477
Internat. J. Math. & Math. Scl.
Vol. 8 No. 3 (1985) 477-482
SOME UNSOLVED PROBLEMS ON MEROMORPHIC FUNCTIONS
OF UNIFORMLY BOUNDED CHARACTERISTIC
SHINJI YAMASHITA
Department of Mathematics
Tokyo Metropolitan University
Fukasawa, Setagaya,
Tokyo
158, Japan
(Received February 14, 1985)
ABSTRACT.
The family
UBC(R)
fsmily of
is defined in terms of the Shimizu-Ahlfors characte-
R
ristic in a Rieman surface
ristic function.
of meromorphic functions of uniformly bounded characte-
UBC(R)
There are some natural parallels between
holomorr.hc
BMOA(R),
the
After a survey
BMOA(R).
some open problems are proposed in contrast with
KEY WORDS AND PHRASES.
and
R.
f11nctonz of bounded mesn oscilltion in
characteristic function,
UBC(R), BMOA(R), armonic majorant, F. Riesz’ decomposition, Carleson measure,
classification of
Riemann surface,
Riemann
Shimizu-Ahlfors’
surfaces.
Primary 30D35, 30D45, 30D50, 30D55,
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
Secondary. o0C8{), 30C85, 30F30.
INTRODUCTION
In a series
ppers [19]
[Pl], [3]
30F20.
I.
[5] I have been studying functions
of uniformly bounded characteristic. After a survey I propose some questions which I
have been unable to answer. The adOective "unsolved" in the present title, therefore,
means more precisely "unsolved by the present author".
R
Let
w
in the complex plane.
D U 8D
R
By
D
8D
+
loglz
w
R
such that the closure
consists of a finite number of mutually dis-
For a point
R.
joint, analytic, smple and closed curves in
exp{lim(gD(z,w)
with poles
is identified with its local-parametric image
we always mean a subdomain of
is compact and the boundary
r
gE(z,w)
be a Riemann surface which has the Green functions
As usual, each point of
R.
in
w
of
D
we set
I) },
z-w
where
z
Let
w
D
t
-
within the parametric disk of center
{z g D;
gD(z,w)
log(r/t)},
f#(z)edxdy,
we consider the second-order differential
The Shimizu-Ahlfors characterztc function of
wg D
by
T(D,w,f)
r
0
w.
0 < t < r.
f
For
where
D
t
f#
meromorphic in
If’I/(l
is defined for a pair
# (z)2dxdy]dt"
t- i [f
f
+
R
Ill2).
w, D
with
478
S. Yamashlta
T(r,f)
T(D,w,f),
the usual Shimizu-Ahlfors characteristic function of
f.
Returning to general
R
we
now set
T(R,w,f)
T(D,w,f)
lim
w6D+E
E > 0
this means that given
IT(R)
T(D)
.nown
< g
D
for all
D
K
n-illR f#(z)2gR(z,wldxdy
It is
T(R,w,f) <
If
to be of bounded
R
R.
w
Thus, f
R.
<
for all
f
We call
T(R,w,f)
in notation, if
if and only if
R.
w
<
f
for
is Lindelfian and mero-
in the sense of Maurice Heins.
By definition, f
T(R,f)
sup
w6R
Each meromorphic
fl
T(R,w,f) <
f
and
UBC m UBC(R)
is of uniformly bounded characteristic, f
notation, if
functions
BC(R)
BC
BC
in
T(R,w,f)
R, then
w
characteristic, f
(hence for each)
morphic in
for a
w 6 R,
-if# (z)2dxdy
Green potential of the measure
R
in
f2
in
.
can be expressed as a quotient
with no common zero in
is a finite-valued subharmonic function in
f
.
T(R)
with the obvious change in case
that
T(R,w,f)
;
K c R (w 6 K) such that
we may find a compact set
R
R.
f
fl/f2
of holomorphic
Therefore,
because
A(z)dxdy
hf#(z)2dxdy.
If
BC, then
CR^(w)
where
@(w)
A
@R
2TCR,w,f),
w 6 R,
(i)
is the least harmonic majorant of
@
harmonic majorants of
in
R.
Therefore, the function
potential part of the F. Riesz decomposition of
and only if
A
@R
exists
In the special case
of center
w 6
and the radii
T((w,r),w,f)
where
(namely, f
R
f (z)
w
R, the smallest
in
@
-I
D
p:
T(, a,fop).
w
is the
of
Apparently, f 6 UBC
is bounded in
if
R.
the non-Euclidean disks
T(r,fw)
f((z + w)/(l + wz)),
R
T(R,w,f)
all the
r, 0 < r < i, so that
(2)
z 6 4.
