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

Nonlinear Analysis 115 (2015) 62–70

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Lengths, areas and Lipschitz-type spaces of planar harmonic mappings Shaolin Chen a , Saminathan Ponnusamy b , Antti Rasila c,∗ a

Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008, People’s Republic of China b

Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India c

Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 Aalto, Finland

article

info

Article history: Received 24 September 2014 Accepted 10 December 2014 Communicated by Enzo Mitidieri

abstract In this paper, we give bounds for length and area distortion for harmonic K -quasiconformal mappings, and investigate certain Lipschitz-type spaces on harmonic mappings. © 2014 Elsevier Ltd. All rights reserved.

MSC: primary 30H05 30H30 secondary 30C20 30C45 Keywords: Harmonic and Bloch mappings Quasiconformal mappings Hardy and Lipschitz spaces Length Area function

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 its real and the imaginary parts of f are real harmonic in D. It is known that every harmonic mapping f defined in D admits a decomposition f = h + g, where h and g are analytic in D. Since the Jacobian Jf of f is given by Jf = |fz |2 − |fz |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 [16]). Let H (D) denote the class of all sense-preserving harmonic mappings in D. We refer to [7,9] 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 Dr to denote the disk D(0, r ) and D, the open unit disk D1 . For a harmonic mapping f defined on D, we use the following standard notations:

Λf (z ) = max |fz (z ) + e−2iθ fz (z )| = |fz (z )| + |fz (z )| 0≤θ≤2π

∗

Corresponding author. E-mail addresses: [email protected] (S. Chen), [email protected], [email protected] (S. Ponnusamy), [email protected] (A. Rasila).

http://dx.doi.org/10.1016/j.na.2014.12.005 0362-546X/© 2014 Elsevier Ltd. All rights reserved.

S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

63

and

  λf (z ) = min |fz (z ) + e−2iθ fz (z )| =  |fz (z )| − |fz (z )| . 0≤θ≤2π

We recall that a function f ∈ H (D) is said to be K -quasiregular, K ∈ [1, ∞), if for z ∈ D, Λf (z ) ≤ K λf (z ). In addition, if f is univalent in D, then f is called a K -quasiconformal harmonic mapping in D. Let Ω be a domain of C, with non-empty boundary. Let dΩ (z ) be the Euclidean distance from z to the boundary ∂ Ω of Ω . In particular, we always use d(z ) to denote the Euclidean distance from z to the boundary of D. The normalized area of a set G ⊂ C is denoted by A(G). It means that A(G) = area (G)/π , where area (G) is the area of G. The area problem of analytic functions has attracted much attention (see [1,22–24]). We investigate the area problem of harmonic mappings and obtain the following result. Theorem 1. Let Ω1 and Ω2 be two proper and simply connected subdomains of C containing the point of origin. Then for a sense-preserving and K -quasiconformal harmonic mapping f defined in Ω1 with f (0) = 0, KA f (Ω1 ) ∩ Ω2 + A(f −1 (Ω2 )) ≥ min{d2Ω1 (0), d2Ω2 (0)}.





(1)

Moreover, if K = 1, then the estimate of (1) is sharp. We remark that Theorem 1 is a generalization of [22, Theorem]. A planar harmonic mapping f defined on D is called a harmonic Bloch mapping if

βf =

sup

z ,w∈D, z ̸=w

|f (z ) − f (w)| < ∞. ρ(z , w)

Here βf is called the Lipschitz number of f , and

ρ(z , w) =

1 2

 log

1 + |(z − w)/(1 − z w)| 1 − |(z − w)/(1 − z w)|



   z−w    = arctanh  1 − zw 

denotes the hyperbolic distance between z and w in D. It is known that

  βf = sup (1 − |z |2 )Λf (z ) . z ∈D

Clearly, a harmonic Bloch mapping f is uniformly continuous as a map between metric spaces, f : (D, ρ) → (C, | · |), and for all z , w ∈ D we have the Lipschitz inequality

|f (z ) − f (w)| ≤ βf ρ(z , w) . A well-known fact is that the set of all harmonic Bloch mappings, denoted by the symbol H B , forms a complex Banach space with the norm ∥ · ∥ given by

