HYPERBOLIC DERIVATIVES AND GENERALIZED SCHWARZ-PICK ESTIMATES

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 11, Pages S (4) Article electronically published on May 12, 24 HYPERBOLIC DERIVATIVES AND GENERALIZED SCHWARZ-PICK ESTIMATES PRATIBHA GHATAGE AND DECHAO ZHENG (Communicated by Joseph A. Ball) Abstract. In this paper we use the beautiful formula of Faa di Bruno for the nth derivative of composition of two functions to obtain the generalized Schwarz-Pick estimates. By means of those estimates we show that the hyperbolic derivative of an analytic self-map of the unit disk is Lipschitz with respect to the pseudohyperbolic metric. 1. Introduction For each z D, letϕ z denote the Möbius transformation of D, ϕ z = z w 1 zw, for w D. The pseudo-hyperbolic distance on D is defined by ρ(z,w) = ϕ z (w), z,w D. The pseudohyperbolic distance is Möbius invariant, that is, ρ(gz, gw) =ρ(z,w), for all g Aut(D), the Möbius group of D, andallz,w D. It has the following useful property: (1.1) 1 ρ(z,w) 2 = (1 z 2 )(1 w 2 ) 1 zw 2. The Bergman metric on D is the hyperbolic metric whose element of length is ds = dz 1 z 2. In this metric every rectifiable arc γ has a length dz γ 1 z 2. It is easy to show that the induced distance on D is given by for z,w D. β(z,w) = 1 2 log 1+ρ(z,w) 1 ρ(z,w), Received by the editors July 9, 23 and, in revised form, August 12, Mathematics Subject Classification. Primary 3C8. The second author was supported in part by the National Science Foundation. 339 c 24 American Mathematical Society

2 331 PRATIBHA GHATAGE AND DECHAO ZHENG Let ϕ be an analytic self-map of the unit disk. By the classical Schwarz-Pick Lemma [2], [5], ϕ decreases the hyperbolic distance between two points and the hyperbolic length of an arc. The explicit inequality is ϕ(z 1) ϕ(z 2 ) 1 ϕ(z 1 )ϕ(z 2 ) z 1 z 2 1 z 1 z 2 for any z 1,z 2 in D. Inparticular, ϕ (z) (1.2) 1 ϕ(z) z 2 for z in D. Let τ ϕ (z) = (1 z 2 )ϕ (z) 1 ϕ(z) 2. Then τ ϕ (z) 1, for all z D. Nontrivial equality τ ϕ (z) =1holdsforsomez D only when ϕ is a fractional linear transformation e iθ ϕ a (z). For each z D, the hyperbolic derivative of ϕ at z is defined by β(ϕ(z),ϕ(w)). β(z,w) β(z,w) In Section 3 we will show that the hyperbolic derivative of ϕ equals τ ϕ (z) and that τ ϕ (z) is Lipschitz with respect to the pseudohyperbolic metric. Hyperbolic derivatives have been used in studying composition operators on the Bloch space [7], [9] and [1]. Recently, MacCluer, Stroethoff, and Zhao [8] used the formula of Faa di Bruno and the theory of the weighted composition operators [11] to obtain the generalized Schwarz-Pick estimates: (1.3) sup z D (1 ϕ(z) 2 < ) for each analytic self-map ϕ of the unit disk. We obtain the following generalized Schwarz-Pick estimates: for each <r<1 and each positive integer n, thereisa positive constant M n,r such that for each analytic self-map ϕ of the unit disk: (1 w 2 ) ϕ (w) (1.4) 1 ϕ(z) 2 M n,r max ρ(w,z)<r 1 ϕ(w) 2, for z in D. Clearly, Combining (1.2) with (1.4) gives (1.3). Moreover, (1.4) directly leads to the result [8] that if ϕ is in the little Bloch class, then z 1 1 ϕ(z) 2 = for each n. The main tool is the beautiful formula of Faa di Bruno [13] for the nth derivative of the composition of two functions. Based on the generalized Schwarz-Pick estimates we will show in Section 3 that τ ϕ,n (z) = (1 z 2 ) n ϕ (n) (z) (1 ϕ(z) 2 ) is Lipschitz with respect to the pseudohyperbolic metric. Thus τ ϕ,n (z) admits a continuous extension to the set of nontrivial Gleason parts of the maximal ideal space of H.