It is not difficult to observe that f 6 UBC(R)
where
R.
BC) and the potential part
we can choose as
tanh
in
8nong
fop 6
is the universal covering projection.
This fact reduces many problems on
UBC(A),
Actually, T(R,p(a),f)
UBC(R)
to those on UBC(), yet,
in some cases, there are subtle differences; see the problem (I)
below, for example.
Hitherto we have been concerned with meromorphic functions f in R. If f is
UNSOLVED PROBLEMS ON MEMOMORPHIC FUNCTIONS
,ot
..
’tee further,
If’l.
can propose cimilar considet’ati
ther, we
-lfrt-l[ffD
0
T*(R,w,f)
we obtain
II
s,
by
(1)
an analogue of
(If
(w)
2r*(R,w,f),
w
[21], [23].
holds
w 6 R, then
for a
(Ifl>^
f 6
H2(R),
R, so that, T*(R,w,f)
in
is
To observe these we remember that
e R,
(3)
(3) coincides
The left-hand side of
with
w 6 R.
(2),
The obvious analogue of
f()le)A(0)
A
(Ifw
<
The converse is also true.
2)^(w),
f(w)
T*(R,w,f)
If
mono maont
t st
H2(R), then
)R(w) -]fl
f 6
If’(z)12dxdy]dt,
t
and others.
w 6 R.
finite for all
if
f#
on re:,!ncing
Jns
Thus, beginning wiLh
T*(D,w,f)
that
479
(3),
together with
A
yields in
the identity
e*(A,f).
A ho!omorphic function
f
T*(R,f)
<
R
in
is called
BMOA, f
BMOA(R)
BMOA
in notation,
if
T*(R,w,f)
sup
w6R
BMOA(R)
the notion of
was first introduced by Thomas A. Metzger, and the present au-
A, this coincides with
the known class BMOA(A). The inclusion formula BMOA(R)
UBCA(R) is obvious,
where UBCA(R) is the family of all the holomorphic functions in UBC(R). This is a
consequence of the obvious inequality T(R,f)
T*(R,f), yet a better estimate
thor extended it to many-valued functions
[23].
In case
R
c
2-11og{2T*(R,f)
T(R,f)
can be proved.
+ 1}
There exists
It is known that if
f
f 6
UBCA(A)
BMOA(A),
BMOA(A).
then
f
B(A)
is Bloch, f
in notation, in
the sense that
sup(
zA
Izl)If’(z)l
<
Similarly, it is known [19] that if
UBC(A), then f is normal in the sense of
N(A) in notation, that is,
f 6
Olli Lehto and Kaarlo I. Virtanen, f 6
Izl2)f#(z)
sup(l
zA
Both B(A) and N(A)
<
.
can be extended to
B(R) and N(R) because R has the hyperBMOA(R) c B(R) and UBC(R) c N(R) are proper in the case
[i]
[18],[22], [25], on normal meromorphic
functi6ns and the related topics.)
Algebraically, BMOA(R) is closed for summation,
while UBC(R) is not; UBC resembles N
at this point.
bolic
2.
metric.
The inclusion formulae
R
A. (See the papers
PROBLEMS
Now, the problems
(I)
equation
For
6
f
in
{Izl
*
R.
cap{a 6 *; n(a,f)
where
"cap"
} we let
n(e,f)
be the number of the roots of the
Suppose that there exists an integer
k
0
such that
k} > 0,
denotes the elliptic capacity.
Is it true that
f 6
UBC(R) ?
This is
480
S. YAMASHITA
R
valid for
unsolved
Be(R).
in case
k > 0
for
(II)
A(R,f)
Let
*
prjection to
O, and is valid for
k
on general
A
R
and
k
>:
Th# ,.b ..m
0.
R. Earlier and well-known conclusion is that
be the spherical area of the image
of the Riemannian image of
R
by
f.
f(R)
R, that is, the
Is there a constant k > 0
of
swh that
T(R,f)
<
kA(R,f)
A(R,f) <w
Since
(5)
is considered to have the radius
always holds if the sphere
1/2, the answer is "no" in general. We must therefore add the condition that A(R,f)
< ; in this case T(R,f) <
is obvious; see (I). The celebrated Herbert J.
Alexander-B. A. Taylor-Joseph L. Ullman inequality teaches us that for holomorphic
T*(R,f) <_-
f,
(2w)-IA(R,f),
A*(R,f)
where, in this case,
f(R).
is the Euclidean area of
See the problem (VII)
below.
(III)
As usual, let
0
X
be the family of open Riemann surfaces
or of those which are hyperbolic with
X(R)
()
R
of class
0
G
It is easy to observe that
OUBCA c OBMOA
To prove that the inclusion is proper we make a few modifications to the argument in
[23].
