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Boundary Schwarz lemma for pluriharmonic mappings between unit balls
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J. Math. Anal. Appl. ••• (••••) •••–•••
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Boundary Schwarz lemma for pluriharmonic mappings between unit balls Yang Liu a,b,∗ , Shaoyu Dai c , Yifei Pan d,e a
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Southeast University, Nanjing 210096, China c Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China d Department of Mathematical Sciences, Indiana University–Purdue University Fort Wayne, Fort Wayne, IN 46805-1499, USA e School of Mathematics and Informatics, Jiangxi Normal University, Nanchang 330022, China b
a r t i c l e
i n f o
Article history: Received 20 May 2015 Available online xxxx Submitted by E.J. Straube Keywords: Schwarz lemma Boundary Schwarz lemma Pluriharmonic mapping Unit ball
a b s t r a c t In this paper, we get a Schwarz lemma for pluriharmonic mappings between the unit balls of any dimensions, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. As an application of the Schwarz lemma obtained, a boundary Schwarz lemma is established for pluriharmonic mappings between unit balls with any dimensions. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The Schwarz lemma is one of the most important results in complex analysis. A variant of the Schwarz lemma is known as the Schwarz–Pick lemma, which essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric. It was generalized to the derivatives of arbitrary order in one complex variable [11,2]. For several complex variables, Rudin [10] gave a first derivative estimate for bounded holomorphic functions on the polydisk, which is really a precursor to Schwarz–Pick estimate in high dimensions. Refs. [1,7] generalized the result for higher order derivative of holomorphic mappings to the unit ball and polydisk in Cn . For the harmonic functions (mappings), there are also some interesting analogues of the Schwarz lemma, see [5].
* Corresponding author at: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China. E-mail address: [email protected] (Y. Liu). http://dx.doi.org/10.1016/j.jmaa.2015.08.008 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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Theorem A. (See [5].) If f : D → D is a harmonic function, and f (0) = 0, then |f (z)| ≤
4 arctan |z|, z ∈ D, π
where D is the unit disk. The condition of f (0) = 0 was removed by Pavlovic [9]. Theorem B. (See [9].) If f : D → D is a harmonic function, then 2 f (z) − 1 − |z| f (0) ≤ 4 arctan |z|, z ∈ D. π 2 1 + |z| Here we will generalize the above result to pluriharmonic mappings between the unit balls of any dimensions. Before listing the main results, we give some notations and definitions first. For any z = (z1 , . . . , zn )T , n w = (w1 , . . . , wn )T ∈ Cn , the inner product and the corresponding norm are given by z, w = j=1 zj wj , 1 z = z, z 2 . Let zj = xj + iyj for 1 ≤ j ≤ n, the real version of z ∈ Cn is denoted by z = (x1 , y1 , . . . , xn , yn )T ∈ R2n . Let B n ⊂ Cn be the unit ball, ∂B n be the boundary of B n . A pluriharmonic function f is complex-valued and defined on a domain Ω ⊂ Cn such that for fixed z ∈ Ω and z0 ∈ ∂B n , f (z + z0 ξ) is harmonic in {ξ : ξ < dΩ (z)}, where dΩ (z) expresses the distance between z and ∂Ω. Especially, it is obtained in [13] that when Ω is a simply connected domain, then f : Ω → C is pluriharmonic if and only if f could be represented by f = η + ζ¯ where η and ζ are holomorphic in Ω. Therefore, a holomorphic function can be regarded as a special pluriharmonic function. f : Ω → CN is called a pluriharmonic mapping if all its components are pluriharmonic functions from Ω to C. In this paper, we first discuss the Schwarz lemma for pluriharmonic mappings f : B n → B N for n, N ≥ 1, which generalizes the corresponding results in [5,9]. Theorem 1.1. If f : B n → B N is a pluriharmonic mapping for n, N ≥ 1, then 2 f (z) − 1 − z f (0) ≤ 4 arctan z, z ∈ B n . π 2 1 + z When n = N = 1, Theorem 1.1 deduces Theorems A and B. On the other hand, Schwarz lemma at the boundary is also an active topic in complex analysis [3]. It has been applied to geometric function theory of one complex variable and several complex variables [6,8,12]. The following result is the classical boundary version of Schwarz lemma for holomorphic functions in one complex variable. Theorem C. (See [3].) Let f be the holomorphic self-mapping of D. If f is holomorphic at z = 1 with f (0) = 0 and f (1) = 1, then f (1) ≥ 1. ¯ denote N × n matrices (∂fi /∂zj ) Let Df and Df N ×n and (∂fi /∂zj )N ×n respectively. The following result is a new Schwarz lemma at the boundary in several complex variables, which could be regarded as a generalization of Theorem C. Theorem D. (See [8].) Let f be a holomorphic self-mapping on B n with n ≥ 1. If f is holomorphic at z0 ∈ aT w0 |2 ∂B n and f (z0 ) = w0 ∈ ∂B n , then there exists a λ ∈ R such that Df (z0 )T w0 = λz0 where λ ≥ |1−¯ 1−a2 > 0 and a = f (0).