∥f ∥HB = |f (0)| + sup{(1 − |z |2 )Λf (z )}. z ∈D

Specially, we use B to denote the set of all analytic functions defined in D which forms a complex Banach space with the norm

∥f ∥B = |f (0)| + sup{(1 − |z |2 )|f ′ (z )|}. z ∈D

The reader is referred to [8, Theorem 2] (see also[5,6]) for a detailed discussion.  For r ∈ [0, 1), the length of the curve C (r ) = f (reiθ ) : θ ∈ [0, 2π ] , counting multiplicity, is defined by

ℓf (r ) =

2π



iθ

|df (re )| = r

0

2π



  fz (reiθ ) − e−2iθ fz (reiθ ) dθ ,

0

where f is a harmonic mapping defined in D. In particular, it is convenient to set

ℓf (1) = sup ℓf (r ). 0
Theorem 2. Let f (z ) = then for n ≥ 1,

|an | + |bn | ≤

∞

n =0

K ℓf (1) 2nπ

an z n +

∞

n =1

bn z be a sense-preserving K -quasiconformal harmonic mapping. If ℓf (1) < ∞, n

(2)

and

√ ℓf (1) K . Λf ( z ) ≤ 2π (1 − |z |)

(3)

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S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

Moreover, f ∈ H B and βf ≤ is f (z ) = z .

√ ℓf (1) K . π

In particular, if K = 1, the estimates of (2) and (3) are sharp, and the extremal function

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 Lω (Ω ) if there is a positive constant C such that

|f (z ) − f (w)| ≤ C ω(|z − w|) for all z , w ∈ Ω .

(4)

For δ0 > 0, let δ



ω(t )

0

t



+∞

dt ≤ C · ω(δ),

0 < δ < δ0 ,

(5)

and

δ

δ

ω(t ) t2

dt ≤ C · ω(δ),

0 < δ < δ0 ,

(6)

where ω is a majorant and C is a positive constant. A majorant ω is said to be regular if it satisfies the conditions (5) and (6) (see [10,11,17–19]). Let G be a proper subdomain of C. We say that a function f belongs to the local Lipschitz space loc Lω (G) if (4) holds, with a positive constant C , whenever z ∈ G and |z − w| < 21 dG (z ) (cf. [12,15]). Moreover, G is said to be a Lω -extension domain if Lω (G) = loc Lω (G). The geometric characterization of Lω -extension domains was first given by Gehring and Martio [12]. Then Lappalainen [15] generalized their characterization, and proved that G is a Lω -extension domain if and only if each pair of points z , w ∈ G can be joined by a rectifiable curve γ ⊂ G satisfying

 γ

ω(dG (ζ )) ds(ζ ) ≤ C ω(|z − w|) dG (ζ )

(7)

with some fixed positive constant C = C (G, ω), where ds stands for the arc length measure on γ . Furthermore, Lappalainen [15, Theorem 4.12] proved that Lω -extension domains exist only for majorants ω satisfying (5). Theorem A (Theorem 3 [13]). f ∈ B if and only if

  (1 − |z |2 )(1 − |w|2 )|f (z ) − f (w)| sup < ∞. |z − w| z ,w∈D,z ̸=w The following result is a generalization of Theorem A. For the related studies of this topic for real functions, we refer to [20,21]. Theorem 3. Let f be a harmonic mapping in D and ω be a majorant. Then the following are equivalent: (a) There exists a constant C1 > 0 such that for all z ∈ D,

Λ f ( z ) ≤ C1 ω



1 d(z )

 ;

