Two hype bolic Schwa z lemmas
by
L. BERNAL-GONZ´
ALEZ and M.C. CALDER´
ON–MORENO∗
Abs ac
In his pape , a sha p e sion o he Schwa z–Pick Lemma o
hype bolic de i a i es is p o ided o holomo phic sel mappings on
he uni disk wi h ixed mul iplici y o he ze o a he o igin, hence
ex ending a ecen esul due o Bea don. A p ope y o p ese -
ing hype bolic dis ances also s udied by Bea don is he e comple ely
cha ac e ized.1
1 In oduc ion and no a ion
The Schwa z Lemma and i s hype bolic e sion (= he Schwa z–Pick
Lemma) con inues a ac ing he a en ion o many ma hema icians. Ou
aim in his pape is o p o e wo sha p e sions o he o me esul s as-
suming ha he mul iplici y o he ze o a he o igin o he holomo phic
unc ion unde conside a ion is ixed. Ou esul s will ex end a ecen one
due o Bea don, see below.
Fi s o all, we need o ix some no a ion. The symbols N,C,R,D,
D(c, ) will deno e, as usual, he se o posi i e in ege s, he complex plane,
he eal line, he open uni disk and he euclidean closed disk {z∈C:
|z−c| ≤ }(c∈C, > 0), espec i ely. As o unc ion spaces, H(D) is he
class o all holomo phic unc ions on Dand Au (D) will s and o he g oup
∗The au ho s ha e been pa ially suppo ed by DGES G an PB96-1348 and he Jun a
de Andaluc´ıa.
12000 Ma hema ics Subjec Classi ica ion: P ima y 30F45. Seconda y 30C80.
Key wo ds and ph ases: Schwa z–Pick Lemma, highe o de hype bolic de i a i e, hy-
pe bolic dis ance, mul iplici y a a poin , m–au omo phism o he uni disk.
1
o con o mal au omo phisms o D. I ∈H(D) and a∈D hen µ( , a) will
ep esen he mul iplici y o he ze o a ao he unc ion (z)− (a). Fo
m∈Nwe in oduce he no a ions
Fm={ ∈H(D) : | |<1, (0) = 0(0) = . . . = (m−1)(0) = 0}
={ ∈H(D) : | |<1, (0) = 0, µ( , 0) ≥m}
and
zmAu (D) = {zm (z) : ∈Au (D)}.
Fo he sake o con enience, we ag ee ha z0Au (D) = Au (D). We say ha
a unc ion is an m- o a ion whene e he e exis s a cons an cwi h |c|= 1
such ha (z) = czm. The se o all m- o a ions will be ep esen ed by Rm.
I is clea ha zm−1Au (D)∩ Fm=Rm. Fo a∈Dwe deno e by ϕa he
special au omo phism ϕa(z) = a−z
1−az . In ac , Au (D) = {kϕa:|a|<1 =
|k|}. No e ha ϕ−1
a=ϕa. We de ine an m-au omo phism o Das a unc ion
o he o m =ψ◦R◦ϕwi h ϕ, ψ ∈Au (D) and R∈ RmWe deno e by
Au m(D) he se o m-au omo phisms o D. Ob iously, Au 1(D) = Au (D).
I is s aigh o wa d o see ha
Au m(D) = {ϕb◦R◦ϕa:a, b ∈D, R ∈ Rm}
and
{ ∈Au m(D) : (a) = b}={ϕb◦R◦ϕa:R∈ Rm}.
The symbol ρwill s and o he hype bolic (o Poinca e’s) dis ance on D,
ha is,
ρ(z, w) = anh−1|ϕz(w)|=1
2log 1 + |z−w
1−zw |
1−| z−w
1−zw |.
I :D→Dis holomo phic hen he hype bolic de i a i e o o de mo
a zas in oduced by Peschl is de ined as
[m](z) = (ϕ (z)◦ ◦ϕz)(m)(0).