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Since B n is equivalent to the unit ball B2n ⊂ R2n , we define the tangent space in terms of real version. For z = (z1 , . . . , zn )T ∈ Cn with zj = xj + iyj for 1 ≤ j ≤ n, denote z by the real version of z and z = (x1 , y1 , . . . , xn , yn )T ∈ R2n . Given z0 ∈ ∂B n , then z0 ∈ ∂B2n , and the tangent space Tz0 (∂B2n ) is defined by T Tz0 (∂B2n ) = β ∈ R2n : z0 β = 0 . For f : B n → B N , denote Jf (z0 ) by the 2N × 2n Jacobian matrix of f at z0 in terms of real coordinates. For a bounded domain V ∈ Cn , C α (V ) for 0 < α < 1 is the set of all functions f on V for which sup
|f (z) − f (w)| : z, w ∈ V¯ |z − w|α
is finite. C k+α (V ) is the set of all functions f on V whose kth order partial derivatives exist and belong to C α (V ) for an integer k ≥ 0. Based on the obtained Theorem 1.1, we get the following boundary Schwarz lemma for pluriharmonic mapping. Theorem 1.2. Let f : B n → B N be a pluriharmonic mapping for n, N ≥ 1. If f is C 1+α at z0 ∈ ∂B n and f (z0 ) = w0 ∈ ∂B N , then we have (I) Jf (z0 )β ∈ Tw0 (∂B2N ) for any β ∈ Tz0 (∂B2n ); T (II) There exists a positive λ ∈ R such that Jf (z0 ) w0 = λz0 , where z0 and w0 are real versions of z0 and w0 respectively, and λ ≥
1−f (0) 22n−1
> 0.
2. Proof of Theorem 1.1 The following lemmas would play an important role in the proof of main results. Lemma 2.1 was given in [1] for p ∈ B n . Lemma 2.1. For given p ∈ B n ∪ ∂B n and q ∈ Cn with q = 0, let L(ξ) = p + ξq for ξ ∈ C. Then L(Dp,q ) ⊂ B n , L(∂Dp,q ) ⊂ ∂B n , where Dp,q = {ξ ∈ C | |ξ − cp,q | < rp,q }, with cp,q =
− p,q q2 ,
rp,q =
1−p2 q2
Proof. Assume L(Dp,q )2 < 1, which means p2 + 2 Re(¯ pT ξq) + ξq2 < 1, and Re(¯ pT qξ) 1 p2 +2 + |ξ|2 < , 2 2 q q q2 i.e., 2 2 2 ξ + p, q < 1 − p + p, q . q2 q2 q2 The proof is finished. 2
2 + p,q 2 q .
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Motivated by [9], we have the following lemma. Lemma 2.2. If f : D → B N is a harmonic mapping for N ≥ 1, then 2 f (z) − 1 − |z| f (0) ≤ 4 arctan |z|, z ∈ D. π 2 1 + |z| Proof. Given 0 ≤ r < 1, 1 − r2 1 f (r) − f (0) = 1 + r2 2π
π −π
1 − r2 1 − r2 − 2 1 + r − 2r cos θ 1 + r2
(1 − r2 )2r 1 = 1 + r2 2π
π −π
f∗ (eiθ )dθ
cos θ f∗ (eiθ )dθ, 1 + r2 − 2r cos θ
(1)
where f∗ (eiθ ) = limρ→1− f (ρeiθ ). Due to f < 1, we have
π 2 2 | cos θ| f (r) − 1 − r f (0) ≤ (1 − r )2r 1 dθ. 2 2 1+r 1+r 2π 1 + r2 − 2r cos θ
(2)
−π
On the other hand, 1 2π
π −π
| cos θ| dθ 1 + r2 − 2r cos θ
1 = 2π 1 = 2π 1 = π
π/2 −π/2
π/2 −π/2
π/2 0
cos θ cos θ + 2 2 1 + r − 2r cos θ 1 + r + 2r cos θ
dθ
2(1 + r2 ) cos θ dθ (1 + r2 )2 − 4r2 cos2 θ
2(1 + r2 ) cos θ dθ (1 − r2 )2 + 4r2 sin2 θ
=
2r 1 + r2 arctan r(1 − r2 ) 1 − r2
=
1 + r2 2 arctan r. r(1 − r2 )
Combining (1) and (3) implies 2 f (r) − 1 − r f (0) ≤ 4 arctan r π 2 1+r for 0 ≤ r < 1. Assume z = reiθ for z ∈ D, then 2 f (zeiθ ) − 1 − |z| f (0) ≤ 4 arctan |z|, z ∈ D. π 2 1 + |z|
(3)
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Let f˜(ξ) = f (ξeiθ ) for ξ ∈ D and eiθ given above, then f˜ is also a harmonic mapping from the unit disk to the unit ball, which gives the desired result. 2 From Lemma 2.2, the following result can be obtained easily, and we omit the proof here. Lemma 2.3. If f : DR → B N is a harmonic mapping for N ≥ 1, then 2 2 f (z) − R − |z| f (0) ≤ 4 arctan z , z ∈ DR , 2 2 R + |z| π R where DR = {z ∈ C : |z| < R}. We now give the proof of Theorem 1.1. Proof. Let f be the one given in the theorem. For any nonzero p ∈ B n , it is obtained that g(ξ) = f (pξ) : 1 D p → B N is a harmonic mapping. From Lemma 2.3, we have g(ξ) −
1 p2 1 p2
ξ 4 1 . g(0) ≤ arctan 1 , ξ ∈ D p p π + |ξ|2 − |ξ|2
1 , one gets Letting ξ = 1 ∈ D p
