777
Internat. J. Math. & Math. Sci.
Vol. i0 No. 4 (1987) 777-786
THE SHARPNESS OF SOME CLUSTER SET RESULTS
D.C. RUNG
Department of Mathematics
The Pennsylvania State University
University Park, PA 16802
and
S.A. OBAID
Department of Mathematics and Computer Science
San Jose State University
San Jose, CA 95192
(Received July 8, 1986 and in revised form October 6, 1986)
Abstract. We show that a recent cluster set theorem of Rung is sharp in a
certain sense. This is accomplished through the construction of an
interpolating sequence whose limit set Is c]osed, totally disconnected and
porous. The results also generalize some of Dolzenko’s cluster set theorems.
Key words.
Cluster sets, interpolating sequences, porous sets. 1980
Primary 30D40, Secondary 30E05.
Mathematics Subject Classification.
1.
INTERPOLATING SEQUENCES.
disconnected set
P
We begin by considering a closed totally
on the boundary
3a
a
of the unit disc
in the complex
=3a-P
is the union of countably many disjoint arcs. Our first
plane. Thus
objective is to construct interpolating sequences on certain curves in the
In a speclal case
unit dlsc a whose llmlt points are all the points of P.
Dolzenko [1] used this construction apparently not realizing that he was
dealing wlth Interpolating sequences. We wish to define an approach to a
point z
a inside a reasonably nice subdomain of a. Let h(t) be a
real-valued function defined for -1 < t < 1. We require that
(1)
(ii)
(iii)
(iv)
(v)
h
be continuous.
h(t)=h(-t).
h(O)=O, h(1)=1, h(t 1) _<
h(t2),
0
_< t _< t 2 _<
1.
(l.l)
h(t) _< t.
h"(t) > O, t
O.
This function h
is said to be a convex approach function.
ie
as follows (See
determines a convex boundary domain (e,h) at z=e
Such an
h
Fig. I).
}(e,h)
(re it :0 _< r _< 1-h(t-e):
It-el _<
I}
For example h(t)=t defines the usual nontangential approach; h(t)=t
defines the horocyc]Ic approach and so on. The boundary of the domain
(1.2)
D.C. RUNG and S.A. OBAID
778
E
consists of two h-curves
Z+
and
Z_
which meet at
r=e iO
and are defined
by
Z+(t,O,h}
[1-h(t-)]e it
Z_(t,O,hl
it
0 < t- <
(1.3)
O<O-t <
and the second
The first curve in (1.3) Is called the right h-curve at
curve in (1.3} is called the left h-curve at
{See Fig. 1). Clearly these
curves are rotations of the corresponding curves at 1.
We construct an Interpolating sequence which has P as its limit set.
Recall that a sequence {Zn} is an interpolating sequence if, for each
bounded sequence of complex numbers
such that
f{Zn}=n
for every
n.
{Wn },
there exists a function
f
We shall use the characterization of
in
SHARPNESS OF SOME CLUSTER SET RESULTS
Garnett [2] for interpolating sequences.
set
a,b E A,
a-b
x(a,b)
the pseudohyp,rbolic distance o.
For
779
A
Tle equenc
{zn}
is interpolating if
and only if
> O"
nm ((Zn,Z m)
(i)
_li__m
(1.4)
There exists a constant
{re iO’l--d _< r
Io-o01
1,
A
such that for any domain
_< d},
Z (1-1Znl) _< Ad.
zED
D
Such a domain
is shown in Fig. 1.
=6-P
Recall that
we denote by (r n,r*In
is the countable unlon of disjoint open arcs whlch
oriented In the usual counterclockwise sense. e
denote the length of this arc by
no construct a sequence
{Zk}
l(r n,r*}[n
Flx such an arc
on the right h-curve
Z+
e
(r n,r*)n
ending at
Intersects each circle
to define
{Zk}
tz;=r,
0 < r < 1,
(e
n
could equally well define these sequences on the left h-burve or both.
simplicity we put the only on the right h-curves}. Because any Z+
For
in at most one point it is enough
by specifying
lZk[. Thus let
htllrn,r)l/16) 1-g
Zol
(1.5)
Consequently, if
zk
zk
{Zk]
e
tO k
lies "over" the interval
10k
k 2 I.
rn=e
and
iO n
then from (1.3} we see that
and is of the form
{rn,V}
[l-h(ek-en)]e
zk
K/2k
k
11-I z01 )/2
I-I Zkl
0 <
ek-e n
< r/8,
Zk
> 1/2.
