The Schwarz Lemma in Clifford Analysis

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 4, April 2014, Pages 1237–1248 S 0002-9939(2014)11854-5 Article electronically published on January 6, 2014

THE SCHWARZ LEMMA IN CLIFFORD ANALYSIS

ZHONGXIANG ZHANG

(Communicated by Richard Rochberg)

Abstract. In this paper, we first give the Cauchy type integral representation for harmonic functions in the Clifford analysis framework, and by using integral representations for harmonic functions in Clifford analysis, the Poisson integral formula for harmonic functions is represented. As its application, the mean value theorems and the maximum modulus theorem for Clifford-valued harmonic functions are presented. Second, some properties of Möbius transformations are given, and a close relation between the monogenic functions and Möbius transformations is shown. Finally, by using the integral representations for harmonic functions and the properties of Möbius transformations, Schwarz type lemmas for harmonic functions and monogenic functions are established.

1. Introduction and preliminaries The Schwarz lemmas play a very important role in classical complex analysis. In [4, 9, 11, 13], Clifford analysis and quaternionic analysis and their applications were systematically studied; they generalize complex analysis to a higher dimension in an elegant and natural way. Naturally, what is the version of the Schwarz lemmas in a Clifford analysis setting? In [17], the result on the Schwarz lemma in Clifford analysis was first given, while the result was built outside the unit ball in Rn+1. What are the versions of the Schwarz lemmas inside the unit ball in Rn+1 which are similar to the classical cases? For the non-commutativity of Clifford algebra, it is difficult to obtain the Schwarz lemmas in Clifford analysis by following the traditional method. Integral representations and Möbius transformations in Clifford analysis were studied by many authors (see [1–3, 5–8, 10, 12, 14–19], etc.); integral representation formulas and Möbius transformations are very important tools. In this paper, by developing some integral representations for harmonic functions which are similar to the above references and using the properties of Möbius transformations, we establish some Schwarz type lemmas inside the unit ball in Rn+1 which are similar to the classical results. Let Vn,0 be an n–dimensional (n ≥ 1) real linear space with basis {e1,e2, ··· ,en}, n and let C(Vn,0)bethe2 –dimensional real linear space with basis

{eA,A= {h1, ··· ,hr}∈PN, 1 ≤ h1 < ···

Received by the editors December 19, 2011 and, in revised form, April 29, 2012. 2010 Mathematics Subject Classification. Primary 30G35. Key words and phrases. Clifford algebra, Poisson integral, Möbius transformation, Schwarz lemma. The author was supported by the DAAD K. C. Wong Education Foundation and the NNSF for Young Scholars of China (No. 11001206).

c 2014 American Mathematical Society Reverts to public domain 28 years from publication 1237

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where N stands for the set {1, ··· ,n} and PN denotes the family of all order-

preserving subsets of N in the above way. We denote e∅ as e0 and eA as eh1···hr for A = {h1, ··· ,hr}∈PN. The product on C(Vn,0)isdefinedby (1.1)⎧ #(A∩B) P (A,B) ⎨⎪eAeB =(−1) (−1) eAB, if A, B ∈ PN, ⎪ ⎩λμ = λAμBeAeB, if λ= λAeA,μ= μBeB, A∈PN B∈PN A∈PN B∈PN where #(A) is the cardinal number of the set A,thenumberP (A, B)= P (A, j), j∈B P (A, j)=#{i, i ∈ A, i > j}, the symmetric difference set AB is also order- preserving in the above way, and λA ∈ R is the coefficient of the eA–component of the Clifford number λ. We also denote λA by [λ]A, λ{i} by [λ]i and λ0 by Reλ.It follows at once from the multiplication rule (1.1) that e0 is the identity element, written now as 1, and, in particular, ⎧ ⎪ 2 − ··· ⎪ei = 1, if i =1, ,n, ⎨⎪ (1.2) e e = −e e , if 1 ≤ i

Thus C(Vn,0) is a real linear, associative, but non-commutative algebra and is called the Cliﬀord algebra over Vn,0. An involution is deﬁned by ⎧ ⎪ − σ(A) ∈ ⎨⎪ eA =( 1) eA, if A PN , (1.3) ⎪ ⎩⎪ λ = λAeA, if λ = λAeA, A∈PN A∈PN where σ(A)=#(A)(#(A)+1)/2. From (1.1) and (1.3), we have ⎧ ⎨ ei = −ei, if i =1, ··· ,n, (1.4) ⎩ λμ = μλ, for any λ, μ ∈ C(Vn,0).