(b) There exists a constant C2 > 0 such that for all z , w ∈ D with z ̸= w ,

  |f (z ) − f (w)| 1 ≤ C2 ω √ ; |z − w| d(z )d(w) (c) There exists a constant C3 > 0 such that for all r ∈ (0, d(z )],  1 1 |f (ζ ) − f (z )| dA(ζ ) ≤ C3 r ω , |D(z , r )| D(z ,r ) r where dA denotes the Lebesgue area measure in D. Note that if ω(t ) = t and f is analytic, then the two way implications (a) ⇐⇒ (b) in Theorem 3 give Theorem A. Krantz [14] proved the following Hardy–Littlewood-type theorem for harmonic functions with respect to the majorant ω(t ) = ωα (t ) = t α (0 < α ≤ 1). Theorem B ([14, Theorem 15.8]). Let u be a real harmonic function in D and 0 < α ≤ 1. Then u satisfies

  ωα d(z ) |∇ u(z )| ≤ C d(z )

for all z ∈ D

S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

65

if and only if

|u(z ) − u(w)| ≤ C ωα (|z − w|)

for all z , w ∈ D.

We have now the generalization of Theorem B in the following form. Theorem 4. Let ω be a majorant satisfying (5), Ω be a Lω -extension domain and f be a harmonic mapping in Ω . Then there exists a constant C4 > 0 such that

Λf (z ) ≤ C4

  ω dΩ (z ) dΩ (z )

for all z ∈ Ω

if and only if there exists a constant C5 > 0 such that

|f (z ) − f (w)| ≤ C5 ω(|z − w|)

for all z , w ∈ Ω .

The proofs of Theorems 1 and 2 will be given in Section 2, and the proofs of Theorems 3 and 4 will be presented in Section 3. 2. Length and area of harmonic mappings For p ∈ (0, ∞], the harmonic Hardy space hp consists of all harmonic functions f such that ∥f ∥p < ∞, where

∥f ∥p =

  sup Mp (r , f )

if p ∈ (0, ∞),

sup |f (z )|

if p = ∞,

0
and 1

2π



|f (reiθ )|p dθ . 2π 0 If f ∈ hp for some p > 0, then the radial limits Mpp

(r , f ) =

f (eiθ ) = lim f (reiθ ) r →1−

exist for almost every θ ∈ [0, 2π ) (cf. [9]). For a harmonic mapping f in D and r ∈ [0, 1), the harmonic area function Sf (r ) of f , counting multiplicity, is defined by Sf ( r ) =



Jf (z ) dσ (z ), Dr

where dσ denotes the normalized Lebesgue area measure on D (cf. [3]). In particular, it is convenient to set Sf (1) = sup Sf (r ). 0
Lemma 1. Let f be a sense-preserving and K -quasiconformal harmonic mapping in D with f (0) = 0. If A(f (D)) < ∞, then f ∈ h2 and

∥f ∥22 ≤ KA(f (D)).

(8)

Moreover, if K = 1, then the estimate of (8) is sharp and the extremal function is f (z ) = z. Proof. Let f be a sense-preserving and K -quasiconformal harmonic mapping in D with f (0) = 0. Then, by the definition of Sf (r ), we see that Sf (1) = A(f (D)) =



Jf (z ) dσ (z ) =



D

≥ ≥ = ≥

1

Λf (z )λf (z ) dσ (z ) D



Λ2f (z ) dσ (z )

K D   1 K

 |fz (z )|2 + |fz (z )|2 dσ (z )

D

∞ 1 

K n =1

n(|an |2 + |bn |2 )

∞ 1 

(|an |2 + |bn |2 )

K n =1 1 = ∥f ∥22 . K

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S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

Then ∥f ∥22 ≤ KA(f (D)). Furthermore, if K = 1, then the function f (z ) = z shows that the estimate of (8) is sharp. The proof of the lemma is complete.  Proof of Theorem 1. Let D1 = f (Ω1 ) ∩ Ω2 and D2 = f −1 (Ω2 ). It is not difficult to see that D2 = f −1 (D1 ). Without loss of generality, we assume that for k = 1, 2, A(Dk ) < ∞. Let E be the component of the open set D2 containing the origin. Then there is a universal covering mapping ϕ such that ϕ : D → E with ϕ(0) = 0. Let F = f ◦ ϕ . It is easy to see that F is also a K -quasiconformal harmonic mapping. By using Lemma 1, we have

