Publicacions Ma em`a iques, Vol 39 (1995), 335–347.
CONTINUITY AND
CONVERGENCE PROPERTIES
OF EXTREMAL-INTERPOLATING DISKS
Pascal J. Thomas
Abs ac
Le abe a sequence o poin s in he uni ball o Cn. E ic Ama
and he au ho ha e in oduced he nonnega i e quan i y ρ(a)=
in αin kj:j=kdG(αj,α
k), whe e dGis he Gleason dis ance in
he uni disk and he fi s infimum is aken o e all sequences αin
he uni disk which map o aby a map om he disk o he ball.
The alue o ρ(a) is ela ed o whe he ais an in e pola ing
sequence wi h espec o analy ic disks passing h ough i , and i
ais an in e pola ing sequence in he ball, hen ρ(a)>0.
In his wo k, we show ha ρ(a) can be ob ained as he limi
o he same quan i y o he unca ed fini e sequences, and ha
ρ(a) depends con inuously on awhen ais fini e. Fu he mo e,
we desc ibe some o he beha io o he minimizing sequences o
maps in ol ed in he ex emal p oblem used o define ρ.
0. In oduc ion.
This a icle is being w i en o p o ide some addi ional p ope ies
o he no ion o ex emal disks in oduced in he join pape [A-T]. I
would no ha e a isen wi hou he s imula ing discussions I had wi h
E ic Ama , o which I wish o hank him. Some o he genesis o his
wo k ook place while I was enjoying he hospi ali y o he Uni e si y o
Cali o nia a Los Angeles and he Uni e si y o Wisconsin-Madison, and
my hanks go o hem as well.
Fi s we ecap and simpli y a ew no a ions om [A-T].
As usual, Ddeno es he uni disk in he complex plane, and Bn he
uni ball in Cn:Bn:= {z∈Cns z·¯z=n
1|zj|2<1}.
Le a={ak,k∈Z∗
+}⊂Bn,α={αk,k∈Z∗
+}⊂D, possibly fini e
sequences. Gi en φa holomo phic map om D o Bn, we shall employ he
abb e ia ed no a ion φ(α)=a o mean ha o all k∈Z∗
+,φ(αk)=ak.
336 P. J. Thomas
When αis such ha a φas abo e does exis , we say ha αmaps
o a, and w i e α→ a o sho . I means ha he map φdesc ibes
an analy ic disk passing h ough he poin s o he sequence awi hou
“slopping o e ” he bounda y o he ball.
Defini ion [A-T]. Fo a⊂Bn,n≥1, k∈Z∗
+,
ρk(a) := in {δk(α):α→ a};
and
ρ(a) := in {δ(α):α→ a},
whe e
δk(α):= 
j:j=k
dD
G(αk,α
j),
δ(α) := in
kδk(α),
and, o zand win a domain Ω, dΩ
Gdeno es he Gleason dis ance:
dΩ
G(z,w) := sup{| (w)|: ∈H∞(Ω) s (z) = 0 and  ∞≤1}.
No e ha , o any nonnega i e in ege n,
1−dBn
G(z,w)2=(1 −|z|2)(1 −|w|2)
|1−z·¯w|2,
in pa icula
dD
G(z,w)= |z−w|
|1−z¯w|.
(see [Ga], [Ru]).
Defini ion [A-T]. We say ha φis an ex emal-in e pola ing disk iff
he e exis s an α⊂Dsuch ha φ(α)=aand δ(α)=ρ(a).
Such disks a e hose o which he p e-image sequence αis, in a sense,
as close oge he as i can wi hin he uni disk ( he Schwa z Lemma is
p e en ing i s poin s om being a bi a ily close o each o he ).
Resul s abou exis ence o ex emal-in e pola ing disks and some o
hei fi s p ope ies we e gi en in [A-T] (whe e hey we e simply called
“ex emal disks”), as well as mo i a ions o he s udy o his no ion.
Essen ially, ρmeasu es whe he ais an in e pola ing sequence wi h e-
spec o holomo phic unc ions bounded on he analy ic disks passing
h ough i (as opposed as being in e pola ing wi h espec o unc ions
bounded on he whole ball).
