Two hyperbolic Schwarz lemmas

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. 3 (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. 6 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, 8 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] 9