3 HYPERBOLIC DERIVATIVES AND SCHWARZ-PICK ESTIMATES Generalized Schwarz-Pick estimates In this section, we will present a proof of the generalized Schwarz-Pick estimates. The generalized Schwarz-Pick estimates will be used in the proof of Theorem 6. The main tool is the beautiful formula of Faa di Bruno, which deals with the nth derivative of composition of an analytic function f on the unit disk with a self-map ϕ of the the unit disk [13]. Theorem 1 (The Formula of Faa di Bruno). If ϕ is an analytic function from the unit disk to the unit disk and if f is an analytic function on the unit disk, then (f ϕ) (n) (z) = n! k 1!k 2! k n! f (k) (ϕ(z)) ( ϕ(j) (z) where k = k k n and the sum is over all k 1,...,k n for which k 1 +2k nk n = n. ) kj The following result is well known [12]. Theorem 2. We include a proof to motivate our Proposition 1. If ϕ is a univalent analytic self-map of D, then for all z D. (1 z 2 ) ϕ (z) 1 ϕ (z) Proof. For a fixed z in D, leth be the Koebe transform of ϕ, w+z ϕ( 1+zw h(w) = ) ϕ(z) (1 z 2 )ϕ (z). Then h() = and h () = 1. It follows from Bieberbach s theorem ([12], page 8) that h () 4. On the other hand, an easy computation gives h () = 1 2 (1 z 2 ) ϕ (z) ϕ (z) z. Hence 1 2 (1 z 2 ) ϕ (z) ϕ z 4. (z) Since z 1, we conclude that (1 z 2 ) ϕ (z) 1 ϕ (z). This completes the proof. (2.1) As a consequence of the proposition, we have (1 z 2 ) 2 ϕ (z) 1 ϕ(z) 2 1(1 z 2 ) ϕ (z) 1 ϕ(z) 2 for all z D if ϕ is a univalent self-map of the unit disk.

4 3312 PRATIBHA GHATAGE AND DECHAO ZHENG Example. Let b be an interpolating Blaschke product with zeros {z n } in the unit disk and ϕ = b 2. Clearly, ϕ (z n ) = and ϕ (z n ) = 2[b (z n )] 2. Let δ = inf zn (1 z n 2 ) b (z n ). Thus (1 z n 2 ) ϕ (z n ) 1 ϕ(z n ) 2 =2(1 z n 2 )[b (z n )] 2 2δ b (z n ) 2δ2 1 z n 2. So the inequality (2.1) does not hold for some analytic self-maps of the unit disk. But by means of the formula of Faa di Bruno we still have the generalized Schwarz- Pick estimates: Theorem 2. For each positive integer n and each number <r<1, thereisa positive constant M n,r such that for each analytic self-map ϕ of the unit disk, (1 w 2 ) ϕ (w) 1 ϕ(z) 2 M n,r max ρ(w,z)<r 1 ϕ(w) 2 for z in D. As we mentioned in the introduction, by the Schwarz-Pick estimates (1.2), we have (1 w 2 ) ϕ (w) 1 ϕ(w) 2 1. Thus Theorem 2 implies the following generalized Schwarz-Pick Estimates [8]. Theorem 3. For each n, there is a positive constant M n such that for each analytic self-map ϕ of the unit disk, 1 ϕ(z) 2 M n, for z in D. If ϕ is in the little Bloch class, i.e., (1 z 2 ) ϕ (z) 1 ϕ(z) 2 as z 1, then noting that for the given < s < 1, for every w D with ρ(w, z) <s, w 1as z 1, Theorem 2 gives the following result in [8]. Theorem 4. Let ϕ be an analytic self-map of the unit disk. If (1 z 2 ) ϕ (z) z 1 1 ϕ(z) 2 =, then for each n, z 1 1 ϕ(z) 2 =. Proof of Theorem 2. For a fixed z in D, letg = ϕ ϕ z. Clearly, g() = ϕ(z). By the formula of Faa di Bruno, we have g (n) (w) = n! k 1!k 2! k n! ϕ(k) (ϕ z (w)) ( ϕ(j) z (w) where k = k k n and the sum is over all k 1,...,k n for which k 1 +2k nk n = n. ) kj

