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Schwarz-Pick type estimates of pluriharmonic mappings in the unit polydisk

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Schwarz-Pick type estimates of pluriharmonic mappings in the unit polydisk
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    a  r   X   i  v  :   1   4   0   9 .   7   8   9   7  v   1   [  m  a   t   h .   C   V   ]   2   8   S  e  p   2   0   1   4 SCHWARZ-PICK TYPE ESTIMATES OF PLURIHARMONICMAPPINGS IN THE UNIT POLYDISK SHAOLIN CHEN AND ANTTI RASILA Abstract.  In this paper, we will give Schwarz-Pick type estimates of arbitraryorder partial derivatives for bounded pluriharmonic mappings defined in the unitpolydisk. Our main results are generalizations of results of Colonna for planarharmonic mappings in [Indiana Univ. Math. J. 38: 829–840, 1989]. 1.  Introduction and main results Let  C n denote the complex Euclidean  n -space. For  z   = ( z  1 ,...,z  n )  ∈  C n , theconjugate of   z  , denoted by  z  , is defined by  z   = ( z  1 ,...,z  n ) .  For  z   and  w  =( w 1 ,...,w n )  ∈  C n , the  inner product   on  C n and the  Euclidean norm   of   z   aregiven by   z,w   :=  nk =1 z  k w k  and   z    :=   z,z   1 / 2 ,  respectively. For  a  ∈  C n , B n ( a,r ) =  { z   ∈  C n :   z   −  a   < r }  is the (open) ball of radius  r  with center  a .Also, we let  B n ( r ) :=  B n (0 ,r ) and use  B n to denote the unit ball  B n (1), and  D  =  B 1 .Let D n =  D ×···× D  ( n  times) be the polydisc in  C n and T n =  T ×···× T ( n  times),where  T  is the unit circle in  C 1 . A multi-index  k  = ( k 1 ,...,k n ) consists of   n  non-negative integers  k  j , where  j  ∈ { 1 ,...,n } . The degree of a multi-index  k  is the sum | k |  =  n j =1 k  j . Given another multi-index  α  = ( α 1 ,...,α n ), let  k α = ( k α 1 1  ,...,k α n n  ).For  z   = ( z  1 ,...,z  n )  ∈  C n , let   z    =  n j =1 | z   j | 2  1 / 2 ,   z   ∞  = max 1 ≤  j ≤ n  | z   j |  and z  k = Π nk =1 z  k j  j  . A continuous complex-valued function  f   defined on a domain Ω  ⊂  C n is said to be pluriharmonic   if for each fixed  z   ∈  Ω and  θ  ∈  ∂  B n , the function  f  ( z  + θζ  ) is harmonicin  { ζ   :   ζ    < d Ω ( z  ) } , where  d Ω ( z  ) denotes the distance from  z   to the boundary  ∂  Ωof Ω (cf. [26]). If Ω  ⊂  C n is a simply connected domain, then a function  f   : Ω  →  C is pluriharmonic if and only if   f   has a representation  f   =  h  +  g,  where  h  and  g  areholomorphic in Ω (see [30]). Let  P  (Ω , C N  ) be the class of all pluriharmonic mappings f   = ( f  1 ,...,f  N  ) from a domain Ω  ⊂  C n to  C N  , where  N   is a positive integerand  f   j  (1  ≤  j  ≤  N  ) are pluriharmonic mappings from Ω into  C .  For a mapping f   ∈ P  (Ω , C N  ) ,  we use  Df   and  Df   to denote the two  N   × n  matrices ( ∂f   j /∂z  m ) N  × n and ( ∂f   j /∂z  m ) N  × n , respectively. We refer to [5, 10, 11, 12, 19, 22] for more details on pluriharmonic mappings. In particular, if   n  = 1, then pluriharmonic mappingsare planar harmonic mappings (cf. [14, 18]). Therefore, pluriharmonic mappings File: pluri.tex, printed: 30-9-2014, 7.35 2000  Mathematics Subject Classification.  Primary: 30C80; Secondary: 32U99. Key words and phrases.  Pluriharmonic mapping, Schwarz-Pick type estimate, polydisk. 