We note that
X
and thus
let
n
(Zk,Zk+
X(Zn,Z m)
_> X (I Zk[ ,I Zk+ll
> 1/3,
nm.
To show (1.4)(ii), for a given domain
be the least index such that
z n e D.
:Z
_< 2
(1-I Zkl
hence the sequence
(z k}
is interpolating.
e
k=n
k=n
Note that
1-I Znl )/ak
Zkl
Z: (1-{
ZkD
D
2(I-I
Znl
< 2d,
want to construct a larger interpolating sequence by taking the union
of all sequences at the points
Vn"
For each arc
(Vn,Vn*)
in
construct
D.C. RUNG and S.A. OBAID
780
a sequent;e on
Z_
it]
We claim tlat th-
exactly the same manner as above
(countal,le) nnJol, S, oI" these sequences is still an interpolating sequence
(Note that any rearrangement of an interpolating sequence is still
interpolating). e first sho that {1.4} is satisfied, beginning ith
(l.4)(i}.
then
a,b E S,
We will show tlt if
(1.7)
i(a,b) > 4-The proof of the above inequality can be done in two steps. First, if a
and b lie on the sane h-curve then we have shown that i(a,b) > 1/3.
Second, if a,b lie on two different h-curves then we use the following
inequality (See [3], p.474) which is valid for any a,b e a
a,b
x(a,b}
la-b|
(1.8)
The right inequality iplies that if
(1.9)
then
x(a,b) > 4--
10)
zmn
be an element of the sequence on
Thus we show that (1.9) is valid.
the h-curve ending at
h-curve ending at
e
ay assue
arg
rj
Let
and let
n
zk
be an eleaent of the sequence on the
as shown in Fig. 1.
n > arg rj.
z
n
Set
arg
zmn=e
m
and
arg
Vn=en
Then it is clear that
zk
>
]z n] sln(O-e n}
(1.11)
>
(1.12)
and
sln(em-en)
Thus
Jzmn-z-rjk
Jzmn
(21r)(em-en)
sin(Om-O n)
1-1 zmnl
where we used (1.1), (1.6), (1.11) and (1.12). This proves that the sequence
S satisfies (1.4)(1).
For (1.4)(ii), let D be the domain specified there. e clat that
r.
aesnD
_
1-1 al _<
(1.13)
5d.
The points of S that lie in D belong to curves that end at the
boundary of D except for at ost to curves which ight end outside D
and B as
(See Fig. 1). Partition the points of S
D into to sets
follows-
zn
A
if and only if
Thus fro (1.1} and (1.5} e have
{r,r)
D
,
otherwise put
Zn
m
B.
SHARPNESS OF SOME CLUSTER SET RESULTS
If
(znm
_C B
then
1).
Let
zn
dent, re the first term
m
2
5[
i=l
has at most two values, say
zlE
2
S
Sl
P C BA.
then
nil 0
(-IZn.l)
4d.
(]-Ial) < 5d,
D
is an interpolating sequence.
POROSITY AND RIGHT h-ANGLES.
on the set
D
Thus
a(
2.
(See Fig.
B
i=1
which implies that
2
(1-1z i
Z
t=1
2
where we used (1.5).
and
Zl
of each sequence lying in
i
(l-iz n ;1 <
E
781
In this section we add another restriction
We assume that
P
The notion of porosity was
is porous.
introduced in 1967 by Dolzenko [1] and later used by Run [4] and Yoshida [5]
to generalize some of the cluster theory results. We note that in 1976
Zajicek [6] generalized the definition of porosity and proved a variety of
interesting properties of porous sets.
Let
P C BA.
For each
largest subarc of the arc
no such arc exists define
porous at e iO if
e ioe a,
let
be the length of the
(e,e,P)
(e i{o-e), e i(o+e)) which does not meet
W(O,z,P)=O. According to Dolzenko [1],
P. If
P is
(2.1)
n(e,P) > o.
A set
P C 0A
90
is porous if it is porous at each
p
P"
P
is o-porous if it
is the finite or countable union of porous sets. A porous set is nowhere
dense and thus a o-porous set is of the first Balre category.
We now define a right h-angle in A at =e i e A. For any positive
c; Then h c is also a convex
x
constant c, set
hC(x)=h[],-c
approach function.
RA(e,a,b,h)
For any constants
(rei#:l-ha(-O)
0 < a
< b,
< r <
1-hb(#-e),
define
0 <
-e
< a).
