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Lengths, areas and Lipschitz-type spaces of planar harmonic mappings

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Lengths, areas and Lipschitz-type spaces of planar harmonic mappings
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  LENGTHS, AREAS AND LIPSCHITZ-TYPE SPACES OF PLANARHARMONIC MAPPINGS SH. CHEN, S. PONNUSAMY  † , AND A. RASILA Abstract.  In this paper, we establish a three circles type theorem, involving theharmonic area function, for harmonic mappings. Also, we give bounds for lengthand area distortion for harmonic quasiconformal mappings. Finally, we will studycertain Lipschitz-type spaces on harmonic mappings. 1.  Introduction and main results Let  D  be a simply connected subdomain of the complex plane  C . A complex-valued function  f   defined in  D  is called a  harmonic mapping   in  D  if and only if both the real and the imaginary parts of   f   are real harmonic in  D . It is known thatevery harmonic mapping  f   defined in  D  admits a decomposition  f   =  h  +  g , where h  and  g  are analytic in  D . Since the Jacobian  J  f   of   f   is given by J  f   =  | f  z | 2 −| f  z | 2 :=  | h ′ | 2 −| g ′ | 2 ,f   is locally univalent and sense-preserving in  D  if and only if   | g ′ ( z  ) |  <  | h ′ ( z  ) |  in D ; or equivalently if   h ′ ( z  )   = 0 and the dilatation  ω  =  g ′ /h ′  has the property that | ω ( z  ) |  <  1 in  D  (see [19]). Let H ( D ) denote the class of all sense-preserving harmonicmappings in  D . We refer to [9, 11] for basic results in the theory of planar harmonic mappings.For  a  ∈ C , let  D ( a,r ) =  { z   :  | z   − a |  < r } . In particular, we use  D r  to denote thedisk  D (0 ,r ) and  D , the open unit disk  D 1 .The classical theorem of three circles [1, 28], also called  Hadamard’s three circles theorem  , states that if   f   is an analytic function in the annulus  B ( r 1 ,r 2 ) =  { z   : 0  <r 1  <  | z  |  =  r < r 2  <  ∞} , continuous on  B ( r 1 ,r 2 ), and  M  1 ,  M  2  and  M   are themaxima of   f   on the three circles corresponding to  r 1 ,  r 2  and  r , respectively, then M  log  r 2 r 1  ≤  M  log  r 2 r 1  M  log  rr 1 2  . Equivalently, we can reformulate this result into a simpler form. That is if   f   isanalytic on the annulus  B ( r 1 , 1) =  { z   : 0  < r 1  <  | z  |  <  1 } , continuous on theclosure, and | f  ( z  ) | ≤  m  =  r α 1 ,  | z  |  =  r 1  and  | f  ( z  ) | ≤  1 ,  | z  |  = 1 , File: Ch-P-R-AreaLength.tex, printed: 15-9-2013, 13.11 2000  Mathematics Subject Classification.  Primary: 30H05, 30H30; Secondary: 30C20, 30C45. Key words and phrases.  Harmonic mapping, three circles theorem, area function. † Corresponding author. This author is on leave from the Department ofMathematics, Indian Institute of Technology Madras, Chennai-600 036, India  . 1  2 Sh. Chen, S. Ponnusamy and A. Rasila then Hadamard’s result states that, for  r 1  ≤  r  ≤  1 , | f  ( z  ) | ≤  m log r log r 1  =  r α ,  | z  |  =  r, where  α  is an integer.The srcinal three circles theorem was given by Hadamard without proof in 1896[15], and comprehensive discussion about the history of this result can be found in[20, pp. 323–325] and [28]. It is a natural question, what results of this type can beproved for other classes of functions and, indeed, there are numerous generalizationsof the thee circles theorem in the literature, see e.g. [3, 21, 26, 30]. In this paper, our first aim is to establish an area version of the three circles theorem (cf. areaversion of Schwarz’ lemma [4]).For a harmonic mapping  f   in  D  and  r  ∈  [0 , 1), the  harmonic area function   S  f  ( r )of   f  , counting multiplicity, is defined by S  f  ( r ) =   D r J  f  ( z  ) dσ ( z  ) , where  dσ  denotes the normalized Lebesgue area measure on D (cf. [8]). In particular,let S  f  (1) = sup 0 <r< 1 S  f  ( r ) . Theorem 1.  