1  2 Sh. Chen and A. Rasila can be understood as the natural generalization of planar harmonic mappings toseveral complex variables.We first recall the classical Schwarz Lemma for analytic functions  f   of   D  intoitself:(1.1)  | f  ′ ( z  ) | ≤  1  − | f  ( z  ) | 2 1  − | z  | 2  , z   ∈  D . In 1920, Sz´asz [29] extended the inequality (1.1) to the following estimate involving higher order derivatives:(1.2)  | f  (2 m +1) ( z  ) | ≤  (2 m  + 1)!(1  − | z  | 2 ) 2 m +1 m  k =0  mk  2 | z  | 2 k , where  m  ∈ { 1 , 2 ,... } .  In 1985, Ruscheweyh (cf. [3, 4, 27]) improved (1.2) to the following sharp estimate:(1.3)  | f  ( n ) ( z  ) | ≤  n !(1  − | f  ( z  ) | 2 )(1  − | z  | ) n (1 +  | z  | ) . Recently, the inequality (1.3) was generalized into a variety of forms (see [1, 2, 4, 16, 17, 24, 31]). In 1989, Colonna established an analogue of the Schwarz-Pick lemma for planarharmonic mappings, which is the following. Theorem A.  ([15, Theorems 3 and 4])  Let   f   be a harmonic mapping of   D  into  D .Then for   z   ∈  D ,  ∂f  ( z  ) ∂z   +  ∂f  ( z  ) ∂z   ≤  4 π 11  − | z  | 2 . This estimate is sharp, and all the extremal functions are  f  ( z  ) = 2 γ π  arg  1 +  ψ ( z  )1  −  ψ ( z  )  , where   | γ  |  = 1  and   ψ  is a conformal automorphism of   D . We refer to [5, 6, 7, 8, 9, 10, 13, 20, 23, 28] for further discussion on this topic. In this paper, we generalize Theorem A to higher dimensional case, and give theestimate for the partial derivatives of arbitrary order. One should note that thehigher dimensional case is very different from the one dimensional situation and,because we are dealing with partial derivatives of arbitrary order, the method of proof from [15] can not be used. By using the coefficient estimates and the Cauchyintegral formula, we prove the following result. Theorem 1.  Let   f   ∈ P  ( D n , D ) .  Then   ∂  α f  ( z  ) ∂z  α 1 1  ··· ∂z  α n n  +  ∂  α f  ( z  ) ∂z  α 1 1  ··· ∂z  α n n  ≤  α !4 π (1 +   z   ∞ ) | α |− n (1  −  z   2 ∞ ) | α |  , where   α  = ( α 1 , ···  ,α n )  is a multi-index with   α  j  >  0 , j  ∈ { 1 ,...,n } .  Schwarz-Pick type estimates of pluriharmonic mappings in the unit polydisk 3 We remark that if   | α |  =  n  = 1, then Theorem 1 coincides with Theorem A.It is well-known that there are no biholomorphic mappings between  D n and  B n (cf. [25, 26]). Hence pluriharmonic mappings between  D n and  B n are particularlyinteresting in the theory of several complex variables. The following result is ananalogue of [5, Theorem 4] for vector-valued pluriharmonic mappings defined in  B n . Theorem 2.  If   f   ∈ P  ( D n , B N  ) , then  (1.4) max θ ∈ C n ,   θ  ∞ =1  Df  ( z  ) θ  +  Df  ( z  ) θ  ≤  4 π (1  −  z   2 ∞ ) , where   θ  is regarded as a column vector.If   f   ∈ P  ( D n , B N  )  with   f  (0) = 0 , then  (1.5)   f  ( z  )  ≤  4 π  arctan  z   ∞ . We remark that if   n  =  N   = 1, then the estimates (1.4) and (1.5) coincide withTheorem A and [21, Lemma], respectively.2.  The proofs of the main results Lemma 1.  Let   m  be a positive integer and   γ   be a real constant. Then     2 π 0 | cos( mθ  +  γ  ) | dθ  = 4 . Proof.  By elementary calculations, we have    2 π 0 | cos( mθ  +  γ  ) | dθ  = 1 m    2 mπ + γ γ  | cos t | dt = 1 m 2 m  k =1    kπ + γ  ( k − 1) π + γ  | cos t | dt = 1 m 2 m  k =1    π 0 | cos t | dt = 2    π 0 | cos t | dt = 4 . The proof of the lemma is complete.    4 Sh. Chen and A. Rasila Proof of Theorem 1.  