aa r X i v : . [ m a t h . C V ] J a n MULTIPOINT JULIA THEOREMS
MARCO ABATE † To Edoardo Vesentini
Abstract.
Following ideas introduced by Beardon-Minda and by Baribeau-Rivard-Wegert in the context of the Schwarz-Pick lemma, we use the iteratedhyperbolic difference quotients to prove a multipoint Julia lemma. As appli-cations, we give a sharp estimate from below of the angular derivative at aboundary point, generalizing results due to Osserman, Mercer and others; andwe prove a generalization to multiple fixed points of an interesting estimatedue to Cowen and Pommerenke. These applications show that iterated hyper-bolic difference quotients and multipoint Julia lemmas can be useful tools forexploring in a systematic way the influence of higher order derivatives on theboundary behaviour of holomorphic self-maps of the unit disk. Introduction
The classical Schwarz-Pick lemma [6, 22, 23, 27] says that every holomorphic self-map of the unit disk D ⊂ C is a weak contraction for the Poincar´e distance ω . Moreprecisely, for every f ∈ Hol( D , D ) and z , w ∈ D we have ω (cid:0) f ( z ) , f ( w ) (cid:1) ≤ ω ( z, w ) , with equality for some z = w if and only if there is equality everywhere if andonly if f is an automorphism of D . In particular, if f / ∈ Aut( D ) then we have ω (cid:0) f ( z ) , f ( w ) (cid:1) < ω ( z, w ) (1)for all z = w .In the century following the appearance of this result many improvements of (1)for non automorphisms have appeared; see, e.g., [3, 4, 10, 12, 14, 16, 17, 26]. Surpris-ingly, in 2004 Beardon and Minda [5] found an elegant unified way to recover allthese results, and more.Their idea is based on the hyperbolic difference quotient f ∗ : D × D → C associatedto a holomorphic self-map f ∈ Hol( D , D ), which is defined as follows: f ∗ ( z, w ) = f ( z ) − f ( w )1 − f ( w ) f ( z ) (cid:30) z − w − wz if z = w ; f ′ ( z ) −| z | −| f ( z ) | if z = w . It is clear that for every w ∈ D the map z f ∗ ( z, w ) is holomorphic. Beardon andMinda observed that (1) is equivalent to saying that, if f is not an automorphism,then f ∗ ( · , w ) is a holomorphic self-map of D for every w ∈ D . But then one can † Partially supported by 2017 PRIN grant “Real and Complex Manifolds: Topology, Geometryand Holomorphic Dynamics”, Ministry of University and Research, Italy, and by 2020 PRA grant“Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni”, Universityof Pisa, Italy.2020 Mathematics Subject Classification: 30C80 (primary); 30E25, 30F45, 30J10 (secondary).
Keywords:
Julia lemma; hyperbolic difference quotient; boundary dilation coefficient; angularderivative; Cowen-Pommerenke estimate. apply the Schwarz-Pick lemma to f ∗ ( · , w ), obtaining the following 3-point Schwarz-Pick lemma: Theorem 1.1 (Beardon-Minda, 2004) . Let f ∈ Hol( D , D ) \ Aut( D ) . Then ω (cid:0) f ∗ ( z, v ) , f ∗ ( w, v ) (cid:1) ≤ ω ( z, w ) for all z , v , w ∈ D . Furthermore equality holds for some z = w and v if andonly if it holds everywhere if and only if f is a Blaschke product of degree 2. Here a
Blaschke product of degree d ≥ D of theform B ( z ) = e iθ d Y j =1 z − a j − a j z , where θ ∈ R and a , . . . , a d ∈ D . In particular, the Blaschke products of degree 1are exactly the automorphisms of D .As mentioned above, Beardon and Minda showed how the apparently innocu-ous Theorem 1.1 can be used to recover many inequalities improving the originalSchwarz-Pick lemma; we refer to their beautiful paper [5] for details.A consequence of Theorem 1.1 is that if f is not a Blaschke product of degree 2then f ∗ ( · , w ) is not an automorphism of D ; therefore its hyperbolic difference quo-tient again is a holomorphic self-map of D , and hence we can apply the classicalSchwarz-Pick lemma to get a 4-point Schwarz-Pick lemma — and then, iteratingthe procedure, a n -point Schwarz-Pick lemma for any n ≥ f ∈ Hol( D , D ) and z , w , . . . , w k ∈ D we define the iteratedhyperbolic difference quotient ∆ w k ,...,w f ( z ) by setting ∆ w f ( z ) = f ∗ ( z, w ) and∆ w k ,...,w f ( z ) = ∆ w k (∆ w k − ,...,w f )( z ). Then we have a multi-point Schwarz-Picklemma: Theorem 1.2 (Baribeau-Rivard-Wegert, 2009) . Given k ≥ , take f ∈ Hol( D , D ) not a Blaschke product of degree at most k . Then ω (cid:0) ∆ w k ,...,w f ( z ) , ∆ w k ,...,w f ( w )) (cid:1) ≤ ω ( z, w ) for all z , w , w , . . . , w k ∈ D . Furthermore equality holds for some z = w and w , . . . , w k if and only if it holds for all z , w , w , . . . , w k ∈ D if and only if f is aBlaschke product of degree k + 1 . Another way of expressing the Schwarz-Pick lemma consists in saying that holo-morphic self-maps of the unit disk send disks for the Poincar´e distance into disksfor the Poincar´e distance. Julia [13] in 1920 noticed that by moving the centers ofthese disks toward the boundary one can get a boundary version of the Schwarz-Picklemma, nowadays known as Julia lemma:
Theorem 1.3 (Julia, 1920) . Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that β f ( σ ) := lim inf z → σ − | f ( z ) | − | z | = α < ∞ . Then there exists a unique τ ∈ ∂ D such that | τ − f ( z ) | − | f ( z ) | ≤ β f ( σ ) | σ − z | − | z | (2) for every z ∈ D . Moreover, equality in (2) holds at one point if and only if it holdseverywhere if and only if f ∈ Aut( D ) . ULTIPOINT JULIA THEOREMS 3
The number β f ( σ ) ∈ (0 , + ∞ ] is the boundary dilation coefficient of f at σ , and itis the absolute value of the angular derivative f ′ ( σ ), the non-tangential limit of f ′ at σ , which is known to exist thanks to the Julia-Wolff-Carath´eodory theorem. Itis well-known that β f ( σ ) > β f ( σ ) < + ∞ then the point τ appearing in the statement of Julia lemma is the non-tangential limit of f at σ ,that we will denote by f ( σ ).The geometrical meaning of (2) is that if β f ( σ ) < + ∞ then f sends horocyclescentered at σ in horocycles centered at f ( σ ), where a horocycle E ( σ, R ) of center σ ∈ ∂ D and radius R > E ( σ, R ) = (cid:26) z ∈ D (cid:12)(cid:12)(cid:12)(cid:12) | σ − z | − | z | < R (cid:27) . Geometrically, E ( σ, R ) is an Euclidean disk or radius R/ ( R + 1) internally tangentat ∂ D in σ .The aim of this paper is to obtain a multipoint version of Julia lemma along thelines of Theorems 1.1 and 1.2. The paper [5] contains a 3-point Julia lemma, but itsstatement does not involve the hyperbolic difference quotient, and it is in a slightlydifferent spirit. Closer to our aims is [18, Proposition 4.1]; but its (Euclidean)statement is quite involved and not easy to use (see Remark 3.8).Our idea then is to obtain a version of Theorem 1.3 involving the iterated hyper-bolic difference quotients. The main difference between the Schwarz-Pick lemmaand the Julia lemma is that the latter works only for maps with finite boundary di-lation coefficient. So the main result allowing our approach to start is the following(see Proposition 3.4): Proposition 1.4.
Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ . Then β ∆ w f ( σ ) = β f ( σ ) 1 − | f ( w ) | | f ( σ ) − f ( w ) | − − | w | | σ − w | for all w ∈ D . In particular β ∆ w f ( σ ) < + ∞ for some w ∈ D if and only if β ∆ w f ( σ ) < + ∞ for all w ∈ D if and only if β f ( σ ) < + ∞ . So if the boundary dilation coefficient is finite for f it remains finite for all theiterated hyperbolic difference quotients of f . This allows us to obtain a multi-pointJulia lemma (see Theorem 4.2): Theorem 1.5.
Given k ≥ , take f ∈ Hol( D , D ) not a Blaschke product of degreeat most k . Let σ ∈ ∂ D be such that β f ( σ ) < + ∞ . Then | ∆ w k ,...,w f ( σ ) − ∆ w k ,...,w f ( z ) | − | ∆ w k ,...,w f ( z ) | ≤ β ∆ wk,...,w f ( σ ) | σ − z | − | z | (3) for all z , w , . . . , w k ∈ D . Moreover, equality occurs for some z , w , . . . , w k ∈ D ifand only if it occurs everywhere if and only if f is a Blaschke product of degree k +1 . We now describe two applications of this theorem. We mentioned before thatthe boundary dilation coefficient is always strictly positive. In some instances it isuseful to have a more explicit bound from below, like the classical one β f ( σ ) ≥ − | f (0) | | f (0) | . (4)In Section 4 we shall prove a much more precise estimate (see Theorem 4.3): Theorem 1.6.