Let   f   =  h  +  g  be harmonic in   D , where   h  and   g  are analytic. If  S  f  ( r 1 )  ≤  m <  1 ,  S  f  (1)  ≤  1  and for all   n  ∈ { 1 , 2 ,... } ,  | g ( n ) (0) | ≤ | h ( n ) (0) | , then for  r 1  ≤  r <  1 , (1.1)  S  f  ( r )  ≤  m log r log r 1 . The estimate of   (1.1)  is sharp and the extremal function is   f  ( z  ) =  αz   +  βz  , where  α  and   β   are constant with   | α | 2 −| β  | 2 = 1 . Corollary 1.1.  Let   f   be analytic in   D  satisfying   S  f  ( r 1 )  ≤  m  and   S  f  (1)  ≤  1 ,  where  0  < r 1  <  1 . Then for   r 1  ≤  r <  1 , (1.2)  S  f  ( r )  ≤  m log r log r 1 . The estimate of   (1.2)  is sharp and the extremal function is   f  ( z  ) =  λz  , where   | λ |  = 1 are constant. For  p  ∈  (0 , ∞ ], the  harmonic Hardy space   h  p consists of all harmonic functions  f  such that   f    p  <  ∞ , where  f    p  =  sup 0 <r< 1 M   p ( r,f  ) if   p  ∈  (0 , ∞ ) , sup z ∈ D | f  ( z  ) |  if   p  =  ∞ ,  and  M   p p ( r,f  ) = 12 π    2 π 0 | f  ( re iθ ) |  p dθ. If   f   ∈  h  p for some  p >  0, then the radial limits f  ( e iθ ) = lim r → 1 − f  ( re iθ )exist for almost every  θ  ∈  [0 , 2 π ) (cf. [11]).  Lengths, areas and Lipschitz-type spaces of planar harmonic mappings 3 We recall that a function  f   ∈ H ( D ) is said to be  K  -quasiregular  ,  K   ∈  [1 , ∞ ), if for  z   ∈  D , Λ f  ( z  )  ≤  Kλ f  ( z  ). In addition, if   f   is univalent in  D , then  f   is called a K  -quasiconformal   harmonic mapping  D .Let Ω be a domain of   C , with non-empty boundary. Let  d Ω ( z  ) be the Euclideandistance from  z   to the boundary  ∂  Ω of Ω. In particular, we always use  d ( z  ) todenote the Euclidean distance from  z   to the boundary of   D .  The area of a set G  ⊂  C  is denoted by  A ( G ). The area problem of analytic functions has attractedmuch attention (see [2, 29, 31, 32]). We investigate the area problem of harmonic mappings and obtain the following result. Theorem 2.  Let   Ω 1  and   Ω 2  be two proper and simply connected subdomains of   C containing the point of srcin. Then for a sense-preserving and   K  -quasiconformal harmonic mapping   f   defined in   Ω 1  with   f  (0) = 0 , (1.3)  KA  f  (Ω 1 ) ∩ Ω 2  +  A ( f  − 1 (Ω 2 ))  ≥  min { d 2Ω 1 (0) ,d 2Ω 2 (0) } . Moreover, if   K   = 1 , then the estimate of   (1.3)  is sharp. We remark that Theorem 2 is a generalization of [29, Theorem].For a harmonic mapping  f   defined on D , we use the following standard notations:Λ f  ( z  ) = max 0 ≤ θ ≤ 2 π | f  z ( z  ) +  e − 2 iθ f  z ( z  ) |  =  | f  z ( z  ) | + | f  z ( z  ) | and λ f  ( z  ) = min 0 ≤ θ ≤ 2 π | f  z ( z  ) +  e − 2 iθ f  z ( z  ) |  =  | f  z ( z  ) |−| f  z ( z  ) |  . Further, a planar harmonic mapping  f   defined on  D  is called a  harmonic Bloch mapping   if  β  f   = sup z,w ∈ D , z  = w | f  ( z  ) − f  ( w ) | ρ ( z,w )  <  ∞ . Here  β  f   is called the  Lipschitz number   of   f  , and ρ ( z,w ) = 12 log  1 + | ( z   − w ) / (1 − zw ) | 1 −| ( z   − w ) / (1 − zw ) |  = arctanh  z   − w 1 − zw  denotes the hyperbolic distance between  z   and  w  in  D . It is known that β  f   = sup z ∈ D  (1 −| z  | 2 )Λ f  ( z  )  . Clearly, a harmonic Bloch mapping  f   is uniformly continuous as a map betweenmetric spaces, f   : ( D ,ρ )  →  ( C , |·| ) , and for all  z,w  ∈ D  we have the Lipschitz inequality | f  ( z  ) − f  ( w ) | ≤  β  f   ρ ( z,w ) . A well-known fact is that the set of all harmonic Bloch mappings, denoted by thesymbol  HB  , forms a complex Banach space with the norm  ·  given by  f   HB  =  | f  (0) | + sup z ∈ D { (1 −| z  | 2 )Λ f  ( z  ) } .  4 Sh. Chen, S. Ponnusamy and A. Rasila Specially, we use  B   to denote the set of all analytic functions defined in  D  whichforms a complex Banach space with the norm  f   B  =  | f  (0) | + sup z ∈ D { (1 −| z  | 2 ) | f  ′ ( z  ) |} . The reader is referred to [10, Theorem 2] (or [5, 6]) for a detailed discussion. For  r  ∈  [0 , 1), the length of the curve  C  ( r ) =  w  =  f  ( re iθ ) :  θ  ∈  [0 , 2 π ]  , countingmultiplicity, is defined by l f  ( r ) =    2 π 0 | df  ( re iθ ) |  =  r    2 π 0  f  z ( re iθ ) − e − 2 iθ f  z ( re iθ )  dθ, where  f   is a harmonic mapping defined in D . In particular, let  l f  (1) = sup 0 <r< 1 l f  ( r ). Theorem 3.  