2 g(1) − 1 − p g(0) ≤ 4 arctan p , π 2 1 + p which means that 2 f (p) − 1 − p f (0) ≤ 4 arctan p . π 2 1 + p When p = 0, the theorem holds obviously. The proof is complete. 2 3. Proof of Theorem 1.2 To prove the main result, we first introduce the Harnack’s inequality as follows. Theorem 3.1. (See [4].) For a nonnegative function f defined on the unit ball in Rn . If f is continuous on the ball and harmonic on its interior, then for any point x with |x| = r < 1, it holds that 1+r 1−r f (0) ≤ f (x) ≤ f (0). n−1 (1 + r) (1 − r)n−1 In the following, we will prove Theorem 1.2 in five steps. Proof. Step 1. Assume z0 = en1 ∈ ∂B n , and f = (f1 , . . . , fn )T is C 1+α in a neighborhood V of z0 . Here en1 = (1, 0, . . . , 0)T from the notation given in Section 1. Assume f (z0 ) = w0 = eN 1 . We consider fj = uj +ivj for 1 ≤ j ≤ N on the ball. Then it is obtained from f that u1 (en1 ) = 1. Moreover, 1 − u1 ≥ 0 is harmonic on the ball. From Theorem 3.1, consider x = (x1 , 0, . . . , 0)T ∈ R2n for x1 near 1, one gets 1 + x1 1 − x1 (1 − u1 (0)) ≤ 1 − u1 (x) ≤ (1 − u1 (0)), 2n−1 (1 + x1 ) (1 − x1 )2n−1
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which gives that 1 − u1 (0) 1 − u1 (x) ≥ . 1 − x1 (1 + x1 )2n−1 Letting x1 → 1− , we have 1 − u1 (0) ∂u1 (en1 ) (1 − u1 (x)) − (1 − u1 (en1 )) = lim ≥ . ∂x1 1 − x1 22n−1 x1 →1−
(4)
Step 2. From Lemma 2.1, let p = z0 , q = (−1 + ik)z0 for any given k ∈ R. Then p + tq = (1 − t + ikt)z0 2 for t ∈ R. p + tq < 1 ⇔ |1 − t + ikt| < 1 ⇔ 0 < t < 1+k 2 , which means that for a given k ∈ R when + n t → 0 , p + tq ∈ B ∩ V . By Theorem 1.1, 2 f (z) − 1 − z f (0) ≤ 4 arctan z, z ∈ B n . π 2 1 + z Therefore, 2 f (p + tq) − 1 − p + tq f (0) ≤ 4 arctan p + tq. π 2 1 + p + tq For t → 0+ , taking the Taylor expansion of f (p + tq),
1−p+tq2 1+p+tq2
(5)
and arctan p + tq at t = 0, we have
¯ (z0 )¯ f (p + tq) = w0 + Df (z0 )qt + Df q t + O(t1+α ), 1 − p + tq2 1 1 = − p¯T qt − pT q¯t + O(t1+α ), 2 1 + p + tq 2 2 π 1 1 arctan p + tq = + p¯T qt + pT q¯t + O(t1+α ). 4 4 4
(6)
Combining (5) and (6), we have 1 1 T T 1+α ¯ (z0 )¯ w0 + Df (z0 )qt + Df f (0)¯ p f (0)p q t − qt − q ¯ t + O(t ) 2 2 1 1 ≤ 1 + p¯T qt + pT q¯t + O(t1+α ), π π which means that
1 1 T T ¯ p q − f (0)p q¯ t + O(t1+α ) Df (z0 )q + Df (z0 )¯ 1+ q − f (0)¯ 2 2 1 1 T p¯ q + pT q¯ t + O(t1+α ). ≤1+2 π π 2 Re w ¯0T
Letting t → 0+ , one gets Re w ¯0T
1 1 1 1 T T ¯ Df (z0 )q + Df (z0 )¯ p q − f (0)p q¯ ≤ p¯T q + pT q¯. q − f (0)¯ 2 2 π π
Substituting z0 = en1 , w0 = eN 1 and p = z0 , q = (−1 + ik)z0 , we have Re
∂f1 (z0 ) 2 ∂f1 (z0 ) (−1 + ik) + (−1 − ik) + f1 (0) ≤ − . ∂z1 ∂z1 π
(7)
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1 (z0 ) 1 (z0 ) 1 (z0 ) Let ∂f∂z = Re ∂f∂z + i Im ∂f∂z and 1 1 1 one gets
− Re
∂f1 (z0 ) ∂z1
7
1 (z0 ) 1 (z0 ) = Re ∂f∂z + i Im ∂f∂z , then from the above inequality, 1 1
∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 − k Im − Re + k Im + Re f1 (0) ≤ − , ∂z1 ∂z1 ∂z1 ∂z1 π
i.e., Re
∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 + Re + k Im − k Im ≥ + Re f1 (0). ∂z1 ∂z1 ∂z1 ∂z1 π
(8)
Since (8) is valid for any k ∈ R, so that Im
∂f1 (z0 ) ∂f1 (z0 ) − Im = 0. ∂z1 ∂z1
(9)
Step 3. Let p = z0 , q = −z0 + ikenj for 2 ≤ j ≤ n and k ∈ R. Then p + tq = (1 − t)z0 + iktenj for t ∈ R. By 2 + Lemma 2.1, p + tq < 1 ⇔ |1 − t|2 + |ikt|2 < 1 ⇔ 0 < t < 1+k 2 . Therefore, given a k ∈ R, when t → 0 , n p + tq ∈ B n ∩ V . From (7), and substituting z0 = en1 , w0 = eN 1 and p = z0 , q = −z0 + ikej for 2 ≤ j ≤ n, we have ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 Re − − + ik − ik + f1 (0) ≤ − . ∂z1 ∂z1 ∂zj ∂zj π From the above inequality, one has −k Im
∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 + k Im ≤ Re + Re − − Re f1 (0). ∂zj ∂zj ∂z1 ∂z1 π
Since (8) is valid for any k ∈ R, so that Im
∂f1 (z0 ) ∂f1 (z0 ) − Im = 0, 2 ≤ j ≤ n. ∂zj ∂zj
(10)
Meanwhile, if we assume p = z0 , q = −z0 + kenj for 2 ≤ j ≤ n and any k ∈ R. It is easy to find Re
∂f1 (z0 ) ∂f1 (z0 ) + Re = 0, 2 ≤ j ≤ n. ∂zj ∂zj
Now rewrite fj = uj + ivj and zj = xj + iyj for 1 ≤ j ≤ n. Since
1 ∂ ∂ + i 2 ∂xj ∂yj , then for 1 ≤ l, j ≤ n, we have ∂fl 1 = ∂zj 2 ∂fl 1 = ∂zj 2
∂ ∂ −i ∂xj ∂yj ∂ ∂ +i ∂xj ∂yj
(ul + ivl ) =
1 2
1 (ul + ivl ) = 2
∂ul ∂vl + ∂xj ∂yj ∂vl ∂ul − ∂xj ∂yj
+i
(11)
∂ ∂zj
1 2
1 +i 2
=
1 2
∂ ∂xj
∂vl ∂ul − ∂xj ∂yj ∂ul ∂vl + ∂xj ∂yj
− i ∂y∂ j
and
∂ ∂zj
=
, .