Ex emal-in e pola ing disks 337
1. Con e gence along fini e subsequences
and semi-con inui y.
Theo em 1.
Le a={ak,k∈Z∗
+}⊂Bn, hen
ρ(a) = lim
N→∞ ρ({ak,1≤k≤N}) = in
N∈Z∗
+
ρ({ak,1≤k≤N}).
Co olla y 1.
The unc ion a→ ρ(a)is uppe semi-con inuous wi h espec o he
opology gi en by he dis ance d(a, b) := supkdG(ak,b
k).
P oo o Co olla y 1:
Lemma 2 in [A-T] p o ed ha he unc ion ρis u.s.c. o e he se o
fini e sequences wi h a gi en numbe o poin s, hus a→ ρ({ak,1≤k≤
N}) is an u.s.c. unc ion, and he g ea es lowe bound o a amily o
u.s.c. unc ions is also u.s.c.
Co olla y 2.
Any sequence a⊂Bn e i ying ρ(a)>0is sepa a ed, i.e. he e exis s
δ>0such ha o any j=k,dG(aj,a
k)≥δ.
This Co olla y is in e es ing in he con ex o in e pola ing sequences
o bounded holomo phic unc ions (see [C], [Ga] o defini ions). Sep-
a a edness is an easy necessa y condi ion o a sequence o poin s o be
an in e pola ing sequence. One also easily sees ha ρ(a)>0 is ano he
necessa y condi ion o a o be an in e pola ing sequence [A-T]. Thus
we see ha his new necessa y condi ion implies a be e -known one.
P oo o Co olla y 2:
By enumbe ing he sequence, ake j=1,k= 2. I is easy o see
(c . [A-T]) ha ρ({a1,a
2})=dG(a1,a
2). Apply Theo em 1 o N=2:
dG(a1,a
2)≥ρ(a)>0.
P oo o Theo em 1:
Gi en any >0, le α⊂Dbe such ha δ(α)<ρ(a)+. Then he e
exis s kand Nsuch ha

j:j=k, 1≤j≤N
dG(αk,α
j)<ρ(a)+2.
Thus by defini ion ρ({ak,1≤k≤N})≤ρ(a)+2, and we ha e
ρ(a)≥in
N∈Z∗
+
ρ({ak,1≤k≤N}).
The es o he Theo em will ollow om he
338 P. J. Thomas
P oposi ion 3.
Suppose ais a sequence in Bnand φa holomo phic map om D o
Bn, and, o 1≤k≤N,αk∈Dsuch ha φ(αk)=ak. Then, gi en any
>0, he e exis s a holomo phic map ψ om D o Bnand a sequence
β⊂Dsuch ha
ψ(β)=aand δ(β)≤δ({αk,1≤k≤N})+.
In pa icula , o any N<M,ρ({ak,1≤k≤N})≥ρ({ak,1≤k≤
M}).
The p oo o P oposi ion 3 will be gi en in Sec ion 4.
End o P oo o Theo em 1:
The las clause o he p oposi ion shows ha he limi in he heo em
exis s and equals he infimum. Fu he mo e, by choosing a sequence α
such ha δ(α)≤ρ({ak,1≤k≤N})+, we ha e
ρ(a)≤δ(β)≤ρ({ak,1≤k≤N})+2,
which p o es he equi ed inequali y.
2. Con e gence o mappings.
The fi s sec ion lends some alida ion o ou app oach (in [A-T]) o
s udying he beha io o ρ(a) mos ly when ais a fini e sequence. In ha
case, since ρis defined as an infimum, i is legi ima e o wonde wha
happens when we ake a sequence o mappings φpand sequences αp
such ha , o any p,φp(αp)=a, and limp→∞ δ(αp)=ρ(a). By Mon el’s
heo em, a subsequence o {φp}pwill con e ge uni o mly on compac
subse s o D, bu no such con e gence is gua an eed o he poin s in he
sequences αp, so he ques ion a ises o wha he ela ionship be ween he
limi o he mappings and he o iginal sequence.