Given k ≥ let f ∈ Hol( D , D ) be not a Blaschke product of degreeat most k . Take σ ∈ ∂ D with β f ( σ ) < + ∞ . Then β f ( σ ) ≥ k X j =0 − | w j +1 | | σ − w j +1 | j Y h =0 | ∆ w h ,...,w f ( σ ) − ∆ w h ,...,w f ( w h +1 ) | − | ∆ w h ,...,w f ( w h +1 ) | (5) MARCO ABATE for every w , . . . , w k +1 ∈ D , where ∆ w h ,...,w f = f when h = 0 . Furthermore wehave equality in (5) for some w , . . . , w k +1 ∈ D if and only if we have equality forall w , . . . , w k +1 ∈ D if and only if f is a Blaschke product of degree k + 1 . In particular we have the following corollary (see Corollary 4.7):
Corollary 1.7.
Given k ≥ let f ∈ Hol( D , D ) be not a Blaschke product of degreeat most k . Take σ ∈ ∂ D with β f ( σ ) < + ∞ . Then β f ( σ ) ≥ k X j =0 j Y h =0 − | ∆ O h f (0) | | ∆ O h f (0) | , where O h = (0 , . . . , ∈ D h is the origin of C h , and ∆ O h f = f when h = 0 .Furthermore we have equality if and only if f is a Blaschke product of degree k + 1 and ∆ O h f (0) = | ∆ O h f (0) | ∆ O h f ( σ ) for all h = 0 , . . . , k . These results when k = 0 recover (4) and when k ≥ f (0) = 0 the estimate (4) implies that β f ( σ ) ≥
1. In 1982, Cowen andPommerenke [9] proved that if moreover f ( σ ) = σ this estimate can be improvedto β f ( σ ) ≥ | − f ′ (0) | − | f ′ (0) | . More precisely, they obtained a sharp estimate valid when f has a fixed point insideand n fixed points on the boundary: Theorem 1.8 (Cowen-Pommerenke, 1982) . Let f ∈ Hol( D , D ) \ Aut( D ) be suchthat f ( z ) = z for some z ∈ D . Assume there exist σ , . . . , σ n ∈ ∂ D distinctpoints with β f ( σ j ) < + ∞ and f ( σ j ) = σ j for j = 1 , . . . , n . Then n X j =1 β f ( σ j ) − ≤ − | f ′ ( z ) | | − f ′ ( z ) | . (6) Furthermore, equality holds if and only if f is a Blaschke product of degree n + 1 . In Section 3 we shall show (see Proposition 3.11) how to obtain this result as aconsequence of our Theorem 1.5 for k = 1. More interestingly, in Section 4 we shallgeneralize the estimate (6) to the case of multiple fixed points (see Theorem 4.12): Theorem 1.9.
Let f ∈ Hol( D , D ) . Given k ≥ , assume that f is not a Blaschkeproduct of degree at most k ≥ and that there exists z ∈ D such that f ( z ) = z and f ′ ( z ) = . . . = f ( k − ( z ) = 0 . Take σ , . . . , σ n ∈ ∂ D distinct points such that β f ( σ j ) < + ∞ and f ( σ j ) = (cid:16) σ j − z − z σ j (cid:17) k + z z (cid:16) σ j − z − z σ j (cid:17) k , for j = 1 , . . . , n . Then n X j =1 (cid:16) ( f ( σ j ) − σ j ) z | f ( σ j ) − z | (cid:17) β f ( σ j ) − k ≤ − (cid:12)(cid:12)(cid:12) f ( k ) ( z ) k ! (1 − | z | ) k − (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − f ( k ) ( z ) k ! (1 − | z | ) k − (cid:12)(cid:12)(cid:12) , with equality if and only if f is a Blaschke product of degree n + k . The proof in the general case is a bit delicate and requires the full force of ourmultipoint Julia lemma. However, the case z = 0 has a simpler statement, andactually a much easier proof (see Corollary 4.13): ULTIPOINT JULIA THEOREMS 5
Corollary 1.10.
Let f ∈ Hol( D , D ) . Given k ≥ , assume that f is not a Blaschkeproduct of degree at most k and that f (0) = · · · = f ( k − (0) = 0 . Take σ , . . . , σ n ∈ ∂ D distinct points such that β f ( σ j ) < + ∞ and f ( σ j ) = σ kj for j = 1 , . . . , n . Then n X j =1 β f ( σ j ) − k ≤ − (cid:12)(cid:12)(cid:12) f ( k ) (0) k ! (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − f ( k ) (0) k ! (cid:12)(cid:12)(cid:12) , with equality if and only if f is a Blaschke product of degree n + k . Summing up, these applications show that iterated hyperbolic difference quo-tients and multipoint Julia lemmas can be an useful tool for exploring in a sys-tematic way the influence of higher order derivatives on the boundary behaviour ofholomorphic self-maps of the unit disk.This paper is organized as follows. In Section 2 we shall collect a number ofpreliminary definitions and results that we shall need later on. In Section 3 weshall discuss 2-point Julia lemmas, proving in particular Proposition 1.4. Finally, inSection 4 we shall introduce our general multipoint Julia lemma and its applications,proving in particular Theorems 1.5, 1.6, 1.9 and Corollaries 1.7 and 1.10.
Ackowledgments.
This paper is respectfully dedicated to the memory of my advisor,Edoardo Vesentini, who, among (many) other things, showed me how beautiful andelegant complex analysis can be, in one, several and infinitely many variables.2.
Preliminaries
In this section we collect a few known results that shall be useful later on.2.1.
Blaschke products.Definition 2.1.
A (finite)
Blaschke product B ∈ Hol( D , D ) is a holomorphic self-map of D continuous up to the boundary with B ( ∂ D ) ⊆ ∂ D . Since a Blaschkeproduct B cannot vanish in a neighbourhood of ∂ D it must have a finite number d ≥ D , counted with respect to their multiplicity. The number d is the degree of B . We shall denote by B d the set of Blaschke products of degree d ≥ B the set of constant functions of modulus 1. Lemma 2.2.
A function B ∈ Hol( D , D ) is a Blaschke product of degree d ≥ ifand only if there are θ ∈ R and a , . . . , a d ∈ D such that B ( z ) = e iθ d Y j =1 z − a j − ¯ a j z . (7) In particular, if γ ∈ Aut( D ) then B ◦ γ and γ ◦ B are still Blaschke products of thesame degree d .Proof. Since σ − a − aσ = σ σ − aσ − a ∈ ∂ D for all σ ∈ ∂ D and a ∈ D , it is clear that all maps of the form (7) are Blaschkeproducts of degree d , with zeroes in a , . . . , a d .Conversely, assume that B is a Blaschke product of degree d ≥
0. If d = 0 thenthe maximum principle applied to 1 /B implies that | B | ≡
1, and hence B ≡ e iθ fora suitable θ ∈ R . MARCO ABATE
Assume d ≥
1, let a , . . . , a d ∈ D be the zeroes of B , listed accordingly to theirmultiplicities, and put B ( z ) = d Y j =1 z − a j − ¯ a j z . Then
B/B is holomorphic without zeroes in D and | B/B | = | B /B | ≡ ∂D .By the maximum principle we get | B/B | , | B /B | ≤
1, and thus
B/B is a constantof modulus 1, as required.If γ ∈ Aut( D ) then B ◦ γ obviously is a Blaschke product of the same degree. Onthe other hand, clearly γ ◦ B is still a Blaschke product. Moreover, if a = γ − (0)then ( γ ◦ B )( z ) = 0 if and only if B ( z ) = a if and only if e iθ d Y j =1 ( z − a j ) = a d Y j =1 (1 − ¯ a j z ) . This is a polynomial equation of degree exactly d ; thus γ ◦ B has exactly d zeroes,counted with multiplicities, and we are done. (cid:3) In particular, the Blaschke products of degree 1 are exactly the automorphismsof D , that is B = Aut( D ). If w ∈ D we shall denote by γ w the automorphism γ w ( z ) = z − w − ¯ wz . Later on we shall need the following
Lemma 2.3.
Let σ , . . . , σ n ∈ ∂ D be distinct points, a , . . . , a n ∈ R + and B ∈ B d for some n ≥ and d ≥ . If B define h : D → C by h ( z )1 − h ( z ) = n X j =1 a j σ j + zσ j − z + 1 + B ( z )1 − B ( z ) . Then h ∈ B n + d .Proof. A quick computation yields h ( z ) = 2 B ( z ) + (cid:0) − B ( z ) (cid:1) S ( z )2 + (cid:0) − B ( z ) (cid:1) S ( z ) , where S ( z ) = n X j =1 a j σ j + zσ j − z ;in particular h is a rational function of degree n + d because numerator and de-nominator have no common factors.When z ∈ D we haveRe 1 + h ( z )1 − h ( z ) = n X j =1 a j Re σ j + zσ j − z +Re 1 + B ( z )1 − B ( z ) = n X j =1 a j − | z | | σ j − z | + 1 − | B ( z ) | | − B ( z ) | > h ( z ) ∈ D , and hence h ( D ) ⊆ D .If σ ∈ ∂ D is different from σ j we have σ j + σσ j − σ = 2 i Im( σσ j ) | σ j − σ | ∈ i R ;thus if σ = σ , . . . σ n we have S ( σ ) = ia ∈ i R and hence setting B ( σ ) = e iφ ∈ ∂ D we have h ( σ ) = 2 e iφ + (1 − e iφ ) ia − e iφ ) ia ∈ ∂ D . ULTIPOINT JULIA THEOREMS 7
To deal with σ = σ j we write h ( z ) = a j ( σ j + z ) (cid:0) − B ( z ) (cid:1) + ( σ j − z ) (cid:0) S j ( z ) + 2 B ( z ) (cid:1) a j ( σ j + z ) (cid:0) − B ( z ) (cid:1) + ( σ j − z ) (cid:0) S j ( z ) + 2 (cid:1) where S j ( σ j ) = 0; therefore we get h ( σ j ) = 1, also when B ( σ j ) = 1.Summing up, we have proved that h is a Blaschke product; being a rationalfunction of degree n + d we get h ∈ B n + d , and we are done. (cid:3) The hyperbolic difference quotient.Definition 2.4.