Since  D n is a simply connected domain in  C n , we see that f   has a representation  f   =  h  +  g , where  h  and  g  are holomorphic in  D n . Let k  = ( k 1 ,...,k n ) be a multi-index. Then  f   can be expressed as a power series asfollows f  ( z  ) =  h ( z  ) +  g ( z  ) =  k a k z  k +  k b k z  k . Claim 1.  For  | k | ≥  1,  | a k |  +  | b k | ≤  4 π . Now we prove Claim 1. Let  z   = ( z  1 ,...,z  n ) = ( r 1 e iθ 1 ,...,r n e iθ n )  ∈  D n , where0  ≤  r  j  <  1 for all  j  ∈ { 1 ,...,n } .  Then for  | k | ≥  1,(2.1)  a k r k 1 1  ··· r k n n  = 1(2 π ) n    2 π 0 ···    2 π 0 f  ( r k 1 1  e iθ 1 ,...,r k n n  e iθ n ) e − i  nj =1  k j θ j dθ 1  ··· dθ n and(2.2)  b k r k 1 1  ··· r k n n  = 1(2 π ) n    2 π 0 ···    2 π 0 f  ( r k 1 1  e iθ 1 ,...,r k n n  e iθ n ) e i  nj =1  k j θ j dθ 1  ··· dθ n . By (2.1) and (2.2), we get r k 1 1  ··· r k n n  ( | a k |  +  | b k | )(2.3)=  1(2 π ) n    2 π 0 ···    2 π 0  e − i  nj =1  k j θ j e − i arg a k + e i  nj =1  k j θ j e i arg b k  f  ( r k 1 1  e iθ 1 ,...,r k n n  e iθ n ) dθ 1  ··· dθ n  ≤  1(2 π ) n    2 π 0 ···    2 π 0  1 +  e (2  nj =1  k j θ j +arg a k +arg b k ) i  ×  f  ( r k 1 1  e iθ 1 ,...,r k n n  e iθ n )  dθ 1  ··· dθ n ≤  2(2 π ) n    2 π 0 ···    2 π 0  cos  n   j =1 k  j θ  j  + (arg a k  + arg b k )2  dθ 1  ··· dθ n . Since  | k | ≥  1, without loss of generality, we assume that  k 1   = 0. By using Lemma1, we see that(2.4)    2 π 0  cos   n   j =1 k  j θ  j  + (arg a k  + arg b k )2  dθ 1  = 4 . Then (2.3) and (2.4) yield that r k 1 1  ··· r k n n  ( | a k |  +  | b k | )  ≤  4 π. For  j  ∈ { 1 ,...,n } , by letting  r  j  →  1 − , we obtain the desired result.  Schwarz-Pick type estimates of pluriharmonic mappings in the unit polydisk 5 For  j  ∈ { 1 ,...,n }  and  z   = ( z  1 ,...,z  n )  ∈  D n ,  let φ ( ζ  ) = ( φ 1 ( ζ  1 ) ,...,φ n ( ζ  n )) , where  ζ   = ( ζ  1 ,...,ζ  n )  ∈  D n and φ  j ( ζ   j ) =  z   j  +  ζ   j 1 +  z   j ζ   j . Then  f   ◦  φ  can be written as following form T  ( ζ  ) =  f  ( φ ( ζ  )) =  H  ( ζ  ) +  G ( ζ  ) =  k c k ζ  k +  k d k ζ  k , where  H   =  h  ◦  φ  and  G  =  g  ◦  φ . By using the proof of Claim 1, we get(2.5)  | c k |  +  | d k | ≤  4 π. For  r  ∈  (0 , 1) and  z   ∈  D n with   z   ∞  < r , by the Cauchy integral formula (cf.[25, 31]), we see that f  ( z  ) = 1(2 πi ) n   | η 1 | = r ···   | η n | = r h ( η 1 ,...,η n )  n j =1 ( η  j  −  z   j ) dη 1 ··· dη n (2.6)+ 1(2 πi ) n   | η 1 | = r ···   | η n | = r g ( η 1 ,...,η n )  n j =1 ( η  j  −  z   j ) dη 1  ··· dη n , which implies that(2.7)  ∂  α f  ( z  ) ∂z  α 1 1  ··· ∂z  α n n =  α !(2 πi ) n   | η 1 | = r ···   | η n | = r h ( η 1 ,...,η n )  n j =1 ( η  j  −  z   j ) α j +1 dη 1  ··· dη n and(2.8)  ∂  α f  ( z  ) ∂z  α 1 1  ··· ∂z  α n n =  α !(2 πi ) n   | η 1 | = r ···   | η n | = r g ( η 1 ,...,η n )  n j =1 ( η  j  −  z   j ) α j +1 dη 1 ··· dη n . For  j  ∈ { 1 ,...,n } , by taking  η  j  =  φ  j ( ζ   j ) =  z j + ζ  j 1+ z j ζ  j , we see from (2.5), (2.7) and (2.8)that ∂  α f  ( z  ) ∂z  α 1 1  ··· ∂z  α n n =  α !(2 πi ) n  n j =1 (1  − | z   j | 2 ) α j ×   | φ 1 ( ζ  1 ) | = r ···   | φ n ( ζ  n ) | = r H  ( ζ  1 ,...,ζ  n )  n j =1 (1 +  z   j ζ   j ) α j − 1  n j =1 ζ  α j +1  j dζ  1 ··· dζ  n =  α !  n j =1 (1  − | z   j | 2 ) α j × α 1 − 1  k 1 =0 ··· α n − 1  k n =0  α 1  −  1 k 1  ···  α n  −  1 k n  c α 1 − k 1 ,...,α n − k n n   j =1 z  k j  j
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