The C (Vn.0)–valued n–diﬀerential forms n n − k N +1 − k N +1 dσ = ( 1) ekdxk , dσ = ( 1) ekdxk k=0 k=0 are exact, where N +1 ∧···∧ ∧ ∧···∧ dxk =dx0 dxk−1 dxk+1 dxn. { ··· } The real linear space with basis e0,e1, ,en is a subspace of C(Vn,0); it is denoted by Cln+1. For all λ = λAeA ∈ C(Vn,0), a norm is deﬁned by |λ| = A 2 λA. The operator D is written as A n ∂ (r) (r−1) D = ek : C (Ω,C(Vn,0)) → C (Ω,C(Vn,0)). ∂xk k=0

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Let f be a function with value in C(Vn,0)deﬁnedinΩ,whereΩisanopen non-empty subset in Rn+1. The operator D acts on the function f from the left and from the right, which is governed by n n ∂fA ∂fA D[f]= ekeA , [f]D = eAek . ∂xk ∂xk k=0 A k=0 A

2. Integral representations for harmonic functions and its applications Suppose Ω and Ω∗ are open, non-empty subsets of Rn+1,andΩ⊂ Ω∗.Letf be ∗ a function with value in C(Vn,0)deﬁnedinΩ. ∗ 2 ∗ Deﬁnition 2.1. A function f is harmonic in Ω if f ∈ C (Ω ,C(Vn,0)) and f ≡ 0 n 2 ∗ ∂ in Ω ,where is the Laplace operator = 2 . Clearly, = DD = DD. k=0 ∂xk 1 ∗ ∗ Lemma 2.2. If f,g ∈ C (Ω ,C(Vn,0)), then for any open bounded subset Ω ⊂ Ω with smooth boundary ∂Ω,

(2.1) fdσg = ([f]Dg + fD[g]) dΩ

∂Ω Ω and

(2.2) fdσg = [f]Dg + fD[g] dΩ.

∂Ω Ω For x ∈ Rn+1 \{0},denote 1 x − 1 1 (2.3) E(x)= n+1 ,H(x)= n−1 , ωn+1 |x| (n − 1)ωn+1 |x|

n+1 1 n where ω =2π 2 stands for the area of the unit sphere S . n+1 n +1 Γ( ) 2 Lemma 2.3. Suppose E(x) and H(x) are as above. Then for x ∈ Rn+1 \{0}, (2.4) D[H](x)=[H]D(x)=E(x), D[H](x)=[H]D(x)=E(x), and (2.5) D[E](x)=[E]D(x)=D[E](x)=[E]D(x)=0.

2 ∗ ∗ Theorem 2.4. Suppose f ∈ C (Ω ,C(Vn,0)), [f]=0in Ω ,andE(x) and H(x) are as above. Then for any open bounded subset Ω ⊂ Ω∗ with smooth boundary ∂Ω,

f(x),x∈ Ω, (2.6) E(y − x)dσyf(y) − H(y − x)dσyD[f](y)= 0,x∈ Rn+1 \ Ω. ∂Ω ∂Ω Remark 2.5. (2.6) is also called a Cauchy integral formula for Cliﬀord-valued harmonic functions. Especially, suppose D[f]=0inΩ∗; then (2.6) is just the Cauchy integral formula for monogenic functions. In this sense, Theorem 2.4 is a general- ization of the Cauchy integral formula for monogenic functions in [4]. Analogous results can also be found in [2, 8, 18].

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Denote B(a, r)asanopenballcenteredata with radius r. For all x ∈ Rn+1, denote S∂B(a,r)x as the symmetric point of x with respect to ∂B(a, r), which is r2 expressed as S x = a+ (x−a). Alternatively, ∀x =(x ,x , ··· ,x ) ∈ ∂B(a,r) |x − a|2 0 1 n n+1 R , x is identical to x = x0e0 + x1e1 + ···+ xnen ∈ Cln+1. The symmetric point r2 of x with respect to ∂B(a, r) can be expressed as S x = a + ,where ∂B(a,r) x − a a = a0e0 + a1e1 + ···+ anen.