Let
E
be a compact set of linear measure zero, yet of positive capacity, ly-
ing on the real axis.
*
R
z
is of
g
OUBCA
2, fh,
H
h(E)
UBCA(R)
Next,
f 6
let
and hence
(IV)
If
a
E
(hence
h(z)
and let
of
OBMOA)
f
X
H2(R).
foh 6
Then
must be a constant.
for
which
f
UBC(R), then
UBC(A)
OUBCA
c 0
#
a), z 6
i/(z
R
yet
h(E)
because it omits the set
a reasonable
find
Let
OH2
is of
0UBCA
X
#
First, the function
of positive capacity, so that
H2(
E).
Therefore, R 6
c
We show that
R
Since
E
OH2
The problem is to
is removable for
OBMOA
f
UBC(R
for each subdomain
of R. Consider
the converse in the special case R
A. Let
0 < r <
Let f be meromorphic in A such that for
each w
A we may choose
0 < r < i such that f
UBa((w,rl). Is it tre that f UBC(A)
The corresponding problem for BOA is solved in the positive
in [21; the extensions to the bordered Riemann surfaces un&er the obvious
technical conditions are now easy.
.
(V)
Each
f
B
and
B
R1
has, as a member of BC(A), the expression: f
are
Blaschke
products with no common zero an F is
1
2
holomorphic
zero-free in A. We observe that F
UBC(A). Conversely suppose that F
where
an
I)
(BI/B2)F
UBC(A)
Is it true that f UBC(A) ? The answer is "no"
19. Actually,
there exists a nonnormal Blaschke quotient
BI/B2 we have only to let F i. What
is a reasonable condition for
BI/B 2 UBC(AI ? An attempt is proposed in [25 in
terms of uniformly separated sequences.
for
f
(VI)
BC(A).
For
W
we set
if
w=O.
4I
UNSOLVED PROBLEMS ON MEMOMORPHIC FUNCTIONS
The length of the arc
SAN SZ(w)
is called a Carleson measure if
is known that
f 6
BMOA(A)
,lwl).
2w(l
(Z(w))/(w)
if and only if
measure (in the differential form).
df(z)
(w)
is
sup
<
A measure
where
w
I proved [2hi that
if
f 6
df
f#2
(VII)
Let
lwl
f 6 UBC(A) ?
UBC0(R
VMOA(R)
It is not difficult to prove that
Furthermore, we see
UBC0(A)
fP
f 6
UBC0(R)
f
A
R
such that
K c R
such
in
T(l,fw)
obtained on replacing
UBC0(R)
fop 6 VMOA(A)
and
because the projection of a closed disk in
uZ{ ?
17qe
difficulty lies in the
be the family of meromorphic functions
in the above.
BMOA(R).
a
3R, namely, given e > 0 we may find a compact
for w E R
K
this can be read in case R
A,
the holomorphic analogue is
i
I note that
is not subharmonic in general.
T(R,w,f) 0 as w
that T(R,w,f) < e
as
then
Izl2)f#(z)2dxdy
(i
CrZeson mesure.
fact that
It
is a Carleson
UBC(A),
is a Carleson measure, and gave a partial answer for the converse.
is
A.
ranges over
Izl2)If’(z)12dxdy
1
c
f 6
UBC(R)
and
0
T
T*
by
VMOA(R)
VMOA(R),
is compact in
R.
Are the
We remark that the following are known:
ffAf#(z)2dxdy
<
f
UBC0(A);
ffAIf’(z)12dxdy
and
<
f
VMOA(A);
see [19] and [2]. Are the eztenson8 te for R ? Metzger, in a communication,
informed me some partial answers on the VMOA part. I do not know further information.
Finally we note that the hyperbolic analogue of
[26],
and a forthcoming paper
log(l
R, and let
If12).
hlf’(z)12/(l If(z)12)2dxdy.
f#
[27].
Let
f
Then
If’I/(l
2(f,0)
+ log h.
(i),
If12).
is the non-Euclidean hyperbolic distance of
with
is possible; see
Ill
f
and others, are true on replacing
tanh-llfl
(f,0)
(f,0) is subharmonic
We note that
and
[21],
< i, in
Az)dxdy
is subharmonic with
The analogue of
by the hyperbolic derivative
UBC(R)
be holomorphic and bounded,
O, and
It is a future task to find some problems on this
family.
The present article depends on the lecture on February
,
1985, at
Mie University,
Tsu, Mie, Japan, where slightly abstract treatment beginning with the F. Riesz decomposition of subharmonic function was discussed.
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