5 HYPERBOLIC DERIVATIVES AND SCHWARZ-PICK ESTIMATES 3313 Evaluating the value of g (n) at gives g (n) () = n! k 1!k 2! k n! ϕ(k) (ϕ z ()) z () ) kj. ( ϕ(j) Noting that ϕ z () = z and ϕ z (j) (w) = (1 z 2 ) z j 1 (1 zw) j 1,wehave g (n) () = n! k 1!k 2! k n! ϕ(k) (z) ( (1 z 2 ) z j 1 ) kj = ( 1) k n! k 1!k 2! k n! ϕ(k) (z)(1 z 2 ) k z n k. The last equality follows from k k n = k and k 1 +2k nk n = n. Thus ( 1) n (1 z 2 ) n ϕ (n) (z) =g (n) () ( 1) k n! k 1!k 2! k n! ϕ(k) (z)(1 z 2 ) k z n k. k<n So + k<n 1 ϕ(z) 2 g(n) () 1 ϕ(z) 2 n! ϕ (k) (z) (1 z 2 ) k z n k k 1!k 2! k n! 1 ϕ(z) 2. Let M k (z) = ϕ(k) (z) (1 z 2 ) k 1 ϕ(z). The above inequality gives 2 M n (z) g(n) () 1 ϕ(z) 2 + n! k 1!k 2! k n! M k(z). k<n We need to estimate g(n) () 1 ϕ(z). 2 Let λ = g(), h= ϕ λ g. Then h is still an analytic self-map of the unit disk, h() =, and h 1. Since ϕ λ ϕ λ (z) =z, we obtain g = ϕ λ h. The formula of Faa di Bruno again gives g (n) (w) = n! n k 1!k 2! k n! ϕ(k) λ (h(w)) ( h(j) (w) ) kj where k = k k n and the sum is over all k 1,...,k n for which k 1 +2k nk n = n. Evaluating g (n) at gives g (n) () = n n! k 1!k 2! k n! ϕ(k) λ () ( h(j) () ) kj since h() =. Noting ϕ (k) λ (w) = (1 λ 2 ) λ k 1 k!(1 λw) k 1, the above equality leads to g (n) () = n! k 1!k 2! k n! [ (1 λ 2 ) λ k 1 k!] ( h(j) () ) kj.

6 3314 PRATIBHA GHATAGE AND DECHAO ZHENG Hence and So Let a n = k<n g (n) () 1 g() 2 n! k 1!k 2! k n! λ k 1 k! n! k 1!k 2! k n!. Sofarwehaveshown ( h(j) () ) kj. M n (z) a n max k<n M k(z)+ n! k 1!k 2! k n! λ k 1 k! ( h(j) () ) kj. Note that h = ϕ λ g, g = ϕ ϕ z, and λ = g() = ϕ(z). Then h (w) = (1 λ 2 )g (w) (1 λg(w)) 2 Let <r<1. Thus h (w) = h (j) () = r (j 1) (j 1)! 1 2π h (j) () r (j 1) (j 1)! 1 2π r (j 1) (j 1)! 1 2π r (j 1) (j 1)! 1 2π h (j) () (j 1)! wj 1. 2π 2π 2π 2π h (re iθ )e i(j 1)θ dθ. h (re iθ ) dθ r (j 1) (1 r 2 ) 1 (j 1)! max w r 1 C r (j 1)! r max (j 1) ρ(z,u) r (1 λ 2 ) g (re iθ ) 1 λg(re iθ ) 2 dθ (1 ϕ λ (g(re iθ ) 2 ) g (re iθ ) 1 g(re iθ ) 2 dθ (1 w 2 ) g (w) 1 g(w) 2 (1 u 2 ) ϕ (u) 1 ϕ(u) 2 for some constant C r >. The third inequality follows from (1 λ 2 )(1 g(re iθ ) 2 ) 1 λg(re =1 ϕ iθ ) 2 λ (g(re iθ )) 2. The last inequality follows from making the change of variable u = ϕ z (w) andthe fact that (1 w 2 ) g (w) =(1 w 2 ) ϕ (ϕ z (w))ϕ z (w) = (1 w 2 )(1 z 2 ) 1 zw 2 ϕ (ϕ z (w)) =(1 ϕ z (w) 2 ) ϕ (ϕ z (w)). Hence h (j) () [jr (j 1) (1 r 2 )] 1 (1 u 2 ) ϕ (u) max ρ(z,u) r 1 ϕ(u) 2. The Schwarz-Pick estimate gives (1 u 2 ) ϕ (u) 1 ϕ(u) 2 1