Lemma 2.6. For all x, y ∈ Cln+1, we have r2 (2.7) S y − S x = |y − x|. ∂B(a,r) ∂B(a,r) |y − a||x − a|

2 ∗ ∗ Theorem 2.7. Suppose f ∈ C (Ω ,C(Vn,0)),and[f]=0in Ω .Then∀x ∈ B(a, R) ⊂ Ω∗,

1 R2 −|x − a|2 (2.8) f(x)= n+1 f(y)dS. ωn+1R |y − x| ∂B(a,R) Proof. For all x ∈ B(a, R) ⊂ Ω∗, by Theorem 2.4, we have

1 R2 − (x − a)(y − a) f(x)= n+1 f(y)dS ωn+1R |y − x| ∂B(a,R) (2.9) 1 y − a + n−1 D[f](y)dS. (n − 1)ωn+1R |y − x| ∂B(a,R) Denote the symmetric point of x with respect to ∂B(a, R)asx∗.Thenx∗ ∈ Rn+1 \ B(a, R), by Theorem 2.4 again,

1 y − x∗ 0= ∗ n+1 dσyf(y) ωn+1 |y − x | ∂B(a,R) (2.10) 1 1 + ∗ n−1 dσyD[f](y), (n − 1)ωn+1 |y − x | ∂B(a,R) and by Lemma 2.6, (2.10) can be rewritten as

1 |x − a|2 − (x − a)(y − a) 0= n+1 f(y)dS ωn+1R |y − x| ∂B(a,R) (2.11) 1 y − a + n−1 D[f](y)dS. (n − 1)ωn+1R |y − x| ∂B(a,R) Combining (2.9) with (2.11) gives (2.8).

Remark 2.8. (2.8) is also called the Poisson integral formula for Cliﬀord-valued harmonic functions which coincides with the classical Poisson integral formula for harmonic functions in Euclidean space Rn+1. Theorem 2.7 is naturally valid for monogenic functions.

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2 ∗ ∗ Corollary 2.9. Suppose f ∈ C (Ω ,C(Vn,0)),and[f]=0in Ω . Then for B(a, R) ⊂ Ω∗,

1 (2.12) f(a)= n f(y)dS. ωn+1R ∂B(a,R)

2 ∗ ∗ Corollary 2.10. Suppose f ∈ C (Ω ,C(Vn,0)),and[f]=0in Ω . Then for B(a, R) ⊂ Ω∗,

1 (2.13) f(a)= n+1 f(y)dV, Vn+1R B(a,R)

n+1 where Vn+1 denotes the volume of the unit ball in R . Proof. By Lemma 2.2, we have

1 n f(y)dS ωn+1R ∂B(a,R ) 1 − = n+1 (y a)dσf(y) (2.14) ωn+1R ∂B (a,R) 1 1 − = n+1 f(y)dV + n+1 (y a)D[f](y)dV. Vn+1R ωn+1R B(a,R) B(a,R) In view of [f]=0inΩ∗, by Lemma 2.2 again,

(y − a)D[f](y)dV B(a,R ) (2.15) |y − a|2 R2 = dσD[f](y)= dσD[f](y)=0. 2 2 ∂B(a,R) ∂B(a,R) Combining (2.14) with (2.15), by Corollary 2.9, (2.13) follows.

Theorem 2.11. Let f be harmonic in the open and connected set Ω.Ifthereexists apointa ∈ Ω such that (2.16) |f(x)|≤|f(a)|, ∀x ∈ Ω, then f must be a constant function in Ω. Proof. By Corollary 2.9 or Corollary 2.10, it can be similarly proved by the method in [4] (see page 56).

Corollary 2.12. Let Ω be a bounded open set in Rn+1 and suppose that f is continuous in Ω and harmonic in Ω.Then (2.17) sup |f(x)| =sup|f(x)|. x∈Ω x∈∂Ω Remark 2.13. Theorem 2.11 and Corollary 2.12 are the generalizations of the results for monogenic functions in [4].