From (9) and (4), it shows that ∂u1 = 0, ∂y1
∂u1 1 − u1 (0) ≥ . ∂x1 22n−1
(12)
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Similarly, from (10) and (11), one gets that ∂u1 = 0, ∂yj
∂u1 = 0, 2 ≤ j ≤ n. ∂xj
(13)
Rewrite z = (z1 , . . . , zn )T ∈ Cn by z = (x1 , y1 , . . . , xn , yn )T ∈ R2n , then e n = (1, 0, . . . , 0, 0)T ∈ R2n . From (12) and (13), it is concluded that 1
Jf (z0 ) w0 = λf z0 T
(14)
1−u1 (0) 1 for w0 = e N , z0 = e n and λf = ∂u ∂x1 ≥ 22n−1 . n Step 4. Now let z0 be any given point at ∂B2n which is not necessary e 1 . Then there exists a real-valued n unitary matrix Uz0 such that Uz0 (z0 ) = e 1 . Let f be the real version of f , and f (z0 ) = w0 where w0 is N N 2N not necessary e 1 at ∂B . Then there is a Uw0 such that Uw0 (w0 ) = e 1 . Denote 1
1
g (z ) = Uw0 ◦ f ◦ Uz0 T (z ), z ∈ B2n , and g(z) = Uw0 ◦ f ◦ Uz0 T (z), z ∈ B n , where Uw0 and Uz0 are complex unitary matrices corresponding to Uw0 and Uz0 , such that Uz0 (z0 ) = en1 n N and Uw0 (w0 ) = eN 1 . Then one gets that g is the real version of g and g(e1 ) = e1 . Moreover, the Jacobian matrix of g could be expressed by Jg (z ) = Uw0 Jf (Uz0 T z )Uz0 T (z ), z ∈ B2n .
(15)
From Step 3, we have n T
Jg (e 1 ) e 1 = λg e 1 for z0 = e 1 and λg ≥ n
1−Re g1 (0) , 22n−1
N
n
which equals to T
Uw0 Jf (Uz0 T e 1 )Uz0 T e 1 = λg e 1 , n
N
n
i.e., Uz0 Jf (z0 ) Uw0 T e 1 = λg e 1 . T
N
n
Multiplying Uz0 T at both sides of the above equation gives Uz0 T Uz0 Jf (z0 ) Uw0 T e 1 = λg Uz0 T e 1 , T
N
n
i.e., Jf (z0 ) w0 = λg z0 , T
(16)
g1 (0) 1−f (0) where λg ≥ 1−Re ≥ 1−g(0) > 0. 22n−1 22n−1 = 22n−1 T T Step 5. For any β ∈ Tz0 (∂B2n ), we have z0 β = 0. Due to (16), Jf (z0 ) w0 = λg z0 , which gives that T T w0 Jf (z0 )β = λg z0 β = 0. Therefore, Jf (z0 )β ∈ Tw0 (∂B2N ). The proof of (I) is finished. 2
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Acknowledgments The work was finished while the first author visited the third author at Department of Mathematical Sciences, Indiana University–Purdue University Fort Wayne. The first author would appreciate the comfortable research environment and all support provided by the institution in the 2014 academic year. The work was supported by the NSF of Zhejiang Province of China under Grant LY14A010008, the NNSF of China under Grants 11101373, 61374077 and 11201199. It was also supported by China Scholarship Council (CSC) under Grant 201308330247. References [1] S. Dai, H. Chen, Y. Pan, The high order Schwarz–Pick lemma on complex Hilbert balls, Sci. China Math. 53 (10) (2010) 2649–2656. [2] S. Dai, Y. Pan, Note on Schwarz–Pick estimates for bounded and positive real part analytic functions, Proc. Amer. Math. Soc. 136 (2) (2008) 635–640. [3] J.B. Garnett, Bounded Analytic Functions, Pure Appl. Math., vol. 96, Academic Press, 1981. [4] A. Harnack, Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, VG Teubner, 1887. [5] E. Heinz, On one-to-one harmonic mappings, Pacific J. Math. 9 (1959) 101–105. [6] S.G. Krantz, The Schwarz lemma at the boundary, Complex Var. Elliptic Equ. 56 (5) (2011) 455–468. [7] Y. Liu, Z. Chen, Schwarz–Pick estimates for holomorphic mappings from the polydisk to the unit ball, J. Math. Anal. Appl. 376 (1) (2011) 123–128. [8] T. Liu, J. Wang, X. Tang, Schwarz lemma at the boundary of the unit ball in Cn and its applications, J. Geom. Anal. 25 (2015) 1890–1914. [9] M. Pavlovic, Introduction to Function Spaces on the Disk, Matematicki Institut SANU, 2004. [10] W. Rudin, Function Theory in Polydiscs, Math. Lecture Note Ser., vol. 41, W.A. Benjamin, New York, 1969. [11] S. Ruscheweyh, Two remarks on bounded analytic functions, Serdica Math. J. 11 (2) (1985) 200–2002. [12] X. Tang, T. Liu, J. Lu, Schwarz lemma at the boundary of the unit polydisk in Cn , Sci. China Math. 58 (8) (2015) 1639–1652. [13] V.S. Vladimirov, L. Ehrenpreis, Methods of the Theory of Functions of Several Complex Variables, MIT Press, 1966.