Let f ∈ Hol( D , D ) be a holomorphic self-map of the unit disk.The hyperbolic derivative f h : D → C of f is given by f h ( z ) = f ′ ( z )1 − | f ( z ) | (cid:30) − | z | = f ′ ( z ) 1 − | z | − | f ( z ) | . The hyperbolic difference quotient f ∗ : D × D → C is given by f ∗ ( z, w ) = f ( z ) − f ( w )1 − f ( w ) f ( z ) (cid:30) z − w − wz if z = w ; f h ( z ) if z = w . It is easy to check that for every w ∈ D the function z f ∗ ( z, w ) is holomorphic.Furthermore, the Schwarz-Pick lemma implies that | f ∗ ( z, w ) | ≤ z , w ) ∈ D × D such that | f ∗ ( z , w ) | = 1 if and only if | f ∗ | ≡ f ∈ Aut( D ). In particular, if γ ∈ Aut( D ) is given by γ ( z ) = e iθ z − a − ¯ az then it is easy to check that γ ∗ ( z, w ) = e iθ − ¯ aw − a ¯ w . This can be seen as a particular case of the following result:
Proposition 2.5.
Let f ∈ Hol( D , D ) and d ≥ . Then f ∈ B d if and only if f ∗ ( · , w ) ∈ B d − for all w ∈ D if and only if f ∗ ( · , w ) ∈ B d − for some w ∈ D .Proof. By definition we have γ w ( z ) f ∗ ( z, w ) = γ f ( w ) (cid:0) f ( z ) (cid:1) (8)for all z , w ∈ D . If f ∗ ( · , w ) ∈ B d − for some w ∈ D then B = γ w f ∗ ( · , w ) is aBlaschke product of degree d , and thus Lemma 2.2 implies that f = γ − f ( w ) ◦ B is aBlaschke product of degree d .Conversely, if f is a Blaschke product of degree d , then for any w ∈ D wehave that γ f ( w ) ◦ f is a Blaschke product of degree d vanishing at w . Therefore γ w is a factor of γ f ( w ) ◦ f , and (8) implies that f ∗ ( · , w ) is a Blaschke product ofdegree d − (cid:3) The classical Julia lemma.Definition 2.6.
The horocycle E ( σ, R ) ⊂ D of center σ ∈ ∂ D and radius R > E ( σ, R ) = (cid:26) z ∈ D (cid:12)(cid:12)(cid:12)(cid:12) | σ − z | − | z | < R (cid:27) . (9)Geometrically, E ( σ, R ) is the euclidean disk of radius R/ ( R + 1) internally tangentto ∂ D in σ . MARCO ABATE
Definition 2.7.
Given f ∈ Hol( D , D ) and σ , τ ∈ ∂ D , set β f ( σ, τ ) = sup z ∈ D (cid:26) | τ − f ( z ) | − | f ( z ) | (cid:30) | σ − z | − | z | (cid:27) . (10)The boundary dilation coefficient of f at σ is given by β f ( σ ) = inf τ ∈ ∂ D β f ( σ, τ ) ∈ [0 , + ∞ ] . (11) Remark . By definition ∀ R > f (cid:0) E ( σ, R ) (cid:1) ⊆ E (cid:0) τ, β f ( σ, τ ) R (cid:1) . In particular, for every f ∈ Hol( D , D ) and σ ∈ ∂ D there is at most one point τ ∈ ∂ D such that β f ( σ, τ ) is finite. Indeed, if we had β f ( σ, τ j ) < + ∞ for two distinct points τ , τ ∈ ∂ D we would get a contradiction choosing R so small that E ( τ , βR ) ∩ E ( τ , βR ) = / (cid:13) , where β = max { β f ( σ, τ ) , β f ( σ, τ ) } .The following well-known result gives us an alternative way to compute theboundary dilation coefficient (for a proof see, e.g., [1, Proposition 1.2.6]): Proposition 2.9.
Take f ∈ Hol( D , D ) and σ ∈ ∂ D . Then β f ( σ ) = lim inf z → σ − | f ( z ) | − | z | . Furthermore (see, e.g., [1, Lemma 1.2.4]):
Lemma 2.10.
Let f : D → D be holomorphic. Then ∀ z ∈ D − | f ( z ) | − | z | ≥ − | f (0) | | f (0) | > . (12) In particular, for all σ ∈ ∂ D we have β f ( σ ) ≥ − | f (0) | | f (0) | > . (13) Moreover, equality in (12) holds at one point z = 0 (and hence everywhere) if andonly if f ( z ) = e iθ z for a suitable θ ∈ R . We can now state the classical Julia lemma [13]:
Theorem 2.11 (Julia lemma) . Let f ∈ Hol( D , D ) , and choose σ ∈ ∂ D so that β f ( σ ) < + ∞ . Let τ ∈ ∂ D be the unique point of ∂ D such that β f ( σ, τ ) < + ∞ .Then | τ − f ( z ) | − | f ( z ) | ≤ β f ( σ ) | σ − z | − | z | , (14) that is ∀ R > f (cid:0) E ( σ, R ) (cid:1) ⊆ E ( τ, β f ( σ ) R ) . (15) Moreover, equality in (14) holds at one point (and hence everywhere) if and only if f ∈ Aut( D ) . As a consequence we have another way for computing the boundary dilationcoefficient:
Corollary 2.12.
Take f ∈ Hol( D , D ) and σ ∈ ∂ D . Then β f ( σ ) = lim r → − − | f ( rσ ) | − r = lim r → − − | f ( rσ ) | − r . ULTIPOINT JULIA THEOREMS 9
Proof.
By Proposition 2.9 we have β f ( σ ) ≤ lim inf r → − − | f ( rσ ) | − r ;in particular the first equality is proven when β f ( σ ) = + ∞ . Assume then that β f ( σ ) < + ∞ . An easy computation shows that rσ ∈ ∂E (cid:0) σ, − r r (cid:1) ; therefore Theo-rem 2.11 yields f ( rσ ) ∈ E (cid:0) τ, β f ( σ ) − r r (cid:1) for a suitable τ ∈ ∂ D . Since E ( τ, R ) is anEuclidean disk of radius R/ ( R + 1) internally tangent to ∂ D in τ it follows that1 − | f ( rσ ) | ≤ | τ − f ( rσ ) | ≤ β f ( σ ) − r r β f ( σ ) − r r . Therefore lim sup r → − − | f ( rσ ) | − r ≤ lim sup r → − β f ( σ )1 + r + β f ( σ )(1 − r ) = β f ( σ )and the first equality is proved.To prove the second equality, first of all notice that21 + r − | f ( rσ ) | − r ≥ | f ( rσ ) | r − | f ( rσ ) | − r = 1 − | f ( rσ ) | − r ≥
11 + r − | f ( rσ ) | − r . (16)From this the second equality immediately follows when β f ( σ ) = + ∞ . If β f ( σ ) < + ∞ then | f ( rσ ) | → r → − , and thus the assertion follows again from (16). (cid:3) Definition 2.13.
Given τ ∈ ∂ D and M >
0, the
Stolz region K ( τ, M ) of vertex τ and amplitude M is K ( τ, M ) = (cid:26) z ∈ D (cid:12)(cid:12)(cid:12)(cid:12) | τ − z | − | z | < M (cid:27) . (17)Note that K ( τ, M ) = / (cid:13) if M ≤
1, for | τ − z | ≥ − | z | . Definition 2.14.
We say that a function f : D → b C has non-tangential (or angular )limit c ∈ b C at σ ∈ ∂ D if f ( z ) → c as z tends to σ within K ( σ, M ) for any M > K -lim z → σ f ( z ) = c and denote c by f ( σ ).We end this preliminary section recalling the famous Julia-Wolff-Carath´eodorytheorem [7, 30]; for a proof see, e.g., [1, Theorem 1.2.7]. Theorem 2.15 (Julia-Wolff-Carath´eodory; 1926) . Let f ∈ Hol( D , D ) and τ , σ ∈ ∂ D . Then K - lim z → σ τ − f ( z ) σ − z = τ ¯ σβ f ( σ, τ ) . (18) If this non-tangential limit is finite then β f ( σ ) = β f ( σ, τ ) < + ∞ , the function f has non-tangential limit τ at σ and K - lim z → σ f ′ ( z ) = τ ¯ σβ f ( σ ) . (19) As anticipated in the introduction, our 2-point Julia lemma will be obtained byapplying the classical Julia lemma to the function f ∗ ( · , w ). To do so we need tocompute the boundary dilation coefficient of the hyperbolic difference quotient; thisis done in the next two results. Lemma 3.1.
Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that f has non-tangentiallimit f ( σ ) ∈ ∂ D at σ . Then K - lim z → σ f ∗ ( z, w ) = f ( σ ) − f ( w )1 − f ( w ) f ( σ ) (cid:30) σ − w − wσ = f ( σ ) σ f ( σ ) − f ( w ) f ( σ ) − f ( w ) σ − wσ − w ∈ ∂ D for all w ∈ D .Proof. It follows immediately from the definition of f ∗ . (cid:3) Definition 3.2.
Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that f has non-tangentiallimit f ( σ ) ∈ ∂ D at σ . Given w ∈ D we set f ∗ ( σ, w ) = K -lim z → σ f ∗ ( z, w ) ;in particular, | f ∗ ( σ, · ) | ≡ Remark . For the sake of completeness, we remark that K -lim w → σ f ∗ ( z, w ) = f ( z ) − f ( σ )1 − f ( σ ) f ( z ) (cid:30) z − σ − σz ≡ f ( σ ) σ for all z ∈ D . Moreover, (18) yields K -lim w → σ f ∗ ( σ, w ) = f ( σ ) σ f ( σ ) σβ f ( σ ) f ( σ ) σβ f ( σ ) = f ( σ ) σ . Proposition 3.4.
Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ .Denote by f ( σ ) ∈ ∂ D the non-tangential limit of f at σ . Then lim inf z → σ − | f ∗ ( z, w ) | − | z | = β f ( σ ) 1 − | f ( w ) | | f ( σ ) − f ( w ) | − − | w | | σ − w | = 1 − | f ( w ) | | f ( σ ) − f ( w ) | (cid:20) β f ( σ ) − | f ( σ ) − f ( w ) | − | f ( w ) | (cid:30) | σ − w | − | w | (cid:21) (20) for all w ∈ D . Moreover, the left-hand side vanishes for some w ∈ D if and onlyif it vanishes for all w ∈ D if and only if f ∈ Aut( D ) .Proof. First of all we have1 − | f ∗ ( z, w ) | = 1 − (cid:12)(cid:12)(cid:12) f ( z ) − f ( w )1 − f ( w ) f ( z ) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) z − w − wz (cid:12)(cid:12)(cid:12) = (cid:18) − (cid:12)(cid:12)(cid:12) f ( z ) − f ( w )1 − f ( w ) f ( z ) (cid:12)(cid:12)(cid:12) (cid:19) − (cid:18) − (cid:12)(cid:12)(cid:12) z − w − wz (cid:12)(cid:12)(cid:12) (cid:19)(cid:12)(cid:12)(cid:12) z − w − wz (cid:12)(cid:12)(cid:12) = (1 −| f ( z ) | )(1 −| f ( w ) | ) | − f ( w ) f ( z ) | − (1 −| z | )(1 −| w | ) | − wz | (cid:12)(cid:12)(cid:12) z − w − wz (cid:12)(cid:12)(cid:12) = (1 − | z | ) −| f ( w ) | | − f ( w ) f ( z ) | −| f ( z ) | −| z | − −| w | | − wz | (cid:12)(cid:12)(cid:12) z − w − wz (cid:12)(cid:12)(cid:12) Recalling Corollary 2.12 and the fact that f has non-tangential limit f ( σ ) ∈ ∂ D at σ we obtainlim inf z → σ − | f ∗ ( z, w ) | − | z | = lim inf r → − − | f ∗ ( rσ, w ) | − r = −| f ( w ) | | − f ( w ) f ( σ ) | β f ( σ ) − −| w | | − wσ | (cid:12)(cid:12)(cid:12) σ − w − wσ (cid:12)(cid:12)(cid:12) = β f ( σ ) 1 − | f ( w ) | | f ( σ ) − f ( w ) | − − | w | | σ − w | , and (20) is proved.Finally, by Theorem 2.11 the right-hand side in (20) vanishes at one point if andonly if it vanishes identically if and only if f ∈ Aut( D ), and we are done. (cid:3) Definition 3.5.
Given f ∈ Hol( D , D ) and σ ∈ ∂ D we put β ∗ f ( σ ; w ) = β f ( σ ) 1 − | f ( w ) | | f ( σ ) − f ( w ) | − − | w | | σ − w | ∈ [0 , + ∞ ]for all w ∈ D . In particular, β ∗ f ( σ, w ) = + ∞ for some w ∈ D if and only if β ∗ f ( σ ; · ) ≡ + ∞ if and only if β f ( σ ) = + ∞ , and β ∗ f ( σ, w ) = 0 for some w ∈ D ifand only if β ∗ f ( σ ; · ) ≡ f ∈ Aut( D ).As a first hint of how it is possible to use this kind of results we show how toimprove (13): Corollary 3.6.
Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ . Then β f ( σ ) 1 − | f ( w ) | | f ( σ ) − f ( w ) | ≥ | w | − (cid:12)(cid:12)(cid:12) f ( w ) − f (0)1 − f ( w ) f (0) (cid:12)(cid:12)(cid:12) | w | + (cid:12)(cid:12)(cid:12) f ( w ) − f (0)1 − f ( w ) f (0) (cid:12)(cid:12)(cid:12) + 1 − | w | | σ − w | for all w ∈ D \ { } and β f ( σ ) ≥
21 + | f h (0) | | f ( σ ) − f (0) | − | f (0) | ≥
21 + | f h (0) | − | f (0) | | f (0) | . (21) Proof.
If we apply (13) to f ∗ ( · , w ) we get β ∗ f ( σ, w ) ≥ − | f ∗ (0 , w ) | | f ∗ (0 , w ) | for all w ∈ D . Since f ∗ (0 , w ) = ( f ( w ) − f (0) w − f ( w ) f (0) if w = 0 ,f h (0) if w = 0 , we immediately get the first inequality for w = 0. When w = 0 we get β f ( σ ) 1 − | f (0) | | f ( σ ) − f (0) | ≥ − | f h (0) | | f h (0) | + 1 = 21 + | f h (0) | (22)and we are done. (cid:3) Notice that when f / ∈ Aut( D ) we have | f h (0) | < Theorem 3.7.
Let f ∈ Hol( D , D ) \ Aut( D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ .Then | f ∗ ( σ, w ) − f ∗ ( z, w ) | − | f ∗ ( z, w ) | ≤ β ∗ f ( σ ; w ) | σ − z | − | z | (23) for all z , w ∈ D . Moreover, equality in (23) occurs for some ( z , w ) ∈ D × D ifand only if it occurs everywhere if and only if f is a Blaschke product of degree 2.Proof. The inequality (23) follows from Theorem 2.11 applied to f ∗ ( · , w ). If wehave equality in (23) for some ( z , w ) ∈ D × D again Theorem 2.11 implies that f ∗ ( · , w ) ∈ Aut( D ), and then Proposition 2.5 implies that f ∈ B . Conversely, f ∈ B implies that f ∗ ( · , w ) ∈ Aut( D ) for all w ∈ D , and thus we have equality in(23) for all z , w ∈ D . (cid:3) Remark . It turns out that (23) implies a (not very illuminating) Euclideanstatement, originally proved by Mercer [18], that in our notations can be expressedas follows: let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ . Take w ∈ D and set Λ = −| z | | σ − z | , ˆ β = β ∗ f ( σ ; w ), φ w ( z ) = w − z − wz and L = 1 − | f ( w ) φ w ( z ) | | − f ( w ) f ∗ ( σ, w ) φ w ( z ) | . Then for all z ∈ D we have | f ( z ) − c w ( z ) | < r w ( z ), where c w ( z ) = f ( σ ) f ∗ ( σ, w ) φ w ( z ) 1 − | f ( w ) | (cid:0) − f ( w ) f ∗ ( σ, w ) φ w ( z ) (cid:1) ˆ β ˆ βL + Λ+ f ( σ ) φ f ( w ) (cid:0) f ∗ ( σ, w ) φ w ( z ) (cid:1) ,r w ( z ) = | φ w ( z ) | − | f ( w ) | (cid:12)(cid:12) − f ( w ) f ∗ ( σ, w ) φ w ( z ) (cid:12)(cid:12) ˆ β ˆ βL + Λ . This can be recovered as follows. Theorem 3.7 says that f ∗ ( z, w ) belongs to thehorocycle of center f ∗ ( σ, w ) and radius β ∗ / Λ, which is an Euclidean disk of center Λ β ∗ +Λ f ∗ ( σ, w ) and radius β ∗ β ∗ +Λ . Since φ w ( z ) f ∗ ( z, w ) = φ f ( w ) (cid:0) f ( z ) (cid:1) , it follows that φ f ( w ) (cid:0) f ( z ) (cid:1) belongs to the Euclidean disk D of center Λ β ∗ +Λ f ∗ ( σ, w ) φ w ( z ) andradius β ∗ β ∗ +Λ | φ w ( z ) | . Using the fact that φ − f ( w ) = φ f ( w ) we get that f ( z ) belongsto the Euclidean disk φ f ( w ) ( D ); computing center and radius of this latter disk weget the assertion.Applying Theorem 2.15 to f ∗ ( · , w ) we get the next corollary: Corollary 3.9.
Let f ∈ Hol( D , D ) \ Aut( D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ .Then K - lim z → σ f ∗ ( σ, w ) − f ∗ ( z, w ) σ − z = f ∗ ( σ, w ) σβ ∗ f ( σ ; w ) and K - lim z → σ ∂f ∗ ∂z ( z, w ) = f ∗ ( σ, w ) σβ ∗ f ( σ ; w ) . As mentioned in the introduction, specializing Theorem 3.7 to the case f ( z ) = z we can recover an estimate due to Cowen and Pommerenke. The main step iscontained in the following Corollary 3.10.