No maliza ions.
Since we a e dealing wi h a fini e sequence a, we may always assume
(a e e-numbe ing) ha ρ(a)=ρ1(a), and shall do so o he emainde
o his sec ion.
Likewise, when ha e a sequence αwhich maps o a, by applying an
au omo phism o he disk, we educe ou sel es o he case whe e α1=0.
We in oduce a class o special holomo phic mappings om he disk
o he ball:
Ex emal-in e pola ing disks 339
Defini ion.
We say ha φ, a holomo phic map om D o Bn, is a ball- alued
Blaschke p oduc o deg ee Niff o ζ∈D,
φ(ζ)=P1(ζ)
Q(ζ),... ,Pn(ζ)
Q(ζ),
whe e P1,... ,P
nand Qa e polynomials wi h max(deg Pj,1≤j≤
n, deg Q)=N,Qhas no ze os in Dand o any ζsuch ha |ζ|=1,
1=|φ(ζ)|2=
n

1
|Pj(ζ)|2
|Q(ζ)|2.
The ollowing was essen ially p o ed in [A-T, Theo em 1]:
Theo em.
I a={ak,1≤k≤N}⊂Bn,α⊂D, wi h φa holomo phic map
om D o Bn, such ha φ(α)=a, and δ1(α)=ρ1(a)(in pa icula
i φgi es an ex emal-in e pola ing disk o a) hen φis a ball- alued
Blaschke p oduc o deg ee no g ea e han N−1, uniquely de e mined
by α.
Only he p ecise o m o he a ional map was no explici ly gi en in
[A-T], bu i is easy o ob ain by ollowing he induc ion pe o med he e,
obse ing ha a each s ep we only pe o m composi ion by M¨obius
au omo phisms o he disc o ball, and mul iplica ion by ζ.
We can now s a e:
Theo em 2.
Le a={ak,1≤k≤N}⊂Bn, and {φp}pa sequence o mappings
om he disk o he ball such ha o each p,φp(αp
k)=ak,1≤k≤N,
whe e αp
k∈D,αp
1=0, and
lim
p→∞ δ1(αp) = lim
p→∞
N

k=2
|αp
k|=ρ(a).
Then he e exis subsequences, deno ed again by {φp}pand {αp}p,a
ball- alued Blaschke p oduc φ, and a sequence α={αk,1≤k≤N}⊂
Dsuch ha
(i) limp→∞ φp=φ, uni o mly on compac se s o D,
(ii) limp→∞ αp
k=αk∈D, and
(iii) φ(αk)=akiff |αk|<1.

340 P. J. Thomas
Le S:= {k∈{1,... ,N}s |αk|<1}. Then ρ({ak,k∈S})=ρ(a)
and φis o deg ee no g ea e han #S−1.
Rema ks.
This heo em says ha we do ha e con e gence owa ds some
ex emal-in e pola ing disk, bu passing only h ough a subsequence o
a.
In he case whe e N= 3 and ai sel does no admi an ex emal-
in e pola ing disk, we see ha he heo em implies ha a subsequence
o he minimizing sequence o mappings has o con e ge o an affine disk
h ough wo o he poin s o he o iginal sequence.
P oo o Theo em 2:
By Mon el’s Theo em and compac ness o D, i is easy o ex ac
subsequences ha ing p ope ies (i) and (ii). The “i ” pa o (iii) ollows
by equicon inui y o he con e ging subsequence o maps.
Now
ρ(a)=ρ1(a) = lim
p→∞ δ1(αp) = lim
p→∞ 

k≥2,k∈S
|αp
k|
k/∈S
|αp
k|

=δ1({αk,k∈S})≥ρ1({ak,k∈S})≥ρ({ak,k∈S}),
which i sel is no less han ρ(a) by P oposi ion 3, so we ac ually ha e
equali y h oughou . Since φ({αk,k ∈S})={ak,k ∈S}, and
δ1({αk,k∈S})=ρ1({ak,k∈S}), φis a ball- alued Blaschke p od-
uc o deg ee no g ea e han #S−1, so ha o k/∈S,|φ(αk)|=1,so
ha φ(αk)=ak, which finally p o es he “only i ” pa o (iii).