Let   f  ( z  ) =  ∞ n =0 a n z  n +  ∞ n =1 b n z  n be a sense-preserving   K  -quasiconformal harmonic mapping. If   l f  (1)  <  ∞ , then for   n  ≥  1 , (1.4)  | a n | + | b n | ≤  Kl f  (1)2 nπ and  (1.5) Λ f  ( z  )  ≤  l f  (1) √  K  2 π (1 −| z  | ) . Moreover,  f   ∈ HB   and   β  f   ≤  l f  (1) √  K π  .  In particular, if   K   = 1 , the estimates of   (1.4) and   (1.5)  are sharp, and the extremal function is   f  ( z  ) =  z. A continuous increasing function  ω  : [0 , ∞ )  →  [0 , ∞ ) with  ω (0) = 0 is called a majorant   if   ω ( t ) /t  is non-increasing for  t >  0. Given a subset Ω of   C , a function f   : Ω  →  C  is said to belong to the  Lipschitz space   L ω (Ω) if there is a positiveconstant  C   such that(1.6)  | f  ( z  ) − f  ( w ) | ≤  Cω ( | z   − w | ) for all  z, w  ∈  Ω . For  δ  0  >  0, let(1.7)    δ 0 ω ( t ) t dt  ≤  C   · ω ( δ  ) ,  0  < δ < δ  0 , and(1.8)  δ     + ∞ δ ω ( t ) t 2  dt  ≤  C   · ω ( δ  ) ,  0  < δ < δ  0 , where  ω  is a majorant and  C   is a positive constant.A majorant  ω  is said to be  regular   if it satisfies the conditions (1.7) and (1.8) (see[12, 13, 22, 23, 24]).Let  G  be a proper subdomain of   C . We say that a function  f   belongs to the local Lipschitz space   loc L ω ( G ) if (1.6) holds, with a fixed positive constant  C  ,whenever  z   ∈  G  and  | z   − w |  <  12 d G ( z  ) (cf. [14, 18]). Moreover,  G  is said to be a L ω -extension domain   if   L ω ( G ) = loc L ω ( G ) .  The geometric characterization of   L ω -extension domains was first given by Gehring and Martio [14]. Then Lappalainen[18] generalized their characterization, and proved that  G  is a  L ω -extension domain  Lengths, areas and Lipschitz-type spaces of planar harmonic mappings 5 if and only if each pair of points  z,w  ∈  G  can be joined by a rectifiable curve  γ   ⊂  G satisfying(1.9)   γ  ω ( d G ( z  )) d G ( z  )  ds ( z  )  ≤  Cω ( | z   − w | )with some fixed positive constant  C   =  C  ( G,ω ), where  ds  stands for the arc lengthmeasure on  γ  . Furthermore, Lappalainen [18, Theorem 4.12] proved that  L ω -extension domains exist only for majorants  ω  satisfying (1.7). Theorem A.  ([16, Theorem 3])  f   ∈ B   if and only if  sup z,w ∈ D ,z  = w   (1 −| z  | 2 )(1 −| w | 2 ) | f  ( z  ) − f  ( w ) || z   − w |  <  ∞ . The following result is a generalization of Theorem A. For the related studies of this topic for real functions, we refer to [25, 27]. Theorem 4.  Let   f   be a harmonic mapping in   D  and   ω  be a majorant. Then the  following are equivalent: (a)  There exists a constant   C  1  >  0  such that for all   z   ∈ D , Λ f  ( z  )  ≤  C  1 ω   1 d ( z  )  ;(b)  There exists a constant   C  2  >  0  such that for all   z,w  ∈ D  with   z    =  w , | f  ( z  ) − f  ( w ) || z   − w | ≤  C  2 ω   1   d ( z  ) d ( w )  ;(c)  There exists a constant   C  3  >  0  such that for all   r  ∈  (0 ,d ( z  )] , 1 | D ( z,r ) |   B n ( z,r ) | f  ( ζ  ) − f  ( z  ) | dA ( ζ  )  ≤  C  3 rω  1 r  , where   dA  denotes the Lebesgue area measure in   D . Note that if   ω ( t ) =  t  and  f   is analytic, then (a) ⇐⇒ (b) in Theorem 4 implies thatTheorem A.Krantz [17] proved the following Hardy-Littlewood-type theorem for harmonicfunctions with respect to the majorant  ω ( t ) =  ω α ( t ) =  t α (0  < α  ≤  1). Theorem B.  ([17, Theorem 15.8])  Let   u  be a real harmonic function in   D  and  0  < α  ≤  1 . Then   u  satisfies  |∇ u ( z  ) | ≤  C ω α  d ( z  )  d ( z  )  for all   z   ∈ D if and only if  | u ( z  ) − u ( w ) | ≤  Cω α ( | z   − w | )  for all   z,w  ∈ D . We generalize Theorem B to the following form.
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