(2.2)
The boundary curve of RA(0,a,b,h) defined by the left inequality will
be called the lower boundary curve of the right h-angle domain and the other
boundary curve is called the upper boundary curve (See Fig. 1). The left
h-angle domain at r, LA(O,a,b,h) is defined by replacing #-O by O-# in
(2.2) with upper and lower boundary curves defined by the same inequality. If
h(t)=t then this represents a typical Stolz angle domain.
a and if E
# then the cluster set of a function f
If E
along E will be denoted by C(f,E). Our final objective is to investigate
the sharpness of the following theorem of Rung [4].
D.C. RUNG and S.A. OBAID
782
.T_ h. e_ 9_ !" _m
r,_
R,
_e ,_c_e p_ __f_o_E .4.2-p.ofous__s__t_ tot_’_a!.y_9_hoice of
C{f,RA(O,a,b,h))
0 < a
C(f,LA(O,a,b,h))
We start hy introducing two well-known results
p.28]-282) and Kerr -l, aWSOlt [8].
Lemma
n,
If there is.a_n
(Garnett)
(1--tan i2
Z
and if
anqD
in the exteded
tak_i(!g._.v_ol.q!
b_e_ deflate d .i._n
f
_< Ad,
,/
> 0
D
c > O,
C{f,ll(o.hc))
d,e to Garnett
such that
where
and
b
(See Koosis
>
(an,a m
for-
.9,
j_tLy_ejL.b_Y_i_l=4_)_ and
A
is a
constant then
y[
inf
X(an,a m) >_
n=l
de.e!Ld_d__on_!!y__q_n
where
Lemma 2 (Kerr-Lawson).
aFe
be the Blaschke function whose zeros
B(z)
{an}. qp_pose
t.he. sequenc
given by
A.
and
Let
[
inf
n=l
6,
that
> O.
X(an,a m)
n
Then given a number
o_j
an___d
6
60
60
> 0
{z.lB(z)] < }
such that the set
of disjoint pseudohype[bolic discs
wit_____h 1-center
a
n
and
B{z)
(i)
(ii)
(iii)
with the following properties"
is defined and analytic in
B(z)
IB(z)l Z
choice of
B(z)
with
z
e
Rh(,,,h)"
there exists either a
RA(fl,a,b,R)
or a
B(z).
(Th____e
which contain infinitely...many zeros of
a
and
b
vary with
e
ifl.)
be the infinite Blaschke product
n:l
{an}
such that for each
i,e,
a_s
e ifl e P,
For each
LA(fl,a,b,h)
Let
> 0
There exists an
li
aa
Then there exists a Blaschke
be a convex approach function.
h
function
where
is contained in the unioa
be a closed totally disconnected porous subset of
P
Le___t
Theorem 1.
Proof.
N{an,50)
which depends only
0"
I-radius
and let
> 0
there exists a number
an l-anZj’
Z qA,
is any arrangement of the interpolating sequence
S
defined in
783
SHARPNESS OF SOME CLUSTER SET RESULTS
P,
relative to the set
section
If) follows.
r=e i E a-P
If
We now prove (li).
is a BJas’hke product then
B(z)
Since
E P
If
then (li) is obvious.
i is an isolate(|
then we first show that (ii) holds for the case when r=e
of an arc
initla]
endpoint
the
r
is
circumstance
this
point of P. In
(r,r*) contained in a-P. Using the results of Rung [4, p.204] and
dstance
Satyanaraya and Weiss [9, p.65] we find that the pseudohyperbo]]c
is at least
(2.2)
domain
h-angle
rlght
the
of
boundarles
the
between
near
dJsta,ce equals to I/4.
above
the
b=3/2,
a=I/2,
Choosing
]b-a/2b+a[.
Thus for large
Lemmas
IB(z)l >_
n, each N(z n
and 2 imply that there exists a positive number
RA(,1/2,1/2,h).
r
nlN(Zn
z $
for
Thus we have
I/8).
P
Consequently if
is a limit point of
for z e
)-P.
then it Is easy to see that
P
so
,h)
RA(/,
still does not meet this
U N(an,I/8)
,
an__d lying over the corresponding interval of
isolated points of
r
IB(z)I >
lim
z+--7
lle on right h-curves ending at
(a n
Recall that the points of
such that
li___m {B(z){ _>
,
when
[I=I
If v=e i is an isolated point then clearly
Finally we prove (iii).
