Open Access

Some properties of starlike harmonic mappings

Journal of Inequalities and Applications20122012:163

https://doi.org/10.1186/1029-242X-2012-163

Received: 10 April 2012

Accepted: 6 July 2012

Published: 23 July 2012

Abstract

A fundamental result of this paper shows that the transformation

F = a z ( h ( z + a 1 + a ¯ z ) + g ( z + a 1 + a ¯ z ) ¯ ) ( h ( a ) + g ( a ) ¯ ) ( z + a ) ( 1 + a ¯ z )

defines a function in S H S 0 whenever f = h ( z ) + g ( z ) ¯ is S H S 0 , and we will give an application of this fundamental result.

MSC:30C45, 30C55.

Keywords

harmonic starlike function growth theorem distortion theorem

1 Introduction

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 +
regular in D , such that p ( z ) is in P if and only if
p ( z ) = 1 + ϕ ( z ) 1 ϕ ( z )
(1.1)

for some function ϕ ( z ) Ω and every z D .

Next, let s 1 ( z ) = z + c 2 z 2 + c 3 z 3 + and s 2 ( z ) = z + d 2 z 2 + d 3 z 3 + be regular 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 subordinated to s 2 ( z ) and we write s 1 ( z ) s 2 ( z ) , then s 1 ( D ) s 2 ( D ) .

Moreover, univalent harmonic functions are generalizations of univalent regular functions; the point of departure is the canonical representation
f = h ( z ) + g ( z ) ¯ , g ( 0 ) = 0
(1.2)
of a harmonic function f in the unit disc D as the sum of a regular function h ( z ) and the conjugate of a regular function g ( z ) . With the convention that g ( 0 ) = 0 , the representation is unique. The power series expansions of h ( z ) and g ( z ) are denoted by
h ( z ) = n = 0 a n z n , g ( z ) = n = 1 b n z n .
(1.3)
If f is a sense-preserving harmonic mapping of D onto some other region, then, by Lewy theorem, its Jacobian is strictly positive, i.e.,
J f ( z ) = | h ( z ) | 2 | g ( z ) | 2 > 0 .
(1.4)

Equivalently [1], the inequality | g ( z ) | < | h ( z ) | holds for all z D . This shows, in particular, that h ( z ) 0 , so there is no loss of generality in supposing that h ( 0 ) = 0 and h ( 0 ) = 1 . The class of all sense-preserving harmonic mappings of the disc with a 0 = b 0 = 0 and a 1 = 1 will be denoted by S H . Thus S H contains the standard class S of regular univalent functions. Although the regular part h ( z ) of a function f S H is locally univalent, it will become apparent that it need not be univalent. The class of functions f S H with g ( 0 ) = 0 will be denoted by S H 0 . At the same time, we note that S H is a normal family and S H 0 is a compact normal family [2].

Finally, let f = h ( z ) + g ( z ) ¯ be an element S H (or S H 0 ). If f satisfies the condition
θ ( Arg f ( r e i θ ) ) = Re ( z h ( z ) z g ( z ) ¯ h ( z ) + g ( z ) ¯ ) > 0
(1.5)
then f is called harmonic starlike function. The class of such functions is denoted by S H S (or S H S 0 ). Also, let f = h ( z ) + g ( z ) ¯ be an element S H (or S H 0 ). If f satisfies the condition
θ ( θ ( Arg f ( r e i θ ) ) ) = Re ( z ( z h ( z ) ) z ( z g ( z ) ) ¯ z h ( z ) + z g ( z ) ¯ ) > 0 ,
(1.6)

then f is called a convex harmonic function. The class of convex harmonic functions is denoted by S H C (or S H C 0 ).

For the aim of this paper, we will need the following lemma and theorem.

Lemma 1.1 ([2], p.51])

If f = h ( z ) + g ( z ) ¯ S H C , then there exist angles α and β such that
Re [ ( e i α h ( z ) + e i α g ( z ) ) ( e i β e i β z 2 ) ] > 0
(1.7)

for all z D .

Theorem 1.2 ([2], p.108])

If f = h ( z ) + g ( z ) ¯ S H is a starlike function and if H ( z ) and G ( z ) are the regular functions defined by z H ( z ) = h ( z ) , z G ( z ) = g ( z ) , H ( 0 ) = G ( 0 ) = 0 , then F = H ( z ) + G ( z ) ¯ is a convex function.

2 Main results

Lemma 2.1 Let f = h ( z ) + g ( z ) ¯ be an element of S H C 0 , then
G ( α , β , r ) ( 1 + r 2 ) 2 | h ( z ) + e 2 i α g ( z ) | G ( α , β , r ) ( 1 r 2 ) 2 ,
(2.1)
where
G ( α , β , r ) = 2 cos ( α + β ) r + 1 + [ 2 cos ( α + β ) ] r 2 + r 4 , cos ( α + β ) > 0 .
Proof Using Theorem 1.2, we write
p ( z ) = ( e i α h ( z ) + e i α g ( z ) ) ( e i β e i β z 2 ) , Re p ( z ) > 0 , p ( 0 ) = ( e i α h ( 0 ) + e i α g ( 0 ) ) ( e i β e i β 0 2 ) = cos ( α + β ) + i sin ( α + β ) .
On the other hand, since
p ( z ) = [ cos ( α + β ) + i sin ( α + β ) ] + p 1 z + p 2 z 2 +
is regular and satisfies the condition Re p ( z ) > 0 , with cos ( α + β ) > 0 , the function
p 1 ( z ) = 1 cos ( α + β ) [ p ( z ) i sin ( α + β ) ]
(2.2)
is an element of P [4]. Therefore, we have
| p 1 ( z ) 1 + r 2 1 r 2 | 2 r 1 r 2 .
(2.3)