By using he ac ϕ0
a( ) = |a|2−1
(1 −a )2 oge he wi h Faa di B uno’s o mula
(see, o ins ance, [3]) o he m h de i a i e o a composi e unc ion i is
no di icul o check ha
[1](z) = (1 −|z|2) 0(z)
1−| (z)|2
2
and ha , o m≥2,
[m](z)=(−1)m+1 (1 −|z|2) (m)(z)
1−| (z)|2+α(z),
whe e α(z) is a ini e sum o e ms each o hem con aining a leas one
ac o among 0(z), . . . , (m−1)(z). Hence i µ( , z)≥m hen α(z) = 0, so
we ha e
[m](z) = (−1)m+1 (1 −|z|2) (m)(z)
1−| (z)|2.
Hype bolic de i a i es a e in a ian in he sense ha |(S◦ ◦T)[m]|=
| [m]|◦Twhene e Sand Ta e con o mal au omo phisms o D.
The Schwa z-Pick Lemma is a non-Euclidean e sion o he classical
Schwa z Lemma. I asse s ha
ρ( (z), (w)) ≤ρ(z, w) and | [1](z)| ≤ 1,
o all z, w ∈Dand e e y ∈H(D). Fu he mo e, he equali y ρ( (z), (w)) =
ρ(z, w) holds o e e y pai z, w ∈Di and only i his equali y holds o
some pai z, w ∈D(z6=w) i and only i | [1](z)|= 1 o e e y z∈Di and
only i | [1](z)|= 1 o some z∈Di and only i ∈Au (D).
Recen ly, Bea don [1] has gi en an in e es ing new e sion o he Schwa z
(o he Schwa z-Pick) Lemma. Speci ically, he p o ed Theo em 1.1 be-
low (see [1, Theo em] and no es ollowing [1, Lemma 1]), which is a non-
Euclidean e sion o a esul due o Dieudonn´e [2] ha es ablishes he ol-
lowing Schwa z Lemma o de i a i es: I ∈ F1 hen
| 0(z)| ≤ (1 i |z| ≤ √2−1
(1+|z|2)2
4|z|(1−|z|2)i |z|>√2−1.
This inequali y is he bes possible in e ms o |z|. We now ansc ibe he
s a emen s o Bea don in ou e minology (he deno ed [1] = ∗). Recall
ha [1](z)∈Di 6∈ Au (D). Bea don ealized ha his allowed o
measu e he hype bolic dis ance be ween wo hype bolic de i a i es.
Theo em 1.1. Assume ha ∈ F1 Au (D). We ha e:
(a) The inequali y
ρ( [1](0), [1](z)) ≤2ρ(0, z) (1)
is sa is ied o all z∈D.
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(b) I equali y holds in (1) o some z∈D {0} hen ∈zAu (D).
(c) I (z) = z2 hen equali y holds in (1) o all z∈D.
2 A p elimina y esul
Be o e s a ing ou heo ems, we need an elemen a y lemma which is an
“m-o de ” gene aliza ion o he Schwa z-Pick Lemma.
Lemma 2.1. Assume ha m∈N, a ∈D, ∈H(D),| |<1on Dand
µ( , a)≥m. Then we ha e

(z)− (a)
1− (a) (z)≤
z−a
1−az 
m
(z∈D) (2)
and | [m](a)| ≤ m!.
Fu he , equali y holds in (2) o all z∈Di and only i i holds o
some z6=ai and only i | [m](a)|=m!i and only i ∈Au m(D).