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3. Mobius¨ transformation The fractional linear transformation is denoted by ax + b (3.1) T (x)= , cx + d n where x = xiei, a, b, c, d ∈ R, and ad − bc = 0. (3.1) is also called a Möbius i=0 transformation. Lemma 3.1. (ax + b)(cx + d)=(cx + d)(ax + b), a, b, c, d ∈ R. (ax + b)(cx + d) (cx + d)(ax + b) Remark 3.2. By Lemma 3.1, T (x)= = . |cx + d|2 |cx + d|2 Remark 3.3. Obviously, each Möbius transformation y = T (x)mapsthehyper- complex space Cln+1 one-to-one onto itself so that there is a well defined inverse mapping x = T −1(y). The inverse mapping T −1 is again a Möbius transformation, given by dy − b (3.2) x = T −1(y)= . −cy + a Proposition 3.4. For a, b, c, d ∈ R,andad − bc =0 , ax + b ax + b ax + b |ax + b| 1 cx + d (3.3) = , = , = . cx + d cx + d cx + d |cx + d| ax + b ax + b cx + d Define the product of two Möbius transformations S and T , which is the com- posite

(3.4) (S ◦ T )(x)=S(T (x)), x ∈ Cln+1. The identity transformation I(x)=x acts as a unit for this multiplication op- eration: I ◦ T = T = T ◦ I. It can be directly deduced that: Proposition 3.5. T ◦ T −1 = I = T −1 ◦ T .

a1x + b1 a2x + b2 Proposition 3.6. Suppose T1(x)= ,andT2(x)= .Then c1x + d1 c2x + d2

(a1a2 + b1c2)x + a1b2 + b1d2 (3.5) T1 ◦ T2(x)= . (c1a2 + d1c2)x + c1b2 + d1d2 Three elementary Möbius transformations are expressed as follows: (I) T (x)= 1 x x + b, b ∈ R.(II)T (x)=ax, a ∈ R. (III) T (x)= = . x |x|2 Proposition 3.7. Every Möbius transformation T can be written as a product of three elementary Möbius transformations. More clearly, ax + b a 1 bc − ad (3.6) T (x)= = + · . cx + d c c cx + d

Theorem 3.8. The image of a hyperplane or sphere in Cln+1 under any M¨obius transformation T is still a hyperplane or sphere in Cln+1. ∂(y ,y , ··· ,y ) For M¨obius transformation y = T (x), denote J(T (x)) = 0 1 n . ∂(x0,x1, ··· ,xn)

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Theorem 3.9. Suppose T is a Möbius transformation. Then (ad − bc)n+1 (3.7) J(T (x)) = . |cx + d|2n+2 Example 3.10. By Theorem 3.8 and Theorem 3.9, Möbius transformation T map- y − a ping ∂B(0, 1) to ∂B(0, 1) with T (a) = 0 is determined by T (y)=± ,where 1 − ay a ∈ R and a = ±1. Moreover, if |a| < 1, then T maps B(0, 1) one-to-one onto itself and maps Rn+1 \ B(0, 1) one-to-one onto itself. If |a| > 1, then T maps B(0, 1) one-to-one onto Rn+1 \ B(0, 1) and maps Rn+1 \ B(0, 1) one-to-one onto B(0, 1). Proposition 3.11. Suppose T is a Möbius transformation. Then bc − ad (3.8) D [T (x)] = (n − 1) . |cx + d|2

n+1 Theorem 3.12. Let Ω∗ be an open non-empty subset of R and T be a M¨obius ax + b transformation, y = T (x)= , denoting Ω = T (Ω∗).Then cx + d cx + d cx + d (3.9) D f(T (x)) =(ad − bc) D [f(y)] , x |cx + d|n+1 |cx + d|n+3 y

where x ∈ Ω∗, y ∈ Ω. Proof. It can be directly proved that ⎧ ⎪ Dx[f(T (x))] = Dy [f(y)] , for T (x)=x + b, ⎪ ⎨⎪ Dx[f(T (x))] = aDy [f(y)] , for T (x)=ax, (3.10) ⎪ ⎪ ⎪ x x 1 ⎩ D [ f(T (x))] = − D [f(y)] , for T (x)= . x |x|n+1 |x|n+3 y x Then by (3.10), we have (3.11) a c2 bc − ad a D [f(y)] = D f(z + ) = D f u + y z − u 2 c bc ad c c c2 v bc − ad 1 a = (|v|n+1v)D f + − v | |n+1 2 ad bc v ⎡c v c ⎛ ⎞⎤ d c2 d d ⎢ x + ⎜bc − ad 1 a⎟⎥ = |x + |n+1(x + ) D ⎣ c f ⎝ + ⎠⎦ . ad − bc c c x d c2 d c |x + |n+1 x + c c In view of (3.6), (3.11) can be rewritten as (3.9); thus the result follows.