I would be nice o be able o desc ibe he subsequence h ough which
an ex emal-in e pola ing disk passes, {ak,k∈S}, in e ms o he se-
quence a. Obse e ha {ak,k∈S}is a sequence wi h he same ρas
he o iginal sequence. I such a subse Sis gi en, we ha e he:
Theo em 3.
Le S⊂{1,... ,N}, such ha 1∈S, be minimal o he p ope y ha
ρ1({ak,k∈S})=ρ1(a)=ρ(a).
Then he e exis s a sequence o fini e sequences in he disk, {αp}pand
a sequence o mappings om he disk o he ball {φp}psuch ha
(i) φp(αp)=a,
(ii) ρ1(a) = limp→∞ δ1(αp),
Ex emal-in e pola ing disks 341
(iii) limp→∞ αp
k=αk∈Dand S={k∈{1,... ,N}s |αk|<1},
(i ) limp→∞ φp=φ, a ball- alued Blaschke p oduc o deg ee no g ea-
e han #S−1, and
( ) φ({αk,k∈S})={ak,k∈S},δ1({αk,k∈S})=ρ1({ak,k∈
S})=ρ1(a)=ρ(a).
P oo o Theo em 3:
The sufficien condi ion o ρ o be a ained gi en in [A-T, Lemma 3]
co e ed he special case whe e he only such se Sis he whole
{1,... ,N}. The p esen p oo will use he same idea.
Fo a gi en p, pick fi s , using he defini ion o ρ1, a sequence {βp
k,k∈
S}so ha
δ1({βp
k,k∈S})≤ρ1({ak,k∈S})+1
p;
hen modi y and comple e his sequence acco ding o P oposi ion 3 o
ge αp={αp
k,1≤k≤N}so ha
ρ1(a)≤δ1({αp})≤ρ1({ak,k∈S})+2
p=ρ1(a)+2
p.
This o ces limp→∞ |αp
k|= 1 o k/∈S.
On he o he hand, o k∈S, by he minimali y p ope y o S,
ρ1({aj,j∈S {k}})>ρ(a). Then
|αp
k|=j∈S|αp
j|
j∈S, j=k|αp
j|≤j∈S|αp
j|
ρ1({aj,j∈S {k}})≤γk<1,
o pla ge enough. Taking subsequences as be o e, we ge he con e -
gence o he poin s αp
k o limi s wi hin he open disk when k∈S, and o
he mappings o a mapping φ. We ob ain ( ) as in he p e ious p oo ,
and he esul ing ex emali y o ces o φ o be a ball- alued Blaschke
p oduc .
Ques ions.
Is he se So Theo em 2 always “minimal”, i.e. o he ype gi en
in Theo em 3? I is clea ha i con ains a “minimal” se S, and ha
any “minimal” se ha con ains i mus be equal o i . Also, he se S
is included in some se S maximal o he p ope y ha an ex emal-
in e pola ing disk does pass h ough {ak,k∈S}; mus S,Sand S
coincide?
Mo e modes ly, a e he e examples o sequences whe e se e al diffe en
minimal se s Scan be ound?
342 P. J. Thomas
3. Con inui y in he fini e case.
Despi e hei limi a ions, he ideas o he p e ious sec ion enable us
o p o e he ollowing con inui y esul :
Theo em 4.
Fo e e y N∈Z∗
+, he unc ion {ak,1≤k≤N}→ρ({ak,1≤k≤
N})is con inuous om (Bn)N o R+.
P oo o Theo em 4:
We shall adop he same no malisa ions as hose in he p e ious sec-
ion.
Since ρ(a) = min1≤k≤Nρk(a), i will be enough o p o e ha ρ1is
a con inuous unc ion. By Lemma 2 in [A-T], we al eady know i o
be uppe semi-con inuous. We shall p oceed by induc ion on N. Since
ρ({a1,a
2})=dG(a1,a
2), he case N= 2 is clea .