Let f ∈ Hol( D , D ) \ Aut( D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ .Assume that f (0) = 0 . Then (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ( z ) z (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) f ( z ) z (cid:12)(cid:12)(cid:12) ≤ (cid:0) β f ( σ ) − (cid:1) | σ − z | − | z | (24) ULTIPOINT JULIA THEOREMS 13 if z = 0 and (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ′ (0) (cid:12)(cid:12)(cid:12) − | f ′ (0) | ≤ β f ( σ ) − . (25) In particular, if furthermore f ( σ ) = σ we get | − f ′ (0) | − | f ′ (0) | ≤ β f ( σ ) − . (26) Moreover, equality occurs in (24) at some z ∈ D \ { } or in (25) if and only if italways occurs if and only f is a Blaschke product of degree 2.Proof. If f (0) = 0 then f ∗ ( z,
0) = ( f ( z ) z if z = 0 ,f ′ (0) if z = 0 ; (27)moreover, f ∗ ( σ,
0) = f ( σ ) /σ and β ∗ f ( σ ; 0) = β f ( σ ) −
1. The assertions then followfrom Theorem 3.7. (cid:3)
When f ( σ ) = σ (26) can be restated as1 β f ( σ ) − ≤ − | f ′ (0) | | − f ′ (0) | = Re 1 + f ′ (0)1 − f ′ (0) . Recalling that f ′ ( σ ) = β f ( σ ) when f ( σ ) = σ , where f ′ ( σ ) is the non-tangentiallimit of f ′ at σ (see Theorem 2.15), using (24) we can now recover a result due toCowen and Pommerenke [9], saying that a similar estimate still holds when thereare several fixed points in the boundary: Proposition 3.11 (Cowen-Pommerenke, 1982) . Let f ∈ Hol( D , D ) \ Aut( D ) besuch that f ( z ) = z for some z ∈ D . Assume there exist σ , . . . , σ n ∈ ∂ D distinctpoints with β f ( σ j ) < + ∞ and f ( σ j ) = σ j for j = 1 , . . . , n . Then n X j =1 β f ( σ j ) − ≤ − | f ′ ( z ) | | − f ′ ( z ) | . (28) Furthermore, equality holds if and only if f is a Blaschke product of degree n + 1 .Proof. First of all, let φ z ( z ) = ( z − z ) / (1 − z z ). Then ˜ f = φ z ◦ f ◦ φ z satisfies˜ f (0) = 0 and ˜ f ′ (0) = f ′ ( z ). Moreover if we put ˜ σ j = φ z ( σ j ) then we have˜ f (˜ σ j ) = ˜ σ j and β ˜ f (˜ σ j ) = β f ( σ j ). Therefore in the proof without loss of generalitywe can assume z = 0.For j = 1 , . . . , n set β j = β f ( σ j ). We would like to prove, by induction on n ,that 1 − | f ( z ) /z | | − f ( z ) /z | = Re (cid:18) f ( z ) /z − f ( z ) /z (cid:19) ≥ n X j =1 β j − (cid:18) σ j + zσ j − z (cid:19) = n X j =1 β j − − | z | | σ j − z | (29)for all z ∈ D , with equality at one point (and hence everywhere) if and only if f ∈ B n +1 . Clearly, when z = 0 the expression f ( z ) /z is replaced by f ′ (0), and thusthe theorem follows taking z = 0.For n = 1 (29) follows from Corollary 3.10. Assume it is true for n −
1. Inparticular we have Re f ( z ) /z − f ( z ) /z − n − X j =1 β j − σ j + zσ j − z ≥ , (30) with equality at one point (and hence everywhere) if and only if f ∈ B n . Thereforewe can find h ∈ Hol( D , C ) with h ( D ) ⊂ D so that1 + f ( z ) /z − f ( z ) /z − n − X j =1 β j − σ j + zσ j − z = 1 + h ( z )1 − h ( z ) . (31)Notice that h ( D ) ⊆ D unless in (30) we have equality at one point (and henceeverywhere); in that case h ≡ e iθ for a suitable θ ∈ R .If h ≡ e iθ , Lemma 2.3 shows that f ( z ) = zB ( z ), where B ∈ B n − . But then f ,being rational of degree n , can have at most n fixed points, whereas we are assumingthat it has n + 1 fixed points, contradiction. Thus h cannot be a constant, and wehave the strict inequality in (30).A quick computation shows that h ( z ) = 2 f ( z ) z − (cid:16) − f ( z ) z (cid:17) S ( z )2 − (cid:16) − f ( z ) z (cid:17) S ( z ) , where S ( z ) = n − X j =1 β j − σ j + zσ j − z , with the usual convention of replacing f ( z ) /z by f ′ (0) when z = 0.Put g ( z ) = zh ( z ). Then g ∈ Hol( D , D ), g (0) = 0 and g ( σ n ) = σ n , because h ( σ n ) = 1. Furthermore we have h ′ ( z ) = 1 (cid:16) − (cid:16) − f ( z ) z (cid:17) S ( z ) (cid:17) (cid:20) z (cid:18) f ′ ( z ) − f ( z ) z (cid:19) + O (cid:18) − f ( z ) z (cid:19)(cid:21) , and thus K -lim z → σ n h ′ ( z ) = σ n ( β n − . Since g ′ ( z ) = h ( z ) + zh ′ ( z ) we get g ′ ( σ n ) = β n . Since h is not a constant we canapply Corollary 3.10 to g obtainingRe (cid:18) h ( z )1 − h ( z ) (cid:19) ≥ β n − (cid:18) σ n + zσ n − z (cid:19) (32)which recalling the definition of h gives exactly (29).If we have equality in one point in (29) we must have equality in one point in(32), and this happens if and only if g is a Blaschke product of degree 2, again byCorollary 3.10. But this occurs if and only if h ∈ Aut( D ); putting this in (31) weget that f ∈ B n +1 by Lemma 2.3.To prove the converse, assume that f ∈ B n +1 with f (0) = 0. Then f ( z ) = zB ( z ),where B ∈ B n , and σ , . . . , σ n ∈ ∂ D are the n distinct solutions of B ( z ) = 1. Let F : C → b C be defined by F ( z ) = 1 + B ( z )1 − B ( z ) − n X j =1 β j − σ j + zσ j − z . Then Re F | ∂ D ≡
0; this implies that Re F (0) = 0, which is exactly1 − | f ′ (0) | | − f ′ (0) | − n X j =1 β j − , and we are done. (cid:3) ULTIPOINT JULIA THEOREMS 15 Multipoint Julia lemmas
Our 2-point Julia lemma has been obtained by applying the classical Julia lemmato the hyperbolic difference quotient f ∗ ( · , w ), which is a holomorphic self-map of D as soon as f is not an automorphism of D . But if we also assume that f is not aBlaschke product of degree 2 then by Proposition 2.5 f ∗ ( · , w ) is not an automor-phism of D , and so its hyperbolic difference quotient is a holomorphic self-map of D to which we may apply the classical Julia lemma, obtaining a 3-point Julia lemma.Clearly this procedure can be iterated; to do so let us introduce some notations. Definition 4.1.
Given k ≥ w , . . . , w k ∈ D the hyperbolic k -th differencequotient ∆ w k ,...,w f of f ∈ Hol( D , D ) is defined by induction by setting ∆ w f ( z ) = f ∗ ( z, w ) and ∆ w k ,...,w f ( z ) = ∆ w k (∆ w k − ,...,w f )( z )for k ≥ w k ,...,w f ∈ Hol( D , D ) as soon as f is not aBlaschke product of degree at most k . Moreover, if σ ∈ ∂ D is such that β f ( σ ) < + ∞ by applying repeatedly Proposition 3.4 we see that β ∆ wk,...,w f ( σ ) is finite. Moreprecisely, β ∆ wk,...,w f ( σ ) can be recursively computed by β ∆ w ,...,wk f ( σ )= β ∆ wk − ,...,w f ( σ ) 1 − | ∆ w k − ,...,w f ( w k ) | | ∆ w k − ,...,w f ( σ ) − ∆ w k − ,...,w f ( w k ) | − − | w k | | σ − w k | , and the non-tangential limit ∆ w k ,...,w f ( σ ) is inductively given by∆ w k ,...,w f ( σ ) = ∆ w k − ,...,w f ( σ ) σ ∆ w k − ,...,w f ( σ ) − ∆ w k − ,...,w f ( w k )∆ w k − ,...,w f ( σ ) − ∆ w k − ,...,w f ( w k ) σ − w k σ − w k . In particular we have a multipoint Julia lemma:
Theorem 4.2.
Given k ≥ , take f ∈ Hol( D , D ) not a Blaschke product of degreeat most k . Let σ ∈ ∂ D be such that β f ( σ ) < + ∞ . Then | ∆ w k ,...,w f ( σ ) − ∆ w k ,...,w f ( z ) | − | ∆ w k ,...,w f ( z ) | ≤ β ∆ wk,...,w f ( σ ) | σ − z | − | z | (33) for all z , w , . . . , w k ∈ D . Moreover, equality occurs for some z , w , . . . , w k ∈ D ifand only if it occurs everywhere if and only if f is a Blaschke product of degree k +1 .Proof. The inequality (33) follows from Theorem 2.11 applied to ∆ w k ,...,w f . If wehave equality in (33) for some z , w , . . . , w k ∈ D again Theorem 2.11 implies that∆ w k ,...,w f ∈ Aut( D ), and then Proposition 2.5 implies that f ∈ B k +1 . Conversely, f ∈ B k +1 implies that ∆ w k ,...,w f ∈ Aut( D ) for all w , . . . , w k ∈ D , and thus wehave equality in (33) for all z , w , . . . , w k ∈ D . (cid:3) The idea is that we can use this multipoint approach to improve known esti-mates by involving higher order derivatives. We shall show two examples of this: astrengthened version of Corollary 3.6 and a generalization of Proposition 3.11.We begin with a reformulation of Theorem 4.2 which gives a far-reaching gener-alization of Corollary 3.6.