n+1 Theorem 3.13. Let Ω∗ be an open non-empty subset of R and let T be a ax + b M¨obius transformation, T (x)= , denoting Ω = T (Ω∗).ThenD[f]=0in Ω cx + d cx + d is equivalent to D f(T (x)) =0in Ω∗. |cx + d|n+1

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Remark 3.14. Theorem 3.13 generalizes the result in [9] (see page 155) in some sense. The first half of the result in Theorem 3.13 can also be found in [10], and an analogous result can be found in [14], while the Dirac operator in [14] is different from this paper. n+1 Theorem 3.15. Let Ω∗ be an open non-empty subset of R and let T be a Möbius ax + b transformation, y = T (x)= , denoting Ω = T (Ω∗).Then cx + d cx + d cx + d 1 (3.12) D f(T (x)) = J(T (x))D [f(y)] , |cx + d|n+1 x |cx + d|n+1 (ad − bc)n y where x ∈ Ω∗, y ∈ Ω. n+1 Theorem 3.16. Suppose Ω∗ is an open non-empty subset of R and T is a ax + b Möbius transformation, y = T (x)= , denoting Ω = T (Ω∗).Then cx + d cx + d cx + d (3.13) g(T (x)) D =(ad − bc)[g(y)] D · , |cx + d|n+1 x y |cx + d|n+3 where x ∈ Ω∗, y ∈ Ω. n+1 Theorem 3.17. Suppose Ω∗ is an open non-empty subset of R and T is a ax + b Möbius transformation, T (x)= , denoting Ω = T (Ω∗).Then[g]D =0in Ω cx +d cx + d is equivalent to g(T (x)) D =0in Ω∗. |cx + d|n+1 n+1 Theorem 3.18. Suppose Ω∗ is an open non-empty subset of R and T is a ax + b Möbius transformation, y = T (x)= , denoting Ω = T (Ω∗).Then cx + d cx + d cx + d 1 (3.14) g(T (x)) D · = J(T (x)) · [g(y)] D , |cx + d|n+1 x |cx + d|n+1 (ad − bc)n y where x ∈ Ω∗, y ∈ Ω.

4. Schwarz lemma Lemma 4.1. Let B(a, R) beaballcenteredata with radius R in Rn+1.Then ∀x ∈ B(a, R),

1 R2 −|x − a|2 (4.1) n+1 dS =1. ωn+1R |y − x| ∂B(a,R)

n+1 2 Theorem 4.2. Let B(0, 1) be an open unit ball in R , f ∈ C (B(0, 1),C(Vn,0)), [f]=0in B(0, 1), |f(x)|≤1, ∀x ∈ B(0, 1) and f(0) = 0.Then∀x ∈ B(0, 1), (4.2) |f(x)|≤M(n)|x|, 4 n+1 n where M(n)=3· ( ) − 1 · 2 2 . 3 Proof. By Theorem 2.7, ∀x ∈ B(0,r), 0

1 r2 −|x|2 (4.3) f(x)= n+1 f(y)dS. ωn+1r |y − x| ∂B(0,r)

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In view of f(0) = 0, by Corollary 2.9,

1 (4.4) 0 = n f(y)dS. ωn+1r ∂B(0,r) Combining (4.3) with (4.4), we have 2 −| |2 r x 1 − 1 (4.5) f(x)= n+1 n+1 f(y)dS. ωn+1r |y − x| |y| ∂B(0,r) It is easy to check that 1 1 − |y − x|n+1 |y|n+1 |x| (4.6) ≤ |y|n + |y|n−1|y − x| + ···+ |y − x|n | − |n+1| |n+1 y x y |x| |y| + |x| |y| + |x| n ≤ 1+ + ···+ . |y − x|n+1|y| |y| |y| By Lemma 4.1, we have

1 r2 −|x|2 (4.7) n+1 dS =1. ωn+1r |y − x| ∂B(0,r) Combining (4.5), (4.6) and (4.7) gives 2 2 n r −|x| 1 1 | |≤2 − | | f(x) 2 n+1 n+1 f(y) dS ωn+1r |y − x| |y| (4.8) ∂B(0,r) n n |x| r + |x| r + |x| ≤ 2 2 1+ + ···+ . r r r Taking r → 1 in (4.8), n n (4.9) |f(x)|≤2 2 |x| (1+(1+|x|)+···+(1+|x|) ) . 1 For 1 > |x| > , obviously 3 (4.10) |f(x)| < 3|x|. 1 For ≥|x|≥0, by (4.9), we have 3 4 n+1 n (4.11) |f(x)|≤3 · ( ) − 1 · 2 2 |x|. 3 Combining (4.10) with (4.11), the result follows.

