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Harmonic function for which the second dilatation is α-spiral

Abstract

Let f=h+ g ¯ be a harmonic function in the unit disc D. We will give some properties of f under the condition the second dilatation is α-spiral.

MSC:30C45, 30C55.

1 Introduction

A planar harmonic mapping in the unit disc D={zC||z|<1} is a complex-valued harmonic function f which maps D onto some planar domain f(D). Since D is simply connected, the mapping f has a canonical decomposition f=h+ g ¯ , where h and g are analytic in D. As usual, we call h the analytic part of f and g the co-analytic part of f. An elegant and complete account of the theory of planar harmonic mapping is given in Duren’s monograph [1].

Lewy [2] proved in 1936 that the harmonic function f is locally univalent in a simply connected domain D 1 if and only if its Jacobian

J f (z)=| h (z) | 2 | g (z) | 2 >0

is different from zero in D 1 . In view of this result, locally univalent harmonic mappings in the unit disc are either sense-reversing if

| g (z)|>| h (z)|

in D 1 or sense-preserving if

| g (z)|<| h (z)|

in D 1 . Throughout this paper, we will restrict ourselves to the study of sense-preserving harmonic mappings. However, since f is sense-preserving if and only if f ¯ is sense-reserving, all the results obtained in this article regarding sense-preserving harmonic mappings can be adapted to sense-reversing ones. Note that f=h+ g ¯ is sense-preserving in D if and only if h (z) does not vanish in the unit disc and the second-complex dilatation w(z)= g ( z ) h ( z ) has the property |w(z)|<1 in D; therefore, we can take h(z)=z+ a 2 z 2 + , g(z)= b 1 z+ b 2 z 2 + . Thus, the class of all harmonic mappings being sense-preserving in the unit disc can be defined by

S H = { f = h ( z ) + g ( z ) ¯ | h ( z ) = z + a 2 z 2 + , g ( z ) = b 1 z + b 2 z 2 + , f  sense-preserving } .

Let Ω be the family of functions ϕ(z) which are regular in D and satisfy the conditions ϕ(0)=0, |ϕ(z)|<1 for all zD. Denote by P the family of functions p(z)=1+ p 1 z+ p 2 z 2 + which are regular in D such that

p(z)= 1 + ϕ ( z ) 1 ϕ ( z )
(1.1)

for some function ϕ(z)Ω for all zD.

Next, let S denote the family of functions s(z)=z+ c 2 z 2 + c 3 z 3 + which are regular in D such that

z s ( z ) s ( z ) =p(z)
(1.2)

for some p(z)P for all zD.

Let s 1 (z)=z+ α 2 z 2 + α 3 z 3 + and s 2 (z)=z+ β 2 z 2 + β 3 z 3 + be analytic functions in D. If there exists ϕ(z)Ω such that s 1 (z)= s 2 (ϕ(z)) for all zD, then we say that s 1 (z) is subordinate to s 2 (z) and we write s 1 (z) s 2 (z), then s 1 (D) s 2 (D).

Now, we consider the following class of harmonic mappings in the plane:

S HPST ( α ) = { f = h ( z ) + g ( z ) ¯ | f S H , h ( z ) S , Re ( e i α w ( z ) ) = Re ( e i α g ( z ) h ( z ) ) > 0 , | α | < π 2 } .
(1.3)

In the present paper, we will investigate the class S HPST (α).

We will need the following lemma and theorem in the sequel.

Theorem 1.1 ([3, 4])

Let h(z) be an element of S , then

r ( 1 + r ) 2 |h(z)| r ( 1 r ) 2 ,

for all |z|=r<1.

1 r ( 1 + r ) 3 | h (z)| 1 + r ( 1 r ) 3 .

These inequalities are sharp because the extremal function is h(z)= z ( 1 z ) 2 .

Lemma 1.2 ([2, 5])

Let h(z) and g(z) be regular in D, h(z) map |z|<1 onto a many-sheeted starlike region, Re( e i α g ( z ) h ( z ) )>0, |α|< π 2 for |z|<1. h(0)=g(0)=0. Then Re( e i α g ( z ) h ( z ) )>0 for |z|<1.

2 Main results

Lemma 2.1 Let f=h(z)+ g ( z ) ¯ be an element of S HPST (α) then

| b 1 | r 1 | b 1 | r | g ( z ) h ( z ) | | b 1 | + r 1 + | b 1 | r
(2.1)

for all |z|=r<1. This inequality is sharp because the extremal function is

e i α g ( z ) h ( z ) = z + b 1 + b ¯ z ,

where b= e i α b 1 .

Proof

Since

then the function

ϕ(z)= W ( z ) W ( 0 ) 1 W ( 0 ) ¯ W ( z ) = W ( 0 ) b 1 b ¯ W ( 0 ) = b b 1 b 2 =0

satisfies the condition of the Schwarz lemma. Using the definition of subordination, we have

W(z)= e i α w(z)= e i α g ( z ) h ( z ) = b + ϕ ( z ) 1 + b ¯ ϕ ( z ) e i α g ( z ) h ( z ) b + z 1 + b ¯ z .

On the other hand, the transformation ( b + z 1 + b ¯ z ) maps |z|<1 onto the disc with the center

C(r)= ( α 1 ( 1 r 2 ) 1 | b 1 | 2 r 2 , α 2 ( 1 r 2 ) 1 | b 1 | 2 r 2 ) ,b= α 1 +i α 2

and the radius

ρ(r)= ( 1 | b 1 | 2 ) r 1 | b 1 | 2 r 2 .