Theorem 4.3.
Given k ≥ , let f ∈ Hol( D , D ) be not a Blaschke product of degreeat most k . Take σ ∈ ∂ D with β f ( σ ) < + ∞ . Then β f ( σ ) ≥ k X j =0 − | w j +1 | | σ − w j +1 | j Y h =0 | ∆ w h ,...,w f ( σ ) − ∆ w h ,...,w f ( w h +1 ) | − | ∆ w h ,...,w f ( w h +1 ) | (34) for every w , . . . , w k +1 ∈ D , where ∆ w h ,...,w f = f when h = 0 . Furthermore wehave equality in (34) for some w , . . . , w k +1 ∈ D if and only if we have equality forall w , . . . , w k +1 ∈ D if and only if f is a Blaschke product of degree k + 1 .Proof. One way to prove the assertion is to obtain by induction a formula for β ∆ wk,...,w f ( σ ) applying repeatedly Proposition 3.4, and then to show that, withthis formula, (34) is equivalent to (33). For the sake of variety we shall describe adifferent proof, relying on the classical Julia lemma.We proceed by induction on k . The case k = 0 is β f ( σ ) ≥ − | w | | σ − w | | f ( σ ) − f ( w ) | − | f ( w ) | (35)which is exactly the classical Julia inequality (14). In particular, we have equalityfor some w ∈ D (and hence for all w ∈ D ) if and only if f ∈ Aut( D ).Assume that (34) holds for k −
1, and take w ∈ D . Since f is not a Blaschkeproduct of degree at most k , by Proposition 2.5 ∆ w f is not a Blaschke product ofdegree at most k −
1. So by the inductive hypothesis we have β ∆ w f ( σ ) ≥ k X j =1 − | w j +1 | | σ − w j +1 | j Y h =1 | ∆ w h ,...,w (∆ w f )( σ ) − ∆ w h ,...,w (∆ w f )( w h +1 ) | − | ∆ w h ,...,w (∆ w f )( w h +1 ) | = k X j =1 − | w j +1 | | σ − w j +1 | j Y h =1 | ∆ w h ,...,w f ( σ ) − ∆ w h ,...,w f ( w h +1 ) | − | ∆ w h ,...,w f ( w h +1 ) | for all w , . . . , w k +1 ∈ D , with equality for some (and hence all) w , . . . , w k +1 ∈ D if and only if ∆ w f is a Blaschke product of degree k , that is, by Proposition 2.5,if and only if f is a Blaschke product of degree k + 1.Now Proposition 3.4 yields β f ( σ ) = | f ( σ ) − f ( w ) | − | f ( w ) | (cid:20) − | w | | σ − w | + β ∆ w f ( σ ) (cid:21) ;therefore β f ( σ ) ≥ | f ( σ ) − f ( w ) | − | f ( w ) | × " − | w | | σ − w | + k X j =1 − | w j +1 | | σ − w j +1 | j Y h =1 | ∆ w h ,...,w f ( σ ) − ∆ w h ,...,w f ( w h +1 ) | − | ∆ w h ,...,w f ( w h +1 ) | = k X j =0 − | w j +1 | | σ − w j +1 | j Y h =0 | ∆ w h ,...,w f ( σ ) − ∆ w h ,...,w f ( w h +1 ) | − | ∆ w h ,...,w f ( w h +1 ) | , with equality for some (and hence all) w , . . . , w k +1 ∈ D if and only if f is a Blaschkeproduct of degree k + 1, and we are done. (cid:3) Remark . It is easy to see that the estimate (34) becomes better and better as k increases.Theorem 4.3 has a number of corollaries that it is worthwhile to state. Corollary 4.5.
Given k ≥ , let f ∈ Hol( D , D ) be not a Blaschke product of degreeat most k . Take σ ∈ ∂ D with β f ( σ ) < + ∞ . Then β f ( σ ) ≥ k X j =0 − | w j +1 | | σ − w j +1 | j Y h =0 − | ∆ w h ,...,w f ( w h +1 ) | | ∆ w h ,...,w f ( w h +1 ) | (36) ULTIPOINT JULIA THEOREMS 17 for every w , . . . , w k +1 ∈ D , where ∆ w h ,...,w f = f when h = 0 . Furthermore wehave equality in (36) if and only if f is a Blaschke product of degree k + 1 and w , . . . , w k +1 ∈ D are such that ∆ w h ,...,w f ( w h +1 ) = | ∆ w h ,...,w f ( w h +1 ) | ∆ w h ,...,w f ( σ ) for all h = 0 , . . . , k .Proof. It follows from (34) using the standard estimate | τ − z | ≥ − | z | valid forall τ ∈ ∂ D and z ∈ D , with equality if and only if z = | z | τ . (cid:3) Remark . If take k = 1 and w = w = z and we assume σ = f ( σ ) = 1 then(36) becomes exactly [20, Theorem 2.1]. Corollary 4.7.
Given k ≥ let f ∈ Hol( D , D ) be not a Blaschke product of degreeat most k . Take σ ∈ ∂ D with β f ( σ ) < + ∞ . Then β f ( σ ) ≥ k X j =0 j Y h =0 | ∆ O h f ( σ ) − ∆ O h f (0) | − | ∆ O h f (0) | ≥ k X j =0 j Y h =0 − | ∆ O h f (0) | | ∆ O h f (0) | (37) where O h = (0 , . . . , ∈ D h is the origin of C h , and ∆ O h f = f when h = 0 .Furthermore we have equality on the left of (37) if and only if f is a Blaschkeproduct of degree k + 1 , and on the right if and only if ∆ O h f (0) = | ∆ O h f (0) | ∆ O h f ( σ ) for all h = 0 , . . . , k .Proof. It follows from Theorem 4.3 taking w h = 0 for h = 1 , . . . , k + 1. (cid:3) Remark . (37) for k = 0 is exactly (13), while for k = 1 it yields (21), because∆ f (0) = f h (0). Corollary 4.9.
Let f ∈ Hol( D , D ) be not a Blaschke product, and σ ∈ ∂ D with β f ( σ ) < + ∞ . Then β f ( σ ) ≥ ∞ X j =0 − | w j +1 | | σ − w j +1 | j Y h =0 | ∆ w h ,...,w f ( σ ) − ∆ w h ,...,w f ( w h +1 ) | − | ∆ w h ,...,w f ( w h +1 ) | ≥ ∞ X j =0 − | w j +1 | | σ − w j +1 | j Y h =0 − | ∆ w h ,...,w f ( w h +1 ) | | ∆ w h ,...,w f ( w h +1 ) | (38) for any sequence { w h } ⊂ D , where ∆ w h ,...,w f = f when h = 0 as usual. Inparticular, β f ( σ ) ≥ ∞ X j =0 j Y h =0 | ∆ O h f ( σ ) − ∆ O h f (0) | − | ∆ O h f (0) | ≥ ∞ X j =0 j Y h =0 − | ∆ O h f (0) | | ∆ O h f (0) | . Proof.
