Open Access

Harmonic function for which the second dilatation is α-spiral

  • Melike Aydog̃an1Email author,
  • Emel Yavuz Duman2,
  • Yaşar Polatog̃lu2 and
  • Yasemin Kahramaner3
Journal of Inequalities and Applications20122012:262

https://doi.org/10.1186/1029-242X-2012-262

Received: 15 June 2012

Accepted: 24 September 2012

Published: 6 November 2012

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.

Keywords

Harmonic functionsgrowth theoremdistortion theoremcoefficient inequality

1 Introduction

A planar harmonic mapping in the unit disc D = { z C | | 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 z D . 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 z D .

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 z D .

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 z D , 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 = 2 s 1 ; s = e i α cos α .

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 cos α ( 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 d θ 1 2 π 0 2 π | G ( r e i θ ) | 2 d θ ,
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 .

 □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Işık University, Meşrutiyet Koyu
(2)
Department of Mathematics and Computer Sciences, İstanbul Kültür University
(3)
Department of Mathematics, İstanbul Ticaret University

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Copyright

© Aydoğan et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.