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Some properties of starlike harmonic mappings

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.

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 zD; 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 zD.

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 zD, 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 zD. 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 zD.

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

References

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Acknowledgements

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

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Correspondence to Melike Aydog̃an.

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Aydog̃an, M., Yemisci, A. & Polatog̃lu, Y. Some properties of starlike harmonic mappings. J Inequal Appl 2012, 163 (2012). https://doi.org/10.1186/1029-242X-2012-163

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Keywords

  • harmonic starlike function
  • growth theorem
  • distortion theorem