Now suppose he p ope y ue o all sequences asuch ha #a≤
N−1. Gi en a∈(Bn)N, assume, o ge a con adic ion, ha he e is
a sequence o sequences ap⊂(Bn)Nsuch ha limp→∞ ρ1(ap)<ρ
1(a).
Then o any p ope subse S⊂{1,... ,N}, by he induc ion hypo hesis,
lim
p→∞ ρ1({ap
k,k∈S})=ρ1({ak,k∈S})≥ρ1(a),
by P oposi ion 3. The e o e o pla ge enough,
ρ1(ap)<min{ρ1({ap
k,k∈S}):S⊂{1,... ,N},#S≤N−1},
so an applica ion o [A-T, Lemma 3] shows ha o each such p he e
exis s a sequence αpin he disk such ha δ1(αp)=ρ1(ap) and a mapping
φpsuch ha φp(αp)=ap.
Now, o small enough, pla ge enough, and any k∈{2,... ,N}, we
ha e
|αp
k|=δ1(αp)
δ1({αp
j,j=k})≤ρ1(ap)
ρ1({ap
j,j=k})≤limp→∞ ρ1(ap)
ρ1(a)+<1.
This implies ha all he poin s αp
k emain in a ela i ely compac disk
wi hin he uni disk, he e o e by ex ac ing a subsequence we may as-
sume ha o each k, limp→∞ αp
k=αk∈D, and limp→∞ φp=φ, wi h
uni o m con e gence on compac subse s o he uni disk. This implies
ha φ(α)=a, bu since
δ1(α) = lim
p→∞ δ1(αp) = lim
p→∞ ρ1(ap)<ρ
1(a),
we ge a con adic ion wi h he defini ion o ρ1.
Ex emal-in e pola ing disks 343
4. P oo o P oposi ion 3.
As usual, we may assume wi hou loss o gene ali y ha α1= 0 and
δ(α)=δ1(α), whe e α={αj}1≤j≤N. Also, enumbe aso ha |aj+1|≥
|aj| o any j≥N+1.
We w i e Mβ o he cons an o in e pola ion o a sequence β, i.e.
Mβ:= in {M>0s ∀a⊂Bn, he e is φ:D→Bn(0,M)s φ(β)=a}.
Pick 0<1 such ha α⊂D(0,
0) and
N

j=2 
αj
0
<δ
1(α)+,
and 1>0 small enough so ha 1≤1−|aN+1|
Mα+2 and φ(D(0,
0)) ⊂
Bn(0,1−(Mα+2)1).
Lemma 4. The e exis s a holomo phic unc ion E∈A(D)(i.e. con-
inuous up o he bounda y) so ha
(i) |E(ζ)|≤1, o any ζ∈D;
(ii) |E(ζ)−1|≤1, o any ζ∈D(0,
0);
(iii) E(1) = 0, bu E(ζ)=0 o any ζ∈D.
P oo :
Fo a∈(−1,+1) le φa(ζ)= ζ−a
1−aζ , and E(ζ)=(1−φa(ζ))/2. Then
E(1) = 0 and E∞≤1.
Bu he disk φa(D(0,
0)) admi s he line segmen [− 0−a
1+a 0; 0−a
1−a 0] o
i s diame e and lima→1− 0−a
1−a 0=−1, so ha o aclose enough o 1,
φa(D(0,
0)) ⊂D(−1,21), which yields (ii).
Lemma 5. The e exis s F∈C
0([ 0,1),Bn)and {αj}j≥N+1 a s ic ly
inc easing sequence such ha
(i) F(αj)=aj o j≥N+1;
(ii) M{αj}j≥1≤M{αj}1≤j≤N+1;
(iii) 1 −|F(x)|≥1(Mα+2)|E(x)|, o any x∈[ 0,1);
(i ) F( 0)=φ( 0).
P oo : Once he αja e gi en, we will define Fby linea in e pola ion:
F(θ 0+(1−θ)αN+1):=θφ( 0)+(1−θ)aN+1
and F(θαj+(1−θ)αj+1):=θaj+(1−θ)aj+1,