P oo . I a∈Dsa is ies µ( , a)≥m hen he unc ion
F( ) = (ϕb◦ ◦ϕa)( )
m( ∈D {0})
has a holomo phic ex ension o Dbecause µ(ϕb◦ ◦ϕa,0) = µ( , a)≥m,
whe e we ha e deno ed b= (a). Fix ∈(0,1). Then |F( )| ≤ 1
mon
| |= , hence an applica ion o he Maximum Modulus P inciple yields
sup{|F( )|:| | ≤ } ≤ 1
m. Le ing →1 we ge sup{|F( )|:| |<1} ≤ 1,
ha is, |F( )| ≤ 1 on Do , equi alen ly, |ϕb◦ ◦ϕa( )|≤| |m( ∈D), which
becomes (2) a e he change o a iable z=ϕa( ). No e ha he alue o
( he ex ension o ) Fa he o igin is
F(0) = lim
→0
1
m·b− (ϕa( ))
1−b (ϕa( )) = lim
z→a
(1 −az)m
1− (a) (z)· (z)− (a)
(z−a)m·(−1)m+1 =
(−1)m+1 ·(1 −|a|2)m
1−| (a)|2·lim
z→a
(z)− (a)
(z−a)m= [m](a)
m!,
a e using he L’Hopi al ule oge he wi h he de ini ion o he hype bolic
de i a i e o o de mand he ac ha 0(a) = . . . = (m−1)(a) = 0. Anew
by he Maximum Modulus P inciple, |F(0)| ≤ 1, whence | [m](a)| ≤ m!.
4
Assume now ha is an m-au omo phism. Since (a) = bwe ha e
=ϕb◦R◦ϕawhe e R∈ Rm, i.e., R( ) = c m o some cwi h |c|= 1.
The e o e F( ) = con D, so |F|= 1 on Dand he equali y holds in (2) o all
z∈D(hence o some z6=a). Mo eo e , | [m](a)|=m!·|F(0)|=m!·|c|=
m!. Con e sely, suppose ha | [m](a)|=m!. Then |F(0)|= 1 and he
Maximum Modulus P inciple ells us ha F( ) = c o some unimodula
cons an c, bu his yields (ϕb◦ ◦ϕa)( ) = c m o all ∈D, which in
u n implies ha =ϕb◦R◦ϕawi h Ras be o e. Consequen ly, is an
m-au omo phism o D.
Finally, assume ha equali y holds in (2) o some z6=a. Then he
change z=ϕa( ) shows ha |ϕb◦ ◦ϕa( )|=| |m o some 6= 0, whence
|F( )|= 1 o some ∈D. Ano he applica ion o he Maximum Modulus
P inciple d i es us o F( ) = con D o some unimodula cons an c, and
his implies as abo e ha ∈Au m(D). This concludes he p oo .
3 Main esul s
We a e now eady o s a e ou heo ems. Like in [1], we can es ima e he
hype bolic dis ance be ween wo no malized hype bolic de i a i es o highe
o de unde ob ious condi ions.
Theo em 3.1. Assume ha ∈ Fm Au m(D). We ha e:
(a) I a∈Dand µ( , a)≥m hen
ρ((−1)m+1 [m](0)
m!, [m](a)
m!)≤2ρ(0, a).(3)
(b) I he e exis s a∈D {0} o which µ( , a)≥msuch ha equali y
holds in (3) hen ∈zmAu (D).
P oo . Obse e ha i µ( , a)≥mand ∈ Fm Au m(D) hen he alues
(−1)m+1 [m](0)
m!, [m](a)
m!a e in Dby Lemma 2.1, so he hype bolic dis ance
be ween hem makes sense. As o (a), since µ( , a)≥m≤µ( , 0) he
unc ions
g(z) = (z)
zm(z∈D {0})
5

and
h(z) = 1−az
a−zm
· (a)− (z)
1− (a) (z)(z∈D {a})
ha e holomo phic ex ensions on he whole Di we se
g(0) = (m)(0)
m!= (−1)m+1 [m](0)
m!
and
h(a) = lim
z→ah(z) = (1 −|a|2)m
1−| (a)|2·(−1)m+1 · (m)(a)
m!= [m](a)
m!.
We may s a wi h a6= 0, since he case a= 0 is i ial.