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Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Boundary Schwarz lemma for pluriharmonic mappings between unit balls Yang Liu a,b,∗ , Shaoyu Dai c , Yifei Pan d,e a
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Southeast University, Nanjing 210096, China c Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China d Department of Mathematical Sciences, Indiana University–Purdue University Fort Wayne, Fort Wayne, IN 46805-1499, USA e School of Mathematics and Informatics, Jiangxi Normal University, Nanchang 330022, China b
a r t i c l e
i n f o
Article history: Received 20 May 2015 Available online xxxx Submitted by E.J. Straube Keywords: Schwarz lemma Boundary Schwarz lemma Pluriharmonic mapping Unit ball
a b s t r a c t In this paper, we get a Schwarz lemma for pluriharmonic mappings between the unit balls of any dimensions, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. As an application of the Schwarz lemma obtained, a boundary Schwarz lemma is established for pluriharmonic mappings between unit balls with any dimensions. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The Schwarz lemma is one of the most important results in complex analysis. A variant of the Schwarz lemma is known as the Schwarz–Pick lemma, which essentially states that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric. It was generalized to the derivatives of arbitrary order in one complex variable [11,2]. For several complex variables, Rudin [10] gave a first derivative estimate for bounded holomorphic functions on the polydisk, which is really a precursor to Schwarz–Pick estimate in high dimensions. Refs. [1,7] generalized the result for higher order derivative of holomorphic mappings to the unit ball and polydisk in Cn . For the harmonic functions (mappings), there are also some interesting analogues of the Schwarz lemma, see [5].
* Corresponding author at: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China. E-mail address: [email protected] (Y. Liu). http://dx.doi.org/10.1016/j.jmaa.2015.08.008 0022-247X/© 2015 Elsevier Inc. All rights reserved.
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Theorem A. (See [5].) If f : D → D is a harmonic function, and f (0) = 0, then |f (z)| ≤
4 arctan |z|, z ∈ D, π
where D is the unit disk. The condition of f (0) = 0 was removed by Pavlovic [9]. Theorem B. (See [9].) If f : D → D is a harmonic function, then 2 f (z) − 1 − |z| f (0) ≤ 4 arctan |z|, z ∈ D. π 2 1 + |z| Here we will generalize the above result to pluriharmonic mappings between the unit balls of any dimensions. Before listing the main results, we give some notations and definitions first. For any z = (z1 , . . . , zn )T , n w = (w1 , . . . , wn )T ∈ Cn , the inner product and the corresponding norm are given by z, w = j=1 zj wj , 1 z = z, z 2 . Let zj = xj + iyj for 1 ≤ j ≤ n, the real version of z ∈ Cn is denoted by z = (x1 , y1 , . . . , xn , yn )T ∈ R2n . Let B n ⊂ Cn be the unit ball, ∂B n be the boundary of B n . A pluriharmonic function f is complex-valued and defined on a domain Ω ⊂ Cn such that for fixed z ∈ Ω and z0 ∈ ∂B n , f (z + z0 ξ) is harmonic in {ξ : ξ < dΩ (z)}, where dΩ (z) expresses the distance between z and ∂Ω. Especially, it is obtained in [13] that when Ω is a simply connected domain, then f : Ω → C is pluriharmonic if and only if f could be represented by f = η + ζ¯ where η and ζ are holomorphic in Ω. Therefore, a holomorphic function can be regarded as a special pluriharmonic function. f : Ω → CN is called a pluriharmonic mapping if all its components are pluriharmonic functions from Ω to C. In this paper, we first discuss the Schwarz lemma for pluriharmonic mappings f : B n → B N for n, N ≥ 1, which generalizes the corresponding results in [5,9]. Theorem 1.1. If f : B n → B N is a pluriharmonic mapping for n, N ≥ 1, then 2 f (z) − 1 − z f (0) ≤ 4 arctan z, z ∈ B n . π 2 1 + z When n = N = 1, Theorem 1.1 deduces Theorems A and B. On the other hand, Schwarz lemma at the boundary is also an active topic in complex analysis [3]. It has been applied to geometric function theory of one complex variable and several complex variables [6,8,12]. The following result is the classical boundary version of Schwarz lemma for holomorphic functions in one complex variable. Theorem C. (See [3].) Let f be the holomorphic self-mapping of D. If f is holomorphic at z = 1 with f (0) = 0 and f (1) = 1, then f (1) ≥ 1. ¯ denote N × n matrices (∂fi /∂zj ) Let Df and Df N ×n and (∂fi /∂zj )N ×n respectively. The following result is a new Schwarz lemma at the boundary in several complex variables, which could be regarded as a generalization of Theorem C. Theorem D. (See [8].) Let f be a holomorphic self-mapping on B n with n ≥ 1. If f is holomorphic at z0 ∈ aT w0 |2 ∂B n and f (z0 ) = w0 ∈ ∂B n , then there exists a λ ∈ R such that Df (z0 )T w0 = λz0 where λ ≥ |1−¯ 1−a2 > 0 and a = f (0).