∥ϕ∥22 ≤ A(E ) ≤ A(D2 ) and

∥F ∥22 ≤ A(f (E )) ≤ KA(f (D2 )) = KA(D1 ), which imply that

∥ϕ∥22 + ∥F ∥22 ≤ A(D2 ) + KA(D1 ). Since ϕ and F belong to h2 , we conclude that the linear measure m(γ ) of

  γ = ξ ∈ ∂ D : both |ϕ(ξ )| and |F (ξ )| are finite is 2π . Let

γϕ = {ξ ∈ γ : |ϕ(ξ )| ≥ dΩ1 (0)} and, similarly, let

γF = {ξ ∈ γ : |F (ξ )| ≥ dΩ2 (0)}. Then

∥ϕ∥22 ≥

1



2π

γϕ

d2Ω1 (0)| dξ | ≥

d2Ω1 (0)m(γϕ )

(9)

2π

and

∥F ∥22 ≥

1 2π

 γF

d2Ω2 (0)| dξ | ≥

d2Ω2 (0)m(γF ) 2π

.

(10)

Claim. γ = γϕ ∪ γF . Suppose γ ̸= γϕ ∪γF . Then there is a point ξ0 ∈ γ \(γϕ ∪γF ) such that ϕ(ξ0 ) ∈ Ω1 with |ϕ(ξ0 )| < dΩ1 (0), and F (ξ0 ) ∈ Ω2 with |F (ξ0 )| < dΩ2 (0). Since f is continuous at ϕ(ξ0 ), we find that F (ξ0 ) = lim F (r ξ0 ) = lim f (ϕ(r ξ0 )) = f (ϕ(ξ0 )). r →1−

r →1−

On the other hand, l = {ϕ(r ξ0 ) : r ∈ [0, 1]} is a curve joining 0 and ϕ(ξ0 ) in Ω1 . Hence ϕ(ξ0 ) ∈ E which is a contradiction with the covering property of D induced by ϕ over E. Hence by (9), (10) and the Claim, we get A(D2 ) + KA(D1 ) ≥ ∥ϕ∥22 + ∥F ∥22 ≥

m(γϕ ) + m(γF ) 2π

min{d2Ω1 (0), d2Ω2 (0)}

= min{d2Ω1 (0), d2Ω2 (0)}. Next we prove the sharpness part. We consider the case that K = 1. Let Ω1 = Ω2 = D and for z ∈ D, let f (z ) = tz, where 0 < t < 1. Then A(D1 ) + A(D2 ) = 1 + t 2 and d(0) = 1. The arbitrariness of t shows that the estimate in (1) is sharp. The proof of the theorem is complete. The following result is well-known (cf. [2]). Lemma C. Among all rectifiable Jordan curves of a given length, the circle has the maximum interior area.



S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

67

Proof of Theorem 2. We first prove (2). By elementary computations, we have

ℓf (r ) = r

2π



  fz (reiθ ) − e−2iθ fz (reiθ ) dθ

0 2π

 ≥r



 |fz (reiθ )| − |fz (reiθ )| dθ

0

≥

2π



r K

Λf (reiθ ) dθ .

(11)

0

Cauchy’s integral formula applied to h′ (z ) = fz (z ) and g ′ (z ) = fz (z ) shows that for n ≥ 1, nan =

fz (z )



1

2π i |z |=r

zn

dz

nbn =

and



fz (z )

2π i |z |=r

zn

1

dz ,

(12)

respectively. By (11) and (12), we get

      fz (z )   fz (z )   + n(|an | + |bn |) = dz dz    n  2π  |z |=r z n |z |=r z  2π 1 r Λf (reiθ ) dθ ≤ 2π r n 0 K ℓf (r ) K ℓf (1) ≤ ≤ , 2π r n 2π r n 1

which implies that K ℓf (1)

|an | + |bn | ≤

.

