-
)
d(z)
<
.
Also,
for thisr:
andall w6D,
+/D
)[f#(,(Z))]2
(1
IW]2)2=(1
IZ]2)
(=-1)
logp 1
din(z)
\(o,)
Ii-zl
’
N
=(/o
+/
)(---)
()-(o,r2) \D(0,r2)
F-om
the condition:f
EN’
itfollows that thereexists apl(0, 1)
such that for[w[ >
pl,r2
16 2{2
(0,2)
(’’’)
din(z) <_
(-)
lgP-t
dt(3.17)
and
f log
D
(...)
rim(z) <
[Ifllv,/D
din(z).
(3.18)
\D(O.,) \D(o.,:)
(1 --Izl)
Combining
(3.16),
(3.17)
and(3.18)
deduces(iii)
right away.Step
5.(iv)
(iii).
Thisisasimple consequence of(3.12).
Thiscompletestheproof.
REMARK.
(i). A
special case of(i)
=>
(iii)
in Threm 3.1 w stated by Wulan and Yah[6].
(ii).
The ce: a 1 of(i)
(iv)
in Theorem 3.1 andn
Theorem 3.2 was givenby Aulriand
Lappan
[3].
(iii).
The ide and examplesof this paperare suitable forthe a-Bloch andlittle a-Blo fuctio(e
[7,9]).
(iv).
It
is anopen question to whichof the results from th paper are lid for a(1, ).
Similar questions may also be ked about corresponding cls ofharmonicfunctions(Cf
[3]).
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