No e ha by Lemma 2.1 (as applied on poin s 0, a) we ge |g| ≤ 1,
|h| ≤ 1 on D, and in ac |g|<1, |h|<1 on Dsince is no an m-
au omo phism. On he o he hand, g(a) = (a)
amand h(0) = (a)
am. I we apply
he Schwa z-Pick Lemma o gand h hen one ob ains ρ(g(0), g(a)) ≤ρ(0, a)
and ρ(h(0), h(a)) ≤ρ(0, a). Consequen ly, obse ing ha g(a) = h(0), he
iangle inequali y yields
ρ((−1)m+1 [m](0)
m!, [m](a)
m!) = ρ(g(0), h(a)) ≤ρ(0, a) + ρ(0, a) = 2ρ(0, a),
which p o es (a). In o de o p o e (b), assume ha equali y in (3) holds
o some a∈D {0}wi h µ( , a)≥m. Then
2ρ(0, a) = ρ(g(0), h(a)) ≤ρ(g(0), g(a)) + ρ(h(0), h(a)) ≤2ρ(0, a),
whence ρ(g(0), g(a)) = ρ(0, a) because bo h e ms in he las sum a e no
g ea e han ρ(0, a). Bu he Schwa z-Pick Lemma ells ha g∈Au (D),
hence ∈zmAu (D). The p oo is inished.
Co olla y 3.2. I mis e en and ∈ Fm Au m(D) hen (m)(0) = 0. In
o he wo ds, ∈ Fm+1.
P oo . F om (3) and he ac ha µ( , 0) ≥mwe ob ain
ρ( [m](0)
m!,− [m](0)
m!)≤2ρ(0,0) = 0,
he e o e [m](0) = − [m](0). Consequen ly, (m)(0) = (−1)m+1 [m](0) =
0.
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I should be no ed ha pa s (a)-(b) o Theo em 1.1 a e co e ed wi h he
case m= 1 in Theo em 3.1 (obse e ha always µ( , a)≥1). An ex ension
o pa (c) o Theo em 1.1 makes no sense because i m≥2 hen he se
{z∈D:µ( , z)≥m}is disc e e in Dexcep o he i ial case ≡0. In
iew o pa s (b)-(c) o Theo em 1.1 one can wonde whe he ∈zAu (D)
implies equali y in (1) o some (o e en o all) z∈D {0}. In ac , we
ha e been able o disco e he exac condi ions unde which equali y holds in
(1). This will be accomplished in he ollowing heo em, which s eng hens
Bea don’s esul .
Theo em 3.3. Suppose ha ∈ F1 Au (D). We ha e:
(a) The inequali y
ρ( [1](0), [1](z)) ≤2ρ(0, z)
is sa is ied o all z∈D.
(b) The equali y
ρ( [1](0), [1](z)) = 2ρ(0, z) (4)
holds o some z∈D {0}i and only i i holds o all poin s o a
diame e o Di and only i ∈zAu (D).
(c) The abo e equali y holds o all z∈D {0}i and only i i holds o
wo nonze o poin s lying in wo dis inc diame e s o Di and only i
is a 2- o a ion.