2nπ Now we are ready to prove the inequality (3). First we observe that Sf ( r ) =



Jf (z ) dσ (z ) ≥ Dr

1



K

Λ2f (z ) dσ (z ).

(13)

Dr

2

For θ ∈ [0, 2π ) and z ∈ D, let Pθ (z ) = fz (z ) + eiθ fz (z ) . By (13) and the subharmonicity of |Pθ |, we have



|Pθ (z )| ≤ ≤ ≤

1−|z |



1

π (1 − |z |)2 

0

1

(1 − |z |)2 Sf (1)K

(1 − |z |)2

2π



|Pθ (z + ρ eiβ )|ρ dβ dρ

0

Λ2f (z ) dσ (z ) D1−|z |

,

and the arbitrariness of θ ∈ [0, 2π ) gives the inequality

Λ2f (z ) ≤

Sf (1)K . (1 − |z |)2

(14)

By Lemma C, we get Sf ( r ) ≤

ℓ2f (r ) 4π 2

.

(15)

By (14) and (15), we have

Λ2f (z ) ≤

ℓ2f (1)K 4π 2 (1 − |z |)2

,

which gives

Λf ( z ) ≤

√ ℓf (1) K . 2π (1 − |z |)

Finally, f ∈ H B easily follows from (16). The proof of this theorem is complete.

(16) 

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S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

3. Bloch and Lipschitz spaces on harmonic mappings The following result easily follows from the monotonicity of ω(t )/t for t > 0. Lemma 2. Let ω be a majorant. For t > 0, if λ ≥ 1, then ω(λt ) ≤ λω(t ). The following lemma is easy to derive (cf. [8, Lemma 1]). Lemma D. Let z , w be complex numbers. Then max |w cos θ + z sin θ | =

θ∈[0,2π ]

1

 |w + iz | + |w − iz | .

2

Proof of Theorem 3. The implications (a) ⇐⇒ (c) easily follow from [4, Theorem 1.1]. We only need to prove (a) ⇐⇒ (b). We first prove that (a) H⇒ (b). Let z , w ∈ D with z ̸= w , and let ϕ(t ) = zt + (1 − t )w , where t ∈ [0, 1]. Since |ϕ(t )| ≤ t |z | + (1 − t )|w|, we see that 1 − |ϕ(t )| ≥ 1 − t |z | − |w| + t |w| ≥ 1 − t + |w|(t − 1) = (1 − t )d(w) and 1 − |ϕ(t )| ≥ 1 − t |z | − |w| + t |w| = 1 − t |z | − |w|(1 − t ) ≥ 1 − t |z | − (1 − t ) = td(z ). Using the last two inequalities, one has

(1 − |ϕ(t )|)2 ≥ (1 − t )td(w)d(z ) and therefore, we get 1 1 − |ϕ(t )|

1

. ≤ √ (1 − t )td(w)d(z )

(17)

By Lemma 2 and the inequality (17), for any z , w ∈ D with z ̸= w , we have

  |f (z ) − f (w)| = 

1 0

  (ϕ(t )) dt  dt

df

   = (z − w) ≤ |z − w|



1

fζ (ϕ(t )) dt + (z − w) 0

≤

=

=

0





 |fζ (ϕ(t ))| + |fζ (ϕ(t ))| dt

1

for some constant C > 0, which gives

  |f (z ) − f (w)| 1 ≤ πCω √ . |z − w| d(w)d(z )

 

fζ (ϕ(t )) dt 

  1 Λf (ϕ(t )) ω dt 1 − |ϕ(t )| 0 ω (1/(1 − |ϕ(t )|))   1  1 C |z − w| ω dt 1 − |ϕ(t )| 0   1  1 C |z − w| ω √ dt (1 − t )td(w)d(z ) 0   1 1 1 C |z − w|ω √ dt √ d(w)d(z ) (1 − t )t 0   π 2 1 2 sin θ cos θ C |z − w|ω √ dθ √ d(w)d(z ) 0 sin2 θ cos2 θ   1 C π |z − w|ω √ , d(w)d(z )

≤ |z − w|

≤

1



1

0

≤

(ζ = ϕ(t ) = w + t (z − w))

S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

69

Now we prove that (b) H⇒ (a). Let f = h + g, where h and g are analytic in D. By Lemma D, we obtain 1

max |fx (z ) cos θ + fy (z ) sin θ | =

θ∈[0,2π]

 |fx (z ) + ify (z )| + |fx (z ) − ify (z )|

2 1

(|2gx (z )| + |2hx (z )|) 2 = |h′ (z )| + |g ′ (z )| =

= Λf (z ).