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Since B n is equivalent to the unit ball B2n ⊂ R2n , we define the tangent space in terms of real version. For z = (z1 , . . . , zn )T ∈ Cn with zj = xj + iyj for 1 ≤ j ≤ n, denote z by the real version of z and z = (x1 , y1 , . . . , xn , yn )T ∈ R2n . Given z0 ∈ ∂B n , then z0 ∈ ∂B2n , and the tangent space Tz0 (∂B2n ) is defined by T Tz0 (∂B2n ) = β ∈ R2n : z0 β = 0 . For f : B n → B N , denote Jf (z0 ) by the 2N × 2n Jacobian matrix of f at z0 in terms of real coordinates. For a bounded domain V ∈ Cn , C α (V ) for 0 < α < 1 is the set of all functions f on V for which sup
|f (z) − f (w)| : z, w ∈ V¯ |z − w|α
is finite. C k+α (V ) is the set of all functions f on V whose kth order partial derivatives exist and belong to C α (V ) for an integer k ≥ 0. Based on the obtained Theorem 1.1, we get the following boundary Schwarz lemma for pluriharmonic mapping. Theorem 1.2. Let f : B n → B N be a pluriharmonic mapping for n, N ≥ 1. If f is C 1+α at z0 ∈ ∂B n and f (z0 ) = w0 ∈ ∂B N , then we have (I) Jf (z0 )β ∈ Tw0 (∂B2N ) for any β ∈ Tz0 (∂B2n ); T (II) There exists a positive λ ∈ R such that Jf (z0 ) w0 = λz0 , where z0 and w0 are real versions of z0 and w0 respectively, and λ ≥
1−f (0) 22n−1
> 0.
2. Proof of Theorem 1.1 The following lemmas would play an important role in the proof of main results. Lemma 2.1 was given in [1] for p ∈ B n . Lemma 2.1. For given p ∈ B n ∪ ∂B n and q ∈ Cn with q = 0, let L(ξ) = p + ξq for ξ ∈ C. Then L(Dp,q ) ⊂ B n , L(∂Dp,q ) ⊂ ∂B n , where Dp,q = {ξ ∈ C | |ξ − cp,q | < rp,q }, with cp,q =
− p,q q2 ,
rp,q =
1−p2 q2
Proof. Assume L(Dp,q )2 < 1, which means p2 + 2 Re(¯ pT ξq) + ξq2 < 1, and Re(¯ pT qξ) 1 p2 +2 + |ξ|2 < , 2 2 q q q2 i.e., 2 2 2 ξ + p, q < 1 − p + p, q . q2 q2 q2 The proof is finished. 2
2 + p,q 2 q .
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Motivated by [9], we have the following lemma. Lemma 2.2. If f : D → B N is a harmonic mapping for N ≥ 1, then 2 f (z) − 1 − |z| f (0) ≤ 4 arctan |z|, z ∈ D. π 2 1 + |z| Proof. Given 0 ≤ r < 1, 1 − r2 1 f (r) − f (0) = 1 + r2 2π
π −π
1 − r2 1 − r2 − 2 1 + r − 2r cos θ 1 + r2
(1 − r2 )2r 1 = 1 + r2 2π
π −π
f∗ (eiθ )dθ
cos θ f∗ (eiθ )dθ, 1 + r2 − 2r cos θ
(1)
where f∗ (eiθ ) = limρ→1− f (ρeiθ ). Due to f < 1, we have
π 2 2 | cos θ| f (r) − 1 − r f (0) ≤ (1 − r )2r 1 dθ. 2 2 1+r 1+r 2π 1 + r2 − 2r cos θ
(2)
−π
On the other hand, 1 2π
π −π
| cos θ| dθ 1 + r2 − 2r cos θ
1 = 2π 1 = 2π 1 = π
π/2 −π/2
π/2 −π/2
π/2 0
cos θ cos θ + 2 2 1 + r − 2r cos θ 1 + r + 2r cos θ
dθ
2(1 + r2 ) cos θ dθ (1 + r2 )2 − 4r2 cos2 θ
2(1 + r2 ) cos θ dθ (1 − r2 )2 + 4r2 sin2 θ
=
2r 1 + r2 arctan r(1 − r2 ) 1 − r2
=
1 + r2 2 arctan r. r(1 − r2 )
Combining (1) and (3) implies 2 f (r) − 1 − r f (0) ≤ 4 arctan r π 2 1+r for 0 ≤ r < 1. Assume z = reiθ for z ∈ D, then 2 f (zeiθ ) − 1 − |z| f (0) ≤ 4 arctan |z|, z ∈ D. π 2 1 + |z|
(3)
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Let f˜(ξ) = f (ξeiθ ) for ξ ∈ D and eiθ given above, then f˜ is also a harmonic mapping from the unit disk to the unit ball, which gives the desired result. 2 From Lemma 2.2, the following result can be obtained easily, and we omit the proof here. Lemma 2.3. If f : DR → B N is a harmonic mapping for N ≥ 1, then 2 2 f (z) − R − |z| f (0) ≤ 4 arctan z , z ∈ DR , 2 2 R + |z| π R where DR = {z ∈ C : |z| < R}. We now give the proof of Theorem 1.1. Proof. Let f be the one given in the theorem. For any nonzero p ∈ B n , it is obtained that g(ξ) = f (pξ) : 1 D p → B N is a harmonic mapping. From Lemma 2.3, we have g(ξ) −
1 p2 1 p2
ξ 4 1 . g(0) ≤ arctan 1 , ξ ∈ D p p π + |ξ|2 − |ξ|2
1 , one gets Letting ξ = 1 ∈ D p