It follows from Theorem 4.3, Corollary 4.7 and Remark 4.4. (cid:3)
A natural question, that we leave open, is whether the first inequality in (38)actually is an equality, at least when f is an infinite Blaschke product.To give an idea of the actual content of (34), let us reformulate it for small valuesof k and particular values of w , . . . , w k +1 .For k = 0 we get (35), that we already noticed to be equivalent to the classicalJulia lemma.For k = 1 we get β f ( σ ) ≥ | f ( σ ) − f ( w ) | − | f ( w ) | (cid:20) − | w | | σ − w | + 1 − | w | | σ − w | | ∆ w f ( σ ) − ∆ w f ( w ) | − | ∆ w f ( w ) | (cid:21) . Since ∆ f (0) = f h (0) and∆ f ( σ ) = f ( σ ) σ f ( σ ) − f (0) f ( σ ) − f (0) , (39)putting w = w = 0 we obtain β f ( σ ) ≥ | f ( σ ) − f (0) | − | f (0) | (cid:12)(cid:12)(cid:12) f ( σ ) σ f ( σ ) − f (0) f ( σ ) − f (0) − f h (0) (cid:12)(cid:12)(cid:12) − | f h (0) | , which is a slightly more precise version of (21). If moreover f (0) = 0 we findagain (25).The case k = 2 with w = w = w = 0 yields β f ( σ ) ≥ | f ( σ ) − f (0) | − | f (0) | × (cid:12)(cid:12)(cid:12) f ( σ ) σ f ( σ ) − f (0) f ( σ ) − f (0) − f h (0) (cid:12)(cid:12)(cid:12) − | f h (0) | (cid:18) | ∆ , f ( σ ) − (∆ f ) h (0) | − | (∆ f ) h (0) | (cid:19) , (40)where we have used the equality ∆ , f (0) = (∆ f ) h (0). To compute (∆ f ) h (0)first all we notice that(∆ w f ) ′ ( z ) = f ′ ( z )( z − w ) − (cid:0) f ( z ) − f ( w ) (cid:1) ( z − w ) · − w z − f ( w ) f ( z )+ f ( z ) − f ( w ) z − w · − w (cid:0) − f ( w ) f ( z ) (cid:1) + (1 − w z ) f ( w ) f ′ ( z ) (cid:0) − f ( w ) f ( z ) (cid:1) , and so(∆ w f ) h (0) = 11 − | ∆ w f (0) | (cid:20) f ( w ) − f (0) − f ′ (0) w w · − f ( w ) f (0)+ f ( w ) − f (0) w f ( w ) f ′ (0) − w (cid:0) − f ( w ) f (0) (cid:1)(cid:0) − f ( w ) f (0) (cid:1) (cid:21) . In particular putting w = 0 we get(∆ f ) h (0) = 11 − | f h (0) | (cid:20) f ′′ (0)2 (cid:0) − | f (0) | (cid:1) + f (0) f ′ (0) (cid:0) − | f (0) | (cid:1) (cid:21) = 11 − | f h (0) | (cid:20) f ′′ (0)2 (cid:0) − | f (0) | (cid:1) + f (0) f h (0) (cid:21) . (41)Applying (39) to ∆ f we also get∆ , f ( σ ) = f ( σ ) f ( σ ) − f (0) f ( σ ) − f (0) ∆ f ( σ ) − f h (0)∆ f ( σ ) − f h (0) , and we have all the terms appearing in (40). In particular, β f ( σ ) ≥ − | f (0) | | f (0) | (cid:20) − | f h (0) | | f h (0) |
21 + | (∆ f ) h (0) | (cid:21) , and thus if f (0) = 0 we obtain β f ( σ ) ≥ − | f ′ (0) | )1 + | f ′ (0) | + | f ′′ (0) | −| f ′ (0) | )ULTIPOINT JULIA THEOREMS 19 that improves (25). If moreover f ′ (0) = 0 we also get β f ( σ ) ≥ | f ′′ (0) | ≥ | f ′′ (0) | = | ∆ , f (0) | ≤ k = 3 and w = w = w = w = 0 yields β f ( σ ) ≥ − | f (0) | | f (0) | (cid:20) − | f h (0) | | f h (0) | (cid:18) − | (∆ f ) h (0) | | (∆ f ) h (0) |
21 + | (∆ , f ) h (0) | + 1 (cid:19) + 1 (cid:21) , where(∆ , f ) h (0) = 11 − | (∆ f ) h (0) | (cid:20) (∆ f ) ′′ (0)2 (cid:0) − | f h (0) | (cid:1) + f h (0)(∆ f ) h (0) (cid:21) , with(∆ f ) ′′ (0) = 11 − | f (0) | (cid:20) f ′′′ (0) + 2 f (0) f h (0) f ′′ (0) + f (0) f h (0) f ′ (0) (cid:21) . We now proceed toward the promised generalization of Proposition 3.11. Letus start with the following reformulation of the case k = 2 of Theorem 4.2 validwhen f (0) = 0: Proposition 4.10.
Let f ∈ Hol( D , D ) , not a Blaschke product of degree at most 2,and σ ∈ ∂ D be such that β f ( σ ) < + ∞ . Assume that f (0) = 0 . Then (cid:12)(cid:12)(cid:12)(cid:12) σ (cid:0) f ( σ ) σ − f ′ (0) (cid:1) − f ′ (0) f ( σ ) σ − z ( f ( z ) z − f ′ (0) ) − f ′ (0) f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) z ( f ( z ) z − f ′ (0) ) − f ′ (0) f ( z ) z (cid:12)(cid:12)(cid:12)(cid:12) ≤ − | f ′ (0) | (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ′ (0) (cid:12)(cid:12)(cid:12) (cid:0) β f ( σ ) − (cid:1) − | σ − z | − | z | (42) for z = 0 , and (cid:12)(cid:12)(cid:12)(cid:12) σ (cid:0) f ( σ ) σ − f ′ (0) (cid:1) − f ′ (0) f ( σ ) σ − f ′′ (0)2 (cid:0) −| f ′ (0) | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) f ′′ (0)2 (cid:0) −| f ′ (0) | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ≤ − | f ′ (0) | (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ′ (0) (cid:12)(cid:12)(cid:12) (cid:0) β f ( σ ) − (cid:1) − . (43) Moreover, equality occurs in (42) at some z ∈ D \ { } or in (43) if and only if italways occurs if and only f is a Blaschke product of degree 3.Proof. We would like to apply Theorem 4.2 with k = 2 and w = w = 0.First of all β ∆ , f ( σ ) = β ∆ f ( σ ) 1 − | ∆ f (0) | | ∆ f ( σ ) − ∆ f (0) | − − | f ′ (0) | (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ′ (0) (cid:12)(cid:12)(cid:12) (cid:0) β f ( σ ) − (cid:1) − , where we used f (0) = 0 and ∆ f (0) = f h (0) = f ′ (0). Next recalling (27) we get∆ , f ( z ) = f ( z ) z − f ′ (0) z − f ′ (0) f ( z ) z if z = 0 , f ′′ (0)2(1 −| f ′ (0) | ) if z = 0where we used (41). The assertion then follows from Theorem 4.2. (cid:3) Notice that (26) is equivalent to saying that the right-hand side of (43) is non-negative; so (43) is an improvement of (26). If f (0) = f ′ (0) = 0 (42) and (43) simplify becoming (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ( z ) z (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) f ( z ) z (cid:12)(cid:12)(cid:12) ≤ (cid:0) β f ( σ ) − (cid:1) | σ − z | − | z | for z = 0 and (cid:12)(cid:12)(cid:12) f ( σ ) σ − f ′′ (0) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12) f ′′ (0) (cid:12)(cid:12) ≤ β f ( σ ) − . These formulas suggest the following
Proposition 4.11.
Let f ∈ Hol( D , D ) and σ ∈ ∂ D be such that β f ( σ ) < + ∞ .Given k ≥ , assume that f is not a Blaschke product of degree at most k ≥ , andthat f (0) = · · · = f ( k − (0) = 0 . Then (cid:12)(cid:12)(cid:12) f ( σ ) σ k − f ( z ) z k (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) f ( z ) z k (cid:12)(cid:12)(cid:12) ≤ (cid:0) β f ( σ ) − k (cid:1) | σ − z | − | z | (44) for z = 0 and (cid:12)(cid:12)(cid:12) f ( σ ) σ k − k ! f ( k ) (0) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12) k ! f ( k ) (0) (cid:12)(cid:12) ≤ β f ( σ ) − k . (45) Moreover, equality occurs in (44) at some z ∈ D \ { } or in (45) if and only if italways occurs if and only f is a Blaschke product of degree k + 1 .Proof. By induction it is easy to prove that∆ O k f ( z ) = ( f ( z ) z k if z = 0 , k ! f ( k ) (0) if z = 0 , (46)and that β ∆ O k f ( σ ) = β f ( σ ) − k . The assertion then follows from Theorem 4.2. (cid:3) In particular, if f ( σ ) = σ k the left-hand sides of (44) and (45) become indepen-dent of σ . This suggests that we might obtain a generalization of Proposition 3.11with multiple fixed points. It turns out that this is easy when the multiple fixedpoint is the origin (see Corollary 4.13 below), but the statement and the proof ofthe general result when the multiple fixed point is not the origin are considerablyharder: Theorem 4.12.
Let f ∈ Hol( D , D ) . Given k ≥ , assume that f is not a Blaschkeproduct of degree at most k and that there exists z ∈ D such that f ( z ) = z and f ′ ( z ) = . . . = f ( k − ( z ) = 0 . Take σ , . . . , σ n ∈ ∂ D distinct points such that β f ( σ j ) < + ∞ and f ( σ j ) = (cid:16) σ j − z − z σ j (cid:17) k + z z (cid:16) σ j − z − z σ j (cid:17) k , (47) for j = 1 , . . . , n . Then n X j =1 (cid:16) ( f ( σ j ) − σ j ) z | f ( σ j ) − z | (cid:17) β f ( σ j ) − k ≤ − (cid:12)(cid:12)(cid:12) f ( k ) ( z ) k ! (1 − | z | ) k − (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − f ( k ) ( z ) k ! (1 − | z | ) k − (cid:12)(cid:12)(cid:12) , (48) with equality if and only if f is a Blaschke product of degree n + k . ULTIPOINT JULIA THEOREMS 21
Proof.