(18) iθ

For r ∈ (0, 1) and θ ∈ [0, 2π ], let w = z + re . Then

  iθ  f (z ) − f (w)   = lim |f (z ) − f (z + re )|  r →0+ r →0+ z−w r lim 

= |fx (z ) cos θ + fy (z ) sin θ |  ≤ C lim ω r →0+

= Cω



1

1





d(z )d(z + reiθ )

 (19)

d(z )

for some constant C > 0. By (18) and (19), we conclude that

Λf (z ) = max |fx (z ) cos θ + fy (z ) sin θ | ≤ max C ω θ∈[0,2π]

θ∈[0,2π]



1 d(z )

Hence (a) ⇐⇒ (b) ⇐⇒ (c). The proof of the theorem is complete.



= Cω



1



d(z )

.



Proof of Theorem 4. We first prove the necessity. Since Ω is a Lω -extension domain, we see that for any z , w ∈ Ω , by using (7), there is a rectifiable curve γ ⊂ Ω joining z to w such that

|f (z ) − f (w)| ≤

 γ

Λf (ζ ) ds(ζ ) ≤ C

 γ

  ω dΩ (ζ ) ds(ζ ) ≤ C ω(|z − w|) dΩ (ζ )

for some constant C > 0. Now we prove the sufficiency. Let z ∈ Ω and r = dΩ (z )/2. For all w ∈ D(z , r ), we get f (w) =

1

2π



2π

P(w, reiθ )f (reiθ + z ) dθ ,

0

where P(w, reiθ ) =

r 2 − |w − z |2

|w − z − reiθ |2

.

By elementary calculations, we have

∂ −(w − z )|w − z − reiθ |2 − (r 2 − |w − z |2 )(w − z − re−iθ ) P(w, reiθ ) = ∂w |w − z − reiθ |4 and

∂ −(w − z )|w − z − reiθ |2 − (r 2 − |w − z |2 )(w − z − reiθ ) P(w, reiθ ) = . ∂w |w − z − reiθ |4 Then for all w ∈ D(z , r /2),

   ∂  |w − z | |w − z − reiθ |2 + (r 2 − |w − z |2 )|w − z − reiθ | iθ   P (w, re )  ∂w ≤ |w − z − reiθ |4 ≤

(r /2)(9r 2 /4) + r 2 (3r /2) 21 = r 4 /4 2r

and

   21  ∂ iθ   P (w, re )  ∂w  ≤ 2r ,

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S. Chen et al. / Nonlinear Analysis 115 (2015) 62–70

which imply that

Λf (w) =

≤ ≤ ≤

 1  2π ∂

2π 1 2π

21

     2π ∂    P(w, reiθ ) f (z + reiθ ) − f (z ) dθ  +  P(w, reiθ ) f (z + reiθ ) − f (z ) dθ   ∂w ∂w 0 0  2π    ∂     ∂    P(w, reiθ ) +  P(w, reiθ ) f (z + reiθ ) − f (z )dθ  ∂w ∂w 0   2π  f (z + reiθ ) − f (z ) dθ

2π 0 21C ω(r ) 2π

r

=

r 21C ω (dΩ (z )/2)

π

dΩ (z )

.

Since ω(t ) is increasing on t ∈ (0, ∞), we conclude that

Λf (z ) ≤

21C ω (dΩ (z )/2)

π

dΩ ( z )

21C ω dΩ (z )



≤

π

dΩ (z )

 ,

for some constant C > 0. The proof of the theorem is complete.



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