n+1 1 Corollary 4.3. Let B(0, 1) be an open unit ball in R , f ∈ C (B(0, 1),C(Vn,0)), D[f]=0in B(0, 1), |f(x)|≤1, ∀x ∈ B(0, 1) and f(0) = 0.Then∀x ∈ B(0, 1), (4.12) |f(x)|≤M(n)|x|, 4 n+1 n where M(n)=3· ( ) − 1 · 2 2 . 3

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Remark 4.4. Theorem 4.2 and Corollary 4.3 are the so-called Schwarz lemmas for harmonic functions and monogenic functions in higher dimension respectively. Obviously, the above Schwarz lemma still holds for functions which satisfy [f]D =0 in B(0, 1). n Lemma 4.5. Let x = xiei ∈ Cln+1,andμ = μAeA ∈ C(Vn,0).Then i=0 A |xμ| = |μx| = |x||μ|. Proof. In view of the following equalities: | |2 · · | |2 | |2| |2 (4.13) μx =[μx (x μ)]0 = x [μμ]0 = x μ and | |2 | · |2 · · | |2 | |2| |2 (4.14) xμ = μ x =[(μ x) xμ]0 = x [μμ]0 = x μ , the result holds.

n+1 1 Theorem 4.6. Let B(0, 1) be an open unit ball in R , f ∈ C (B(0, 1),C(Vn,0)), D[f]=0in B(0, 1), |f(x)|≤1, ∀x ∈ B(0, 1) and f(λ)=0, λ ∈ R and |λ| < 1. Then ∀x ∈ B(0, 1), |x − λ| (4.15) |f(x)|≤M(n)(1 + |λ|)n , |1 − λx|n+1 n n · 4 n+1 − · 2 where x = xiei and M(n)=3 ( 3 ) 1 2 . i=0 Proof. For λ ∈ R and |λ| < 1, taking the M¨obius transformation y = T (x)= x − λ , by Example 3.10, T maps B(0, 1) in Rn+1 one-to-one onto itself with 1 − λx y + λ T (λ)=0.Thenx = T −1(y)= . For all y ∈ B(0, 1), denote 1+λy 1+λy (4.16) ϕ(y)=(1−|λ|)n f(T −1(y)). |1+λy|n+1 By Theorem 3.13, in view of D[f]=0inB(0, 1), we have D[ϕ(y)] = 0 in B(0, 1). It is clear that ϕ(0) = 0, and |ϕ(y)|≤1, ∀y ∈ B(0, 1). By Corollary 4.3, we have, ∀y ∈ B(0, 1), (4.17) |ϕ(y)|≤M(n)|y|,

n · 4 n+1 − · 2 where M(n)=3 ( 3 ) 1 2 . Combining (4.16) with (4.17), by Proposition 3.4 and Lemma 4.5, we have (1 −|λ|)n |x − λ| (4.18) |f(x)|≤M(n) . |1+λy|n |1 − λx| It is easy to check that 1 − λ2 (4.19) |1+λy| = . |1 − λx| Combining (4.18) with (4.19), (4.15) follows.

Remark 4.7. Inthecaseofλ = 0, Theorem 4.6 is just Corollary 4.3.

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n+1 1 Theorem 4.8. Let B(0, 1) be an open unit ball in R , f ∈ C (B(0, 1),C(Vn,0)), [f]D =0in B(0, 1), |f(x)|≤1, ∀x ∈ B(0, 1) and f(λ)=0, λ ∈ R and |λ| < 1. Then ∀x ∈ B(0, 1), |x − λ| (4.20) |f(x)|≤M(n)(1 + |λ|)n , |1 − λx|n+1

n n · 4 n+1 − · 2 where x = xiei and M(n)=3 ( 3 ) 1 2 . i=0

Acknowledgements This paper was done while the author was visiting Freie Universit¨at Berlin in 2011. The author is very grateful to Professor Heinrich Begehr and the reviewer for their suggestions.

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School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China E-mail address: [email protected]

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