The fact that z is multiple fixed point of f of order k is equivalent to sayingthat we can write f ( z ) = z + 1 k ! f ( k ) ( z )( z − z ) k + O (cid:0) ( z − z ) k +1 (cid:1) . We claim that then∆ ( z ) h f ( z ) = 1 k ! f ( k ) ( z )(1 − | z | ) h − ( z − z ) k − h + O (cid:0) ( z − z ) k − h +1 (cid:1) (49)for all h = 1 , . . . , k , where ( z ) h = ( z , . . . , z ) ∈ D h . We proceed by induction. For h = 1 we have∆ z f ( z ) = f ( z ) − z z − z − z z − z f ( z ) = 1 k ! f ( k ) ( z )( z − z ) k − (cid:2) O ( z − z ) (cid:3) = 1 k ! f ( k ) ( z )( z − z ) k − + O (cid:0) ( z − z ) k (cid:1) , as claimed. Assume that (49) holds for 1 ≤ h − ≤ k −
1. Then ∆ ( z ) h − f ( z ) = 0yields∆ ( z ) h f ( z ) = ∆ ( z ) h − f ( z ) z − z (1 − z z )= 1 k ! f ( k ) ( z )(1 − | z | ) h − ( z − z ) k − h (cid:2) − | z | + O ( z − z ) (cid:3) = 1 k ! f ( k ) ( z )(1 − | z | ) h − ( z − z ) k − h + O (cid:0) ( z − z ) k − h +1 (cid:1) , as wanted.Now we claim that β ∆ ( z0 ) h f ( σ j ) = 1 − | z | | σ j − z | " (cid:0) f ( σ j ) − σ j (cid:1) z | f ( σ j ) − z | ! β f ( σ j ) − h (50)for all h = 1 , . . . , k and j = 1 , . . . , n . We again proceed by induction on h . For h = 1 Proposition 3.4 yields β ∆ z f ( σ j ) = 1 − | z | | f ( σ j ) − z | β f ( σ j ) − − | z | | σ j − z | = 1 − | z | | σ j − z | (cid:0) β f ( σ ) − (cid:1) + (cid:18) − | z | | f ( σ j ) − z | − − | z | | σ j − z | (cid:19) β f ( σ )= 1 − | z | | σ j − z | (cid:18) β f ( σ ) − | σ j − z | − | f ( σ j ) − z | | f ( σ j ) − z | β f ( σ j ) (cid:19) = 1 − | z | | σ j − z | " (cid:0) f ( σ j ) − σ j (cid:1) z | f ( σ j ) − z | ! β f ( σ j ) − , as claimed. Assume that (50) holds for 1 ≤ h − ≤ k −
1. Using again the factthat ∆ ( z ) h − f ( z ) = 0 we get β ∆ ( z0 ) h f ( σ j ) = β ∆ ( z0 ) h − f ( σ j ) − − | z | | σ j − z | = 1 − | z | | σ j − z | " (cid:0) f ( σ j ) − σ j (cid:1) z | f ( σ j ) − z | ! β f ( σ j ) − h , and we are done.We need one more preliminary computation. We claim that∆ ( z ) h f ( σ j ) = f ( σ j ) σ hj (cid:18) σ j − z σ j − z (cid:19) h f ( σ j ) − z f ( σ j ) − z (51) for j = 1 , . . . , n and h = 1 , . . . , k . As always, we argue by induction on h . For h = 1 we have∆ z f ( σ j ) = f ( σ j ) − z − z f ( σ j ) 1 − z σ j σ j − z = f ( σ j ) σ j σ j − z σ j − z f ( σ j ) − z f ( σ j ) − z as claimed. Assume that (51) holds for 1 ≤ h − ≤ k −
1. Recalling that∆ ( z ) h − f ( z ) = 0 we obtain∆ ( z ) h f ( σ j ) = ∆ ( z ) h − f ( σ j ) 1 − z σ j σ j − z = f ( σ j ) σ hj (cid:18) σ j − z σ j − z (cid:19) h f ( σ j ) − z f ( σ j ) − z , and (51) is proved. In particular, we have ∆ ( z ) k f ( σ j ) = 1 if and only if1 − f ( σ j ) z f ( σ j ) − z = (cid:18) σ j − z − z σ j (cid:19) k if and only if f ( σ j ) = (cid:16) σ j − z − z σ j (cid:17) k + z z (cid:16) σ j − z − z σ j (cid:17) k . In other words, condition (47) is just another way of writing ∆ ( z ) k f ( σ j ) = 1.We can now apply Theorem 4.2. Recalling the assumption ∆ ( z ) k f ( σ j ) = 1 weget | − ∆ ( z ) k f ( z ) | − | ∆ ( z ) k f ( z ) | ≤ − | z | | σ j − z | " (cid:0) f ( σ j ) − σ j (cid:1) z | f ( σ j ) − z | ! β f ( σ j ) − k | σ j − z | − | z | (52)for all z ∈ D and j = 1 , . . . , n , where∆ ( z ) k f ( z ) = 1 k ! f ( k ) ( z )(1 − | z | ) k − + O ( z − z )by (49). Furthermore, equality in (52) holds in one point (and hence everywhere)if and only if f is a Blaschke product of degree k + 1.We now claim thatRe (cid:18) ( z ) k f ( z )1 − ∆ ( z ) k f ( z ) (cid:19) ≥ n X j =1 | σ j − z | − | z | (cid:16) ( f ( σ j ) − σ j ) z | f ( σ j ) − z | (cid:17) β f ( σ j ) − k Re (cid:18) σ j + zσ j − z (cid:19) , (53)with equality in one point (and hence everywhere) if and only if f is a Blaschkeproduct of degree n + k .We argue by induction on n . For n = 1 (53) is exactly equivalent to (52). Assumethat (53) holds for n −
1. In particular we haveRe g ( z )1 − g ( z ) − n − X j =1 a j σ j + zσ j − z ≥ , (54)with equality in one point (and hence everywhere) if and only if f ∈ B n + k − , where g = ∆ ( z ) k f and a j = | σ j − z | − | z | (cid:16) ( f ( σ j ) − σ j ) z | f ( σ j ) − z | (cid:17) β f ( σ j ) − k = 1 β ∆ ( z0 ) k f ( σ j ) > . ULTIPOINT JULIA THEOREMS 23
Therefore we can find h ∈ Hol( D , C ) with h ( D ) ⊂ D so that1 + g ( z )1 − g ( z ) − n − X j =1 a j σ j + zσ j − z = 1 + h ( z )1 − h ( z ) . Notice that either h ( D ) ⊆ D or h ≡ e iθ ∈ ∂ D , and the latter case occurs if and onlyif we have equality in (54).If h ≡ e iθ Lemma 2.3 implies that g ∈ B n − . So g is a rational function ofdegree n −
1; but we are assuming that the equation g ( z ) = 1 has at least n distinctsolutions, contradiction.So h ∈ Hol( D , D ). Since g ( σ n ) = 1, β g ( σ n ) = a n and g ′ ( σ n ) = σ n /a n , wherethe latter equality follows from (19), a quick computation yields h ( σ n ) = 1 and h ′ ( σ n ) = σ n /a n .Put ˜ h ( z ) = zh ( z ). Then we have ˜ h (0) = 0, ˜ h ( σ n ) = σ n and β ˜ h ( σ n ) = a n + 1. Sowe can apply Corollary 3.10 to ˜ h obtainingRe (cid:18) h ( z )1 − h ( z ) (cid:19) ≥ a n Re (cid:18) σ n + zσ n − z (cid:19) , with equality in one point (and hence everywhere) if and only if h ∈ Aut( D ).Recalling (54) and Lemma 2.3 we see that we have proven (53), with equality inone point (and hence everywhere) implying that g is a Blaschke product of degree n ,and thus that f is a Blaschke product of degree n + k , by Proposition 2.5.In particular, (48) follows taking z = z in (53), and equality there impliesthat f ∈ B n + k .To prove the converse, assume that f ∈ B n + k , so that g = ∆ ( z ) k f ∈ B n , and σ , . . . , σ n ∈ ∂ D are the n distinct solutions of g ( z ) = 1. Let F : C → b C be definedby F ( z ) = 1 + g ◦ φ z ( z )1 − g ◦ φ z ( z ) − n X j =1 a j σ j + φ z ( z ) σ j − φ z ( z ) , where φ z ( z ) = ( z − z ) / (1 − z z ); notice that g ◦ φ z is still a Blaschke productthanks to Lemma 2.2. Then Re F | ∂ D ≡
0; this implies Re F (0) = 0, which givesexactly1 − | g ( z ) | | − g ( z ) | = n X j =1 a j − | z | | σ j − z | = n X j =1 (cid:16) ( f ( σ j ) − σ j ) z | f ( σ j ) − z | (cid:17) β f ( σ j ) − k , and we are done. (cid:3) Corollary 4.13.
Let f ∈ Hol( D , D ) . Given k ≥ , assume that f is not a Blaschkeproduct of degree at most k and that f (0) = · · · = f ( k − (0) = 0 . Take σ , . . . , σ n ∈ ∂ D distinct points such that f ( σ j ) = σ kj and β f ( σ j ) < + ∞ for j = 1 , . . . , n . Then n X j =1 β f ( σ j ) − k ≤ − (cid:12)(cid:12)(cid:12) f ( k ) (0) k ! (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) − f ( k ) (0) k ! (cid:12)(cid:12)(cid:12) , with equality if and only if f is a Blaschke product of degree n + k .Proof. It immediately follows from Theorem 4.12 applied with z = 0.Alternatively, we can apply directly Proposition 3.11 to g = ∆ O k − f . Indeed,(46) shows that g ( z ) = f ( z ) /z k − for z = 0 and g (0) = 0; in particular, g ′ (0) = k ! f ( k ) (0). Moreover, g ( σ j ) = σ j and g ′ ( σ j ) = β f ( σ j ) − ( k −
1) for all j = 1 , . . . , n ;hence the assertion follows immediately from (28). (cid:3) Notice that when k = 1 the condition (47) becomes f ( σ j ) = σ j − z − z σ j + z z σ j − z − z σ j = σ j . So (48) reduces to (28), and thus Theorem 4.12 for k = 1 recovers exactly Propo-sition 3.11. Remark . We have seen that Proposition 3.11 for a generic fixed point z followed immediately from the case z = 0, just replacing the map f by the com-position φ z ◦ f ◦ φ z . Such an approach however does not allow to easily deduceTheorem 4.12 from Corollary 4.13 because the boundary dilation coefficient de-pends in a complicated way on the higher order derivatives, and so we need theiterated hyperbolic difference quotients to keep everything under control. References [1] M. Abate,
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Marco Abate, Dipartimento di Matematica, Universit`a di Pisa, Largo Pontecorvo 5,I-56127 Pisa, Italy.
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