7 HYPERBOLIC DERIVATIVES AND SCHWARZ-PICK ESTIMATES 3315 for each u D. Thus n! k 1!k 2! k n! λ k 1 k! ( h(j) () n! k 1!k 2! k n! λ k 1 k!r k n (1 r 2 ) k (1 u 2 ) ϕ (u) max ρ(z,u) r 1 ϕ(u) 2. Let b n,r = n! k 1!k 2! k n! k!rk n (1 r 2 ) k. The above inequality gives M n (z) a n max M k(z)+b n,r k<n By the induction, we conclude that to complete the proof. M n (z) M n,r max ρ(z,u) r ) kj max ρ(z,u) r (1 u 2 ) ϕ (u) 1 ϕ(u) 2. (1 u 2 ) ϕ (u) 1 ϕ(u) 2 3. Hyperbolic derivatives In this section we will first show that the hyperbolic derivative of an analytic self-map ϕ of the unit disk equals τ ϕ (z). Then we will show that τ ϕ (z) islipschitz with respect to the pseudo-hyperbolic metric. Theorem 5. Let ϕ : D D be an analytic self-map. Then, for each point z D, the hyperbolic derivative of ϕ is equal to β(ϕ(z),ϕ(w)) = τ ϕ (z). β(z,w) β(z,w) Proof. Assume that ϕ is not constant. For each fixed z D, ρ(ϕ(z),ϕ(w)) converges to zero as β(w, z) converges to zero because ϕ is continuous in D and ϕ(z) < 1. Thus β(ϕ(z),ϕ(w)) β(ϕ(z),ϕ(w)) ρ(ϕ(z),ϕ(w)) ρ(z,w) = β(z,w) β(z,w) β(z,w) ρ(ϕ(z),ϕ(w)) ρ(z,w) β(z,w). Both the first and third factors of the product on the right converge to one. Now the second factor is ρ(ϕ(z),ϕ(w)) ϕ(z) ϕ(w) 1 zw = ρ(z,w) z w 1 ϕ(z)ϕ(w). Thus So This completes the proof. ρ(ϕ(z),ϕ(w)) = ϕ (z) (1 z 2 ) β(z,w) ρ(z,w) 1 ϕ(z) 2. β(ϕ(z),ϕ(w)) β(z,w) β(z,w) = ϕ (z) (1 z 2 ) 1 ϕ(z) 2. For each n, define τ ϕ,n (z) = (1 z 2 ) n ϕ (n) (z) 1 ϕ(z) 2.

8 3316 PRATIBHA GHATAGE AND DECHAO ZHENG Theorem 6. Let ϕ be an analytic self-map of the unit disk D. Then for each n, τ ϕ,n (z) is Lipschitz with respect to the pseudohyperbolic metric. More precisely, τ ϕ,n (z) τ ϕ,n (w) C n ρ(z,w) for any z,w D. HereC n is a positive constant only depending on n. Proof. Suppose that f is a differentiable function on the unit disk. Let z f = f z and z f = f z. Note that τ ϕ,n(z) is differentiable on the unit disk. Easy calculations give z τ ϕ,n (z) = zn(1 z 2 ) n 1 ϕ (n) (z)(1 ϕ(z) 2 )+(1 z 2 ) (n) ϕ (n) (z)ϕ (z)ϕ(z) (1 ϕ(z) 2 ) 2 and Thus z τ ϕ,n (z) = z τ ϕ,n (z) 1 (1 ϕ(z) 2 ) 2 {[(1 z 2 ) n ϕ (n+1) (z) zn(1 z 2 ) n 1 ϕ (n) (z)] (1 ϕ(z) 2 )+(1 z 2 ) n ϕ (n) (z)ϕ (z)ϕ(z)}. 1 1 z 2 [n(1 z 2 ) n ϕ (n) (z) 1 ϕ(z) 2 + ϕ(z) [ (1 z 2 ) ϕ (z) 1 ϕ(z) 2 ][ (1 z 2 ) n ϕ (n) (z) 1 ϕ(z) 2 ] (n +1)M n 1 z 2, where the last inequality follows from Theorem 3, and z τ ϕ,n (z) 1 1 z 2 { (1 z 2 ) n+1 ϕ (n+1) (z) 1 ϕ(z) 2 + n (1 z 2 ) n ϕ n (z) 1 ϕ(z) 2 + ϕ(z) [ (1 z 2 ) n ϕ (n) (z) 1 ϕ(z) 2 ][ (1 z 2 ) ϕ (z) 1 ϕ(z) 2 ]} M n+1 +(n +1)M n 1 z 2, where the last inequality follows from Theorem 3. Given z and w in D, letγ(t) : [, 1] D be a smooth curve to connect z and w, i.e., 1 dτ ϕ,n (γ(t)) τ ϕ,n (z) τ ϕ,n (w) = dt dt 1 1 d dt τ ϕ,n(γ(t)) dt [ z τ ϕ,n (γ(t)) dγ(t) dt where the last inequality follows from the first chain rule: d dt τ ϕ,n(γ(t)) = z τ ϕ (γ(t)) dγ(t) dt + z τ ϕ,n (γ(t)) dγ(t) ]dt, dt + z τ ϕ,n (γ(t)) dγ(t). dt Combining the above estimates gives M n+1 +2(n +1)M n τ ϕ,n (z) τ ϕ,n (w) 1 γ(t) 2 d γ(t). γ