I Is a
1 3
contains infinitely many zeros of B(z). Suppose =e
RA(,,,h)
limit point of Isolated points
integer
such that the right h-angle
2
m
We shall show that there exists an
e P.
Vn
_
be the union of the arcs
Let
S.
infinitely many points of
U (’n,’rn*)
n=l
and without loss of generality we take 8=0.
i
i8
and subarcs (e k
of arcs (e k
eik
eiOk
im
k-o
k-ak
k
there is a sequence
By (2.1)
Ba-P
with
(2.3)
> O.
We consider two cases according as to whether the subsequence
approach 1 from above or below.
contains
RA(,l,m,h)
domain
(e
l= k
elk
If both, then select a subsequence
ta
approaching from one side of 1. The case of (e k
from
approaching
below is slightly more complicated and so we prove this case. Ne suppose to
the contrary that none of the left h angle domains LA(0,1,m,h), m=2,3,4
contain infinitely many points of S. For each m, there must be an infinite
i
ifl
subsequence of the arcs {e k ,e k
such that the corresponding Blaschke
elk
{zk)}
sequence
boundary curve
on
Z+(t,k,h)
Z_(t,O,h)
of
has its first term,
Z+(t,ak,h)
Z_(t,flk,h)
and
(e
between
Z_(t,k,h
e
Because arg z k)
and
a
ak
1.
a
(See
Then note that
meet at a point on the radius to the midpoint of
.....
the arc
between the upper
LA(O,l,m,h) (defined in (2.2)} and
To see this we first may assume that
Fig. 1}.
zk),
then z
lles
18 <
2
and certainly between Z_(t,0,h), the lower
D.C. RUNG and S.A. OBAID
784
boundary curve of
upper boundary curve of
km ik
(k m)
z0
(k m)
]z 0
Thus the point
win=
(k m)
z0
radius as
Wml e
I.ml
<
Ok
(e
m
be
choose
ink s
has
iqkm,eiflk m
Z_(t,0,hm)
on
which lies on the same
satisfies
lz 0
> I-
lWm
h(#m/m)
Now
For each
woul(|
According to (1.5) the first
associated with e
h(I
i@m
.
0, m
km
must lie between the
and 5A else there
inside LA(O,l,m,h).
S
infinitely many points of
io
m
satisfying
e
(e
Blaschke sequence term
Z_(t,O,hm),
LA(O,l,m,h),
Z(o k)
But then
and BZ.
LA(O,l,m,h),
and so the properties of
h
L--]--J
h -1 together with the above
two Inequalities imply that
#km-km (flkm-km
16
m
.
This contradicts (2.3).
and this last expression tends to 0 as m
is replaced by
When the intervals approach 1 from above the left angle at
and the proof proceeds along the same lines
the corresponding right angle at
as before. This completes the proof of the theorem.
appearing in property (ii) of Theorem
Note that the constant
Remark 1.
50=1/8,
depends only on the three constants
v=l/4,, A=5 (which appear in
(1.7), (1.10) and (1.13) respectively) so that any Blaschke
product whose
zeros are an interpolating sequence with these three
constants satisfies
[B(z)[ >
for a single constant
.
A slight modification of a result of Dolzenko [1,
p.8] gives the
following lemma.
Lemma 3.
A set
P_C
P:
U
n=l
where
Pn
P n’
are closed, .totally.dlsconnected and. porous sets can
be written
the form
P=OFk
k=l
where
Fk
the sets
ar-e dl.sJint., c!os.e.d,..and_porouff,
Fp
with respect
an___d
to
Fq
Moreover if lq then each of
l___ies., en_t_irely on an arc compl.ementary to the other
SHARPNESS OF SOME CLUSTER SET RESULTS
Theorem 2.
P=
t)
s(,t
._Gj_.v__e_J_,. c-[)._or___ous
which
t)a
785
be w,-itte as
,:an
lPn
where f’ n are closed and porous. ’!l(c__e_.ELELE a bounde(]
n
with the following_Zo_o_pertiesholomorphic functipji f(z) i__qn
(i)
(ii)
f(z)
a
j continuous from within
For each point
at each point of
there exist to
P,
such that the cluster- sets of
le tomal_.; .s
h
at
along these a_n_gles are
different.
Proof.
Pn"
P=
Consider the set
n
Lemma 3 implies that there exist mutually
disjoint, closed, and porous sets
Fk
P=
such that
lies entirely in an arc complementary to
for
Fj
F k.