2 g(1) − 1 − p g(0) ≤ 4 arctan p , π 2 1 + p which means that 2 f (p) − 1 − p f (0) ≤ 4 arctan p . π 2 1 + p When p = 0, the theorem holds obviously. The proof is complete. 2 3. Proof of Theorem 1.2 To prove the main result, we first introduce the Harnack’s inequality as follows. Theorem 3.1. (See [4].) For a nonnegative function f defined on the unit ball in Rn . If f is continuous on the ball and harmonic on its interior, then for any point x with |x| = r < 1, it holds that 1+r 1−r f (0) ≤ f (x) ≤ f (0). n−1 (1 + r) (1 − r)n−1 In the following, we will prove Theorem 1.2 in five steps. Proof. Step 1. Assume z0 = en1 ∈ ∂B n , and f = (f1 , . . . , fn )T is C 1+α in a neighborhood V of z0 . Here en1 = (1, 0, . . . , 0)T from the notation given in Section 1. Assume f (z0 ) = w0 = eN 1 . We consider fj = uj +ivj for 1 ≤ j ≤ N on the ball. Then it is obtained from f that u1 (en1 ) = 1. Moreover, 1 − u1 ≥ 0 is harmonic on the ball. From Theorem 3.1, consider x = (x1 , 0, . . . , 0)T ∈ R2n for x1 near 1, one gets 1 + x1 1 − x1 (1 − u1 (0)) ≤ 1 − u1 (x) ≤ (1 − u1 (0)), 2n−1 (1 + x1 ) (1 − x1 )2n−1
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which gives that 1 − u1 (0) 1 − u1 (x) ≥ . 1 − x1 (1 + x1 )2n−1 Letting x1 → 1− , we have 1 − u1 (0) ∂u1 (en1 ) (1 − u1 (x)) − (1 − u1 (en1 )) = lim ≥ . ∂x1 1 − x1 22n−1 x1 →1−
(4)
Step 2. From Lemma 2.1, let p = z0 , q = (−1 + ik)z0 for any given k ∈ R. Then p + tq = (1 − t + ikt)z0 2 for t ∈ R. p + tq < 1 ⇔ |1 − t + ikt| < 1 ⇔ 0 < t < 1+k 2 , which means that for a given k ∈ R when + n t → 0 , p + tq ∈ B ∩ V . By Theorem 1.1, 2 f (z) − 1 − z f (0) ≤ 4 arctan z, z ∈ B n . π 2 1 + z Therefore, 2 f (p + tq) − 1 − p + tq f (0) ≤ 4 arctan p + tq. π 2 1 + p + tq For t → 0+ , taking the Taylor expansion of f (p + tq),
1−p+tq2 1+p+tq2
(5)
and arctan p + tq at t = 0, we have
¯ (z0 )¯ f (p + tq) = w0 + Df (z0 )qt + Df q t + O(t1+α ), 1 − p + tq2 1 1 = − p¯T qt − pT q¯t + O(t1+α ), 2 1 + p + tq 2 2 π 1 1 arctan p + tq = + p¯T qt + pT q¯t + O(t1+α ). 4 4 4
(6)
Combining (5) and (6), we have 1 1 T T 1+α ¯ (z0 )¯ w0 + Df (z0 )qt + Df f (0)¯ p f (0)p q t − qt − q ¯ t + O(t ) 2 2 1 1 ≤ 1 + p¯T qt + pT q¯t + O(t1+α ), π π which means that
1 1 T T ¯ p q − f (0)p q¯ t + O(t1+α ) Df (z0 )q + Df (z0 )¯ 1+ q − f (0)¯ 2 2 1 1 T p¯ q + pT q¯ t + O(t1+α ). ≤1+2 π π 2 Re w ¯0T
Letting t → 0+ , one gets Re w ¯0T
1 1 1 1 T T ¯ Df (z0 )q + Df (z0 )¯ p q − f (0)p q¯ ≤ p¯T q + pT q¯. q − f (0)¯ 2 2 π π
Substituting z0 = en1 , w0 = eN 1 and p = z0 , q = (−1 + ik)z0 , we have Re
∂f1 (z0 ) 2 ∂f1 (z0 ) (−1 + ik) + (−1 − ik) + f1 (0) ≤ − . ∂z1 ∂z1 π
(7)
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1 (z0 ) 1 (z0 ) 1 (z0 ) Let ∂f∂z = Re ∂f∂z + i Im ∂f∂z and 1 1 1 one gets
− Re
∂f1 (z0 ) ∂z1
7
1 (z0 ) 1 (z0 ) = Re ∂f∂z + i Im ∂f∂z , then from the above inequality, 1 1
∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 − k Im − Re + k Im + Re f1 (0) ≤ − , ∂z1 ∂z1 ∂z1 ∂z1 π
i.e., Re
∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 + Re + k Im − k Im ≥ + Re f1 (0). ∂z1 ∂z1 ∂z1 ∂z1 π
(8)
Since (8) is valid for any k ∈ R, so that Im
∂f1 (z0 ) ∂f1 (z0 ) − Im = 0. ∂z1 ∂z1
(9)
Step 3. Let p = z0 , q = −z0 + ikenj for 2 ≤ j ≤ n and k ∈ R. Then p + tq = (1 − t)z0 + iktenj for t ∈ R. By 2 + Lemma 2.1, p + tq < 1 ⇔ |1 − t|2 + |ikt|2 < 1 ⇔ 0 < t < 1+k 2 . Therefore, given a k ∈ R, when t → 0 , n p + tq ∈ B n ∩ V . From (7), and substituting z0 = en1 , w0 = eN 1 and p = z0 , q = −z0 + ikej for 2 ≤ j ≤ n, we have ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 Re − − + ik − ik + f1 (0) ≤ − . ∂z1 ∂z1 ∂zj ∂zj π From the above inequality, one has −k Im
∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) ∂f1 (z0 ) 2 + k Im ≤ Re + Re − − Re f1 (0). ∂zj ∂zj ∂z1 ∂z1 π
Since (8) is valid for any k ∈ R, so that Im
∂f1 (z0 ) ∂f1 (z0 ) − Im = 0, 2 ≤ j ≤ n. ∂zj ∂zj
(10)
Meanwhile, if we assume p = z0 , q = −z0 + kenj for 2 ≤ j ≤ n and any k ∈ R. It is easy to find Re
∂f1 (z0 ) ∂f1 (z0 ) + Re = 0, 2 ≤ j ≤ n. ∂zj ∂zj
Now rewrite fj = uj + ivj and zj = xj + iyj for 1 ≤ j ≤ n. Since
1 ∂ ∂ + i 2 ∂xj ∂yj , then for 1 ≤ l, j ≤ n, we have ∂fl 1 = ∂zj 2 ∂fl 1 = ∂zj 2
∂ ∂ −i ∂xj ∂yj ∂ ∂ +i ∂xj ∂yj
(ul + ivl ) =
1 2
1 (ul + ivl ) = 2
∂ul ∂vl + ∂xj ∂yj ∂vl ∂ul − ∂xj ∂yj
+i
(11)
∂ ∂zj
1 2
1 +i 2
=
1 2
∂ ∂xj
∂vl ∂ul − ∂xj ∂yj ∂ul ∂vl + ∂xj ∂yj
− i ∂y∂ j
and
∂ ∂zj
=
, .
