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An investigation on a new class of harmonic mappings

Abstract

In the present paper, we give an extension of the idea which was introduced by Sakaguchi (J. Math. Soc. Jpn. 11:72-75, 1959), and we give some applications of this extended idea for the investigation of the class of harmonic mappings.

MSC:30C45, 30C55.

1 Introduction

Let D={z|z|<1} be the open unit disc in the complex plane . A complex valued harmonic function f:DC has the representation

f=h(z)+ g ( z ) ¯ ,
(1.1)

where h(z) and g(z) are analytic in and have the following power series expansions:

h(z)= n = 0 a n z n ,g(z)= n = 0 b n z n ,zD,

where a n , b n C, n=0,1,2, . Choose g(0)=0 (i.e., b 0 =0), so the representation (1.1) is unique in and is called the canonical representation of f in . It is convenient to make further normalization (without loss of generality) h(0)=0 (i.e., a 0 =0) and h (0)=1 (i.e., a 1 =1). The family of such functions f is denoted by S H [1]. It is known that if f is a sense-preserving harmonic mapping of onto some other region, then by Lewy’s theorem its Jacobian is strictly positive, i.e.,

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

Equivalently, the inequality | g (z)|<| h (z)| holds for all zD. The family of all functions f S H with the additional property that g (0)=0 (i.e., b 1 =0) is denoted by S H 0 [1]. Observe that the classical family of univalent functions consists of all functions f S H 0 such that g(z)=0 for all zD. Thus, it is clear that S S H 0 S H .

Let Ω be the family of functions ϕ(z) regular in the open unit disc and satisfy the conditions ϕ(0)=0, |ϕ(z)|<1 for all zD.

Denote by the family of functions p(z)=1+ p 1 z+ p 2 z 2 + regular in such that p(z) in if and only if

p(z)= 1 + ϕ ( z ) 1 ϕ ( z )
(1.2)

for some ϕ(z)Ω and every zD.

If s(z)=z+ c 2 z 2 + is regular in the open unit disc and satisfies the condition

Re ( e i α z s ( z ) s ( z ) ) >0,zD
(1.3)

for some real α, |α|<π/2, then s(z) is said to be an α-spirallike function in [2, 3]. Such functions are known to be univalent [4]. The class of such functions is denoted by S α .

Let F 1 (z)=z+ α 2 z 2 + α 3 z 3 + and F 2 (z)=z+ β 2 z 2 + β 3 z 3 + be analytic functions in . If there exists a function ϕ(z)Ω such that F 1 (z)= F 2 (ϕ(z)) for every zD, then we say that F 1 (z) is subordinate to F 2 (z) and we write F 1 F 2 . We also note that if F 1 F 2 , then F 1 (D) F 2 (D) [2, 3]. We also note that w(z)= g (z)/ h (z) is the analytic second dilatation of f and |w(z)|<1 for every zD.

In this paper we investigate the class of harmonic mappings defined by

S H ( α ) = { f = h ( z ) + g ( z ) ¯ | w ( z ) 1 + z 1 z , h ( z ) S α } .

For this aim we need the following theorems and lemmas.

Theorem 1.1 [5]

Let s(z) S α , then

rF(cosα,r) | s ( z ) | rF(cosα,r)
(1.4)

and

[ ( 1 r ) cos α ( 1 + r ) sin α ] F ( cos α , r ) | s ( z ) | [ ( 1 + r