After simple calculations from (2.3), we get (2.1). □

Corollary 2.2 Let f = h ( z ) + g ( z ) ¯ be an element of S H C 0 , then
G ( α , β , r ) ( 1 + r 2 ) 2 ( 1 r ) | h ( z ) | G ( α , β , r ) ( 1 r ) 3 ( 1 + r ) 2 ,
(2.4)
| w ( z ) | G ( α , β , r ) ( 1 + r 2 ) 2 ( 1 r ) | g ( z ) | r G ( α , β , r ) ( 1 r ) 3 ( 1 + r ) 2 .
(2.5)
Proof Since f S H C 0 , then g ( z ) = h ( z ) w ( z ) and the second dilatation w ( z ) satisfies the condition of Schwarz lemma, then the inequality (2.1) can be written in the form
G ( α , β , r ) | 1 + e 2 i α w ( z ) | ( 1 + r 2 ) 2 ( 1 r ) | h ( z ) | G ( α , β , r ) | 1 + e 2 i α w ( z ) | ( 1 r 2 ) 2
(2.6)

which is given in (2.4) and (2.5). □

Corollary 2.3 Let f = h ( z ) + g ( z ) be an element of S C H 0 , then
r G ( α , β , r ) ( 1 + r 2 ) 2 ( 1 r ) | h ( z ) | r G ( α , β , r ) ( 1 r ) 3 ( 1 + r ) 2 ,
(2.7)
| w ( z ) | r G ( α , β , r ) ( 1 + r 2 ) 2 ( 1 r ) | g ( z ) | r 2 G ( α , β , r ) ( 1 r ) 3 ( 1 + r ) 2 .
(2.8)

Proof Using Theorem 1.2 and Corollary 2.2, we obtain (2.7) and (2.8). □

Theorem 2.4 If f = h ( z ) + g ( z ) ¯ is in S H S 0 and a is in D , then
F = a z ( h ( z + a 1 + a ¯ z ) + g ( z + a 1 + a ¯ z ) ¯ ) ( h ( a ) + g ( a ) ¯ ) ( z + a ) ( 1 + a ¯ z )
(2.9)

is likewise in S H S 0 .

Proof For ρ real, 0 < ρ < 1 , let
F ρ = a z ( h ( ρ ( z + a 1 + a ¯ z ) ) + g ( ρ ( z + a 1 + a ¯ z ) ) ¯ ) ( h ( ρ a ) + g ( ρ a ) ¯ ) ( z + a ) ( 1 + a ¯ z )
(2.10)
then we have
(2.11)
Letting z = e i θ and w = ρ ( z + a 1 + a ¯ z ) in (2.11) and after the straightforward calculations, we obtain
Re ( z F z z ¯ F z ¯ F ) = 1 | a | 2 | a + e i θ | 2 Re ( w h ( w ) w ρ ( w ) ¯ h ( w ) + ρ ( w ) ¯ ) > 0 ,
(2.12)
and we conclude that
F ρ = a z ( h ( ρ ( z + a 1 + a ¯ z ) ) + g ( ρ ( z + a 1 + a ¯ z ) ) ¯ ) ( h ( ρ a ) + g ( ρ a ) ¯ ) ( z + a ) ( 1 + a ¯ z )

is in S H S 0 for every admissible ρ. From the compactness of S H S 0 [2] and (2.11), we infer that F = lim ρ 1 F ρ is in S H S 0 . We also note that this theorem is a generalization of the theorem of Libera and Ziegler [3]. □

Corollary 2.5 Let f = h ( z ) + g ( z ) ¯ be an element of S H S 0 , then
(2.13)
(2.14)
Proof Using Theorem 2.4, we have
{ F = a . z . h ( z + a 1 + a ¯ z ) ( h ( a ) + g ( a ) ¯ ) ( z + a ) ( 1 + a ¯ z ) + a . z . g ( z + a 1 + a ¯ z ) ¯ ( h ( a ) + g ( a ) ¯ ) ( z + a ) ( 1 + a ¯ z ) F = H ( z ) + G ( z ) ¯ .
(2.15)
If we apply Corollary 2.3 to H ( z ) and G ( z ) by taking
u = z + a 1 + a ¯ z z = u a 1 + a ¯ u

a = k u , 1 < k < 1 and after straightforward calculations, we get (2.13) and (2.14). □

Declarations

Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions. They were very helpful for our paper.

Authors’ Affiliations

(1)
Department of Mathematics, Işık University
(2)
Department of Mathematics and Computer Science, Kültür University

References

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Copyright

© Aydogĝan et al.; licensee Springer 2012

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