From (9) and (4), it shows that ∂u1 = 0, ∂y1
∂u1 1 − u1 (0) ≥ . ∂x1 22n−1
(12)
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Similarly, from (10) and (11), one gets that ∂u1 = 0, ∂yj
∂u1 = 0, 2 ≤ j ≤ n. ∂xj
(13)
Rewrite z = (z1 , . . . , zn )T ∈ Cn by z = (x1 , y1 , . . . , xn , yn )T ∈ R2n , then e n = (1, 0, . . . , 0, 0)T ∈ R2n . From (12) and (13), it is concluded that 1
Jf (z0 ) w0 = λf z0 T
(14)
1−u1 (0) 1 for w0 = e N , z0 = e n and λf = ∂u ∂x1 ≥ 22n−1 . n Step 4. Now let z0 be any given point at ∂B2n which is not necessary e 1 . Then there exists a real-valued n unitary matrix Uz0 such that Uz0 (z0 ) = e 1 . Let f be the real version of f , and f (z0 ) = w0 where w0 is N N 2N not necessary e 1 at ∂B . Then there is a Uw0 such that Uw0 (w0 ) = e 1 . Denote 1
1
g (z ) = Uw0 ◦ f ◦ Uz0 T (z ), z ∈ B2n , and g(z) = Uw0 ◦ f ◦ Uz0 T (z), z ∈ B n , where Uw0 and Uz0 are complex unitary matrices corresponding to Uw0 and Uz0 , such that Uz0 (z0 ) = en1 n N and Uw0 (w0 ) = eN 1 . Then one gets that g is the real version of g and g(e1 ) = e1 . Moreover, the Jacobian matrix of g could be expressed by Jg (z ) = Uw0 Jf (Uz0 T z )Uz0 T (z ), z ∈ B2n .
(15)
From Step 3, we have n T
Jg (e 1 ) e 1 = λg e 1 for z0 = e 1 and λg ≥ n
1−Re g1 (0) , 22n−1
N
n
which equals to T
Uw0 Jf (Uz0 T e 1 )Uz0 T e 1 = λg e 1 , n
N
n
i.e., Uz0 Jf (z0 ) Uw0 T e 1 = λg e 1 . T
N
n
Multiplying Uz0 T at both sides of the above equation gives Uz0 T Uz0 Jf (z0 ) Uw0 T e 1 = λg Uz0 T e 1 , T
N
n
i.e., Jf (z0 ) w0 = λg z0 , T
(16)
g1 (0) 1−f (0) where λg ≥ 1−Re ≥ 1−g(0) > 0. 22n−1 22n−1 = 22n−1 T T Step 5. For any β ∈ Tz0 (∂B2n ), we have z0 β = 0. Due to (16), Jf (z0 ) w0 = λg z0 , which gives that T T w0 Jf (z0 )β = λg z0 β = 0. Therefore, Jf (z0 )β ∈ Tw0 (∂B2N ). The proof of (I) is finished. 2
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Acknowledgments The work was finished while the first author visited the third author at Department of Mathematical Sciences, Indiana University–Purdue University Fort Wayne. The first author would appreciate the comfortable research environment and all support provided by the institution in the 2014 academic year. The work was supported by the NSF of Zhejiang Province of China under Grant LY14A010008, the NNSF of China under Grants 11101373, 61374077 and 11201199. It was also supported by China Scholarship Council (CSC) under Grant 201308330247. References [1] S. Dai, H. Chen, Y. Pan, The high order Schwarz–Pick lemma on complex Hilbert balls, Sci. China Math. 53 (10) (2010) 2649–2656. [2] S. Dai, Y. Pan, Note on Schwarz–Pick estimates for bounded and positive real part analytic functions, Proc. Amer. Math. Soc. 136 (2) (2008) 635–640. [3] J.B. Garnett, Bounded Analytic Functions, Pure Appl. Math., vol. 96, Academic Press, 1981. [4] A. Harnack, Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, VG Teubner, 1887. [5] E. Heinz, On one-to-one harmonic mappings, Pacific J. Math. 9 (1959) 101–105. [6] S.G. Krantz, The Schwarz lemma at the boundary, Complex Var. Elliptic Equ. 56 (5) (2011) 455–468. [7] Y. Liu, Z. Chen, Schwarz–Pick estimates for holomorphic mappings from the polydisk to the unit ball, J. Math. Anal. Appl. 376 (1) (2011) 123–128. [8] T. Liu, J. Wang, X. Tang, Schwarz lemma at the boundary of the unit ball in Cn and its applications, J. Geom. Anal. 25 (2015) 1890–1914. [9] M. Pavlovic, Introduction to Function Spaces on the Disk, Matematicki Institut SANU, 2004. [10] W. Rudin, Function Theory in Polydiscs, Math. Lecture Note Ser., vol. 41, W.A. Benjamin, New York, 1969. [11] S. Ruscheweyh, Two remarks on bounded analytic functions, Serdica Math. J. 11 (2) (1985) 200–2002. [12] X. Tang, T. Liu, J. Lu, Schwarz lemma at the boundary of the unit polydisk in Cn , Sci. China Math. 58 (8) (2015) 1639–1652. [13] V.S. Vladimirov, L. Ehrenpreis, Methods of the Theory of Functions of Several Complex Variables, MIT Press, 1966.