9 HYPERBOLIC DERIVATIVES AND SCHWARZ-PICK ESTIMATES 3317 If we choose γ to be a geodesic to connect z and w, then the above inequality gives τ ϕ,n (z) τ ϕ,n (w) (M n+1 +2(n +1)M n )β(z,w) (M n+1 +2(n +1)M n )ρ(z,w) 1 ρ(z,w) 2. The last inequality comes from the fact that for all <x<1, 1 1+x ln 2 1 x x 1 x 2. If ρ(z,w) < 1/8, the above inequality gives τ ϕ,n (z) τ ϕ,n (w) 2(M n+1 +2(n +1)M n )ρ(z,w). If ρ(z,w) 1/8, we have 8 ρ(z,w) 1, and τ ϕ,n (z) τ ϕ,n (w) max{ τ ϕ,n (z), τ ϕ,n (w) } M n 8M n ρ(z,w). Choosing C n =max{2(m n+1 +2(n +1)M n ), 8M n },wehave τ ϕ,n (z) τ ϕ,n (w) C n ρ(z,w) to complete the proof. Theorem 6 has an application to closed-range composition operators on the Bloch space. Hoffman [6] showed that (1 z 2 ) n ϕ (n) (z) continuously extends to the maximal ideal space of H. Let G be the subset of the maximal ideal space of H consisting of nontrivial Gleason parts. As a corollary of a result in [1] and Theorem 1.2 [4], we have the following result. Corollary 1. Suppose that ϕ is an analytic self-mapping of the unit disk. Then τ ϕ,n (z) admits a continuous extension to G. Addendum After we finished this paper, we obtained K. Stroethoff s paper [14], which showed that ρ( τ ϕ (z), τ ϕ (w) ) 2ρ(z,w), for z,w D. This generalizes Beardon s result [3]: If ϕ() =, then ρ(τ ϕ (),τ ϕ (w)) < 2ρ(,w) for w D. We thank K. Stroethoff. References [1] S.AxlerandK.Zhu,Boundary behavior of derivatives of analytic functions, Michigan Math. J. 39 (1992), MR 93e:373 [2] L.V.Ahlfors,Conformal invariants: topics in geometric function theory, McGraw-Hill, New York, MR 5:1211 [3] A. Beardon, The Schwarz-Pick Lemma for derivatives, Proc. Amer. Math. Soc. 125 (1997), MR 97m:362 [4] A. Brudnyi, Topology of the maximal ideal space of H, J. Funct. Analysis 189 (22), MR 23c:4666 [5] J. Garnett Bounded analytic functions, Academic Press, New York, MR 83g:337 [6] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. Math. 86 (1967), MR 35:5945 [7] P. Ghatage, J. Yan and D. Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc. 129 (21), MR 22a:4734

10 3318 PRATIBHA GHATAGE AND DECHAO ZHENG [8] B. MacCluer, K. Stroethoff, and R. Zhao, Generalized Schwarz-Pick estimates, Proc. Amer. Math. Soc. 131 (23), MR 23g:338 [9] K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (1995), MR 95i:4761 [1] A. Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pacific J. Math. 188 (1999), MR 2d:4744 [11] S. Ohno, K. Stroethoff and R. Zhao, Weighted composition operators between Bloch-type space, Rocky Mountain J. Math. 33 (23), MR 24d:4758 [12] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag 299, New York, MR 95b:38 [13] S. Roman, The Formula of Faa di Bruno, Amer. Math. Monthly 87 (198), MR 82d:263 [14] K. Stroethoff, Lecture notes on The Schwarz-Pick Lemma for derivatives, Preprint. Department of Mathematics, Cleveland State University, Cleveland, Ohio address: p.ghatage@csuohio.edu Department of Mathematics, Vanderbilt University, Nashville, Tennessee address: zheng@math.vanderbilt.edu