Therefore, we can write

| e i α g ( z ) h ( z ) b 1 ( 1 r 2 ) 1 | b 1 | 2 r 2 | ( 1 | b 1 | 2 ) r 1 | b 1 | 2 r 2
(2.2)

which gives (2.1). □

Corollary 2.2 Let f S HPST (α), then

(2.3)
(2.4)

for all |z|=r<1.

Proof

Using Lemma 1.2 and Lemma 2.1, then we can write

(2.5)
(2.6)

If we use Theorem 1.1 in the inequalities (2.5) and (2.6), we get (2.3) and (2.4). □

Corollary 2.3 Let f=h(z)+ g ( z ) ¯ be an element of S H P T S (α), then

( 1 | b 1 | 2 ) ( 1 r ) 3 ( 1 + r ) 5 ( 1 + | b 1 | r ) 2 J f ( z ) ( 1 | b 1 | 2 ) ( 1 + r ) 3 ( 1 r ) 5 ( 1 + | b 1 | r ) 2
(2.7)

for all |z|=r<1.

Proof

Since

J f ( z ) =| h (z) | 2 | g (z) | 2 =| h (z) | 2 ( 1 | w ( z ) | 2 ) ,
(2.8)

using Lemma 2.1 and Theorem 1.1 in the equality (2.8) and after simple calculations, we get (2.7). □

Corollary 2.4 If f=h(z)+ g ( z ) ¯ is an element of S H P T S (α), then

(2.9)

where a=| b 1 | for all |z|=r<1.

Proof

Using Corollary 2.2 and Theorem 1.1, we obtain

( | h ( z ) | | g ( z ) | ) ( 1 r 4 ) ( 1 + | b 1 | r ) ( 1 + r 4 ) ( | b 1 | + r ) ( 1 r ) 3 ( 1 + r ) 3 ( 1 + | b 1 | r ) ,

and

( | h ( z ) | + | g ( z ) | ) ( 1 + r ) 2 ( 1 + | b 1 | ) ( 1 r ) 3 ( 1 + | b 1 | r ) .

Therefore, we have

(2.10)

Integrating the last inequality (2.10), we get (2.9). □

Theorem 2.5 Let f=h(z)+ g ( z ) ¯ be an element of S H P T S (α), then

k = 1 n | A k | 2 | t + 1 | 2 + k = 1 n | B k | 2
(2.11)

where A k =(k+1)( b k + 1 b 1 a k + 1 ); B k =(k+1)( b k + 1 b 1 +t a k + 1 ); a k and b k are the coefficients of the functions h(z) and g(z); k=1,2,3,,n; t=2s1; s= e i α cosα.

Proof

Since

g(z)= b 1 z+ b 2 z 2 + b 3 z 3 + g (z)= b 1 +2 b 2 z+3 b 3 z 2 +.

We denote by G(z)= 1 b 1 g(z)

then we have

{ 1 cos α ( e i α 1 b 1 g ( z ) h ( z ) i sin α ) = p ( z ) e i α 1 b 1 g ( z ) h ( z ) = cos α p ( z ) + i sin α , 1 b 1 g ( z ) h ( z ) = 1 + e i α cos α ( p ( z ) 1 ) .
(2.12)

Since p(z) is in P, there is a function ϕ(z) satisfying the conditions of the Schwarz lemma such that

p(z)= 1 + ϕ ( z ) 1 ϕ ( z ) p(z)1= 2 ϕ ( z ) 1 ϕ ( z ) .
(2.13)

Using this equation in (2.12) and after the following calculations given above

1 b 1 g ( z ) h ( z ) =1+ e i α cosα ( p ( z ) 1 ) =1+s ( 2 ϕ ( z ) 1 ϕ ( z ) ) ,

we get the following equality:

1 b 1 g (z) h (z)= ( t h ( z ) + 1 b 1 g ( z ) ) .
(2.14)

If ϕ(z)= c 1 z+ c 2 z 2 + c 3 z 3 + , we have

k = 1 n A k z k + k = n + 1 D k z k = [ ( 1 + t ) + k = 1 n B k z k ] ϕ(z),
(2.15)

where

k = n + 1 D k z k = k = n + 1 A k z k ( c 1 B n z n + 1 + c 1 B n + 1 z n + 2 + ) .

Therefore, the equality (2.15) can be considered in the following form:

F(z)=G(z)ϕ(z).
(2.16)

Using the Clunie method [6], then we can write

1 2 π 0 2 π | F ( r e i θ ) | 2 dθ 1 2 π 0 2 π | G ( r e i θ ) | 2 dθ,

which gives

k = 1 n | A k | 2 r 2 k + k = n + 1 | D k | 2 r 2 k ( | t + 1 | 2 + k = 1 n | B k | 2 r 2 k ) .
(2.17)

Eventually, we will let r 1 , then we have

k = 1 n | A k | 2 | t + 1 | 2 + k = 1 n | B k | 2 .

 □

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Correspondence to Melike Aydog̃an.

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Aydog̃an, M., Duman, E.Y., Polatog̃lu, Y. et al. Harmonic function for which the second dilatation is α-spiral. J Inequal Appl 2012, 262 (2012). https://doi.org/10.1186/1029-242X-2012-262

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Keywords

  • Harmonic functions
  • growth theorem
  • distortion theorem
  • coefficient inequality