Furthermore
Fk
Corresponding to each
kj.
set F k we construct a function Bk(Z) (Bk(Z)=B(z), P:F k) as we have done in
Theorem 1. Following Dolzenko we define f(z) as the infinite series
E a2k Bk(Z),
f(z)
(2.4)
z e a,
where
value
is the fixed constant appearing in Theorem 2.
(Recall that this
obtained in property (ii) of Theorem
is independent of the
particular Bk(Z}; see Remark
after Theorem 1. There is no loss of
generality in assuming
0
<
< 1/2.)
The series (2.4) is clearly
uniformly and absolutely convergent on compact subsets of a and so f(z) is
analytic and bounded on a.
It is also clear that f(z) is continuous from
within a at each point
C-P. This proves (i) of the theorem.
e
r=e ifl
point
now proceed to prove (it).
Fko.
v.
Then for
oreover
kk 0
the functions
we have for
z
k=ko+
Consider a fixed
ko-1
limit.
Set
E
a
k=l
Bk{Z)
2k Bk(Z)l
at
2kBk().
and let
are continuous at the
Bk(Z
a
z
/e now use Theorem 1, specifically
and the continuity of
Fko
_<
2(ko,
/(1- 2 ).
property (il) of
for
v
Thus for
k < k0
z e
Bko(Z),
property (2.5),
to show the following
RA(/i’,,,h)
we find
li____ If(z)
1i___ ]
zr
>
2 k0
2k 0
ko-1
Bko(Z)
zrli- - ]Bko(Z)[
z-r
k0
Z
k=l
zr
2k [Bk(Z)-Bk(r)]
Z
ko+l
2k
Bk(Z)
k0
k=OZ e2k[Bk(Z)-Bk(r)]]
(2.6)
786
D.C. RUNG and S.A. OBAID
’{1-
2.
Note that (2.6} implies that there are no points of
the disc
if(z} a]
r O.
/.
that there exists a sequence
angle domain at
C(f,RA{fl,-,1/2,h}}
within
Oil the other hand property (ii]} of Theorem
such that
(Zn}
of zeros of
zn
Bko(Z
contained 1 an
h
Consequently using (2.5) and the
continuity of the finite su we have
2k
Now
r
(
r0
[Bk(Zn)-Bk(Z)]
(2.7)
because
> 0.
rO-r
(2.8)
Expressions 12.7) and 12.8) imply states that the cluster sot of f(z) along
the h angle domain containing the zeros of
contains points in the
Bko(Z
circle
Iflzl-a] < r O.
This completes the proof of (ii).
not assumed to be porous then the zeros of
Bko(Z
If the sets
so the best that can be said is that the total cluster set of
different from the cluster set of
f
along
Pn
are
only accumulate at
f
at
and
is
Rh(fl,1,1,h }.
Thus we have generalized Dolzenko’s results and have shown the sharpness
of Theorem R when the exceptional a-porous set ls the union of closed porous
sets.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
DOLZENKO, E.P. Boundary Properties of Arbitrary Functions, Izv. Acad. Nauk SSSR
31, 3-14, (1967) (Russian);" English translation; Math. USSR-Izv. I, 1-12,
(1967).
GARNETT, J. Interpolating Sequences for Bounded Harmonic Functions, Indiana Univ.
Math. J. 21(1971), 187-192.
BELNA, C., OBAID, S.A. and RUNG, D.C. Geometric Conditions for Interpolation,
Proc. Amer. Math. Soc. 88(1983), 469-475.
RUNG, D.C. Meier Type Theorems for General Boundary Approach and -Porous
Exceptional Sets, Pacific J. Math. 76(1978), 201-213.
YOSHIDA, H. On Some Generalizations of Meter’s Theorems, Pacific J. Math. 46
(1973), 609-621.
ZAJICEK, L. Sets of -Poroslty and Sets of -Poroslty (q), Casopis Pest. Math.
I01(1976), 350-359.
p
KOOSIS, P. Introduction to H Spaces, London Math. Soc. Lecture Note Ser(2),
Press, Cambridge, 1980.
Univ.
Vol. 40, Cambridge
8.
KERR-LAWSON, A. Some Lemmas on Interpolating Blaschke Products and a Correction,
Canad. J. Math. 17(1969), 531-534.
9.
STAYAMARAYANA, U.V. and WEISS, M.L. The Geometry of Convex Curves Tending to
in the Unit Disc, Proc. Amer. Math. Soc. 41(1973), 159-166.