P oo . Pa (a) is as in Theo em 1.1. I has been ansc ibed o he sake
o comple eness. As o (b)-(c), i equali y (4) holds o some z∈D {0}
hen we al eady know ha ∈zAu (D) by Theo em 1.1(b). Assume now
ha ∈zAu (D). Then is ei he a o a ion kz2(|k|= 1) o a unc ion
o he o m kzϕa(z) wi h 0 <|a|<1 = |k|. Wi hou loss o gene ali y
we can suppose k= 1 because ρ(kz, kw) = ρ(z, w) o all z,w∈Di
|k|= 1. I (z) = z2 hen (4) holds on he uni disk by Theo em 1.1(c). I
(z) = zϕa(z) wi h a6= 0 hen a di ec compu a ion gi es
[1](z) = 1−|z|2
1−za−z
1−az 
2·a−2z+az2
(1 −az)2.(5)
7
On he o he hand, (4) means ha
1
2log
1 + 
[1](0)− [1](z)
1− [1](0) [1](z)
1−
[1](0)− [1](z)
1− [1](0) [1](z)
= log 1 + |z|
1−|z|,
which is equi alen o

a−1−|z|2
1−|za−z
1−az |2·a−2z+az2
(1−az)2
1−a1−|z|2
1−|za−z
1−az |2·a−2z+az2
(1−az)2

=2|z|
1 + |z|2(6)
due o (5) and o he ac ha ψ(|z|)2) = ψ2|z|
1 + |z|2whe e ψis he
unc ion ψ( ) = 1+
1− , which is one- o-one on (0,1). The le -hand side o (6)
can be w i en (a e some minu es o hea y and ca e ul calcula ions) as

a(|1−az|2−|z(a−z)|2)(1 −az)−(1 −|z|2)(1 −az)(a−2z+az2)
(|1−az|2−|z(a−z)|2)(1 −az)−a(1 −|z|2)(1 −az)(a−2z+az2)
=

(1 −|a|2)(1 −|z|2)((2z−a|z|2−az2)
(1 −|a|2)(1 −|z|2)(−az −az|z|2+1+|z|2)
.
The e o e, a e squa ing, (6) is equi alen o
(2z−a|z|2−az2)(2z−a|z|2−az2)(1 + |z|2)2=
(−az −az|z|2+1+|z|2)(−az −az|z|2+1+|z|2)4|z|2.
New hea y simpli ica ions lead us o he equi alence o (6) o
−2|a|2|z|4(|z|4−2|z|2+ 1) + (a2z2+a2z2)(|z|6−2|z|4+|z|2)=0,
o , wha is he same,
|z|2(1 −|z|2)2[−2|a|2|z|2+a2z2+a2z2]=0.
I z6= 0 (o he wise, (6) is i ial) he las equali y is he same as
−2aazz + (az)2+ (az)2= 0,
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ha is, (az −az)2= 0, o , equi alen ly, az =az. In o he wo ds, (6) holds
i and only i az ∈R, which in u n means ha z∈aR, ha is, (6) holds i
and only i zbelongs o he diame e D∩aRpassing h ough a.
Wi h his we ha e p o ed (b) and he ac ha is a 2- o a ion i and
only i (4) holds on all o D. The emaining o (c) is easy, o i (4) holds o
wo nonze o poin s lying in wo dis inc diame e s o D hen mus be in
zAu (D) bu i canno be o he o m kzϕa(z) wi h a6= 0. Consequen ly,
is a 2- o a ion and he heo em is p o ed.
Re e ences
[1] A.F. Bea don, The Schwa z–Pick Lemma o de i a i es, P oc. Ame .
Ma h. Soc. 125 (1997), 3255–3256.
[2] J. Dieudonn´e, Reche ches su quelques p obl`emes ela i s aux polynˆomes
e aux onc ions bo n´ees d’une a iable complexe, Ann. Sci. ´
Ecole
No m. Sup. 48 (1931), 247–358.
[3] W.F. Donoghe, J ., Dis ibu ions and Fou ie T ans o ms, Academic
P ess, New Yo k, 1966.
LUIS BERNAL GONZ´
ALEZ MAR´
IA DEL CARMEN CALDER´
ON MORENO
DEPARTAMENTO DE AN´
ALISIS MATEM´
ATICO DEPARTAMENTO DE AN´
ALISIS MATEM´
ATICO
FACULTAD DE MATEM´
ATICAS, APDO. 1160 FACULTAD DE MATEM´
ATICAS, APDO. 1160
AVENIDA REINA MERCEDES AVENIDA REINA MERCEDES
41080 SEVILLA, SPAIN 41080 SEVILLA, SPAIN
E–mail: lb[email p o ec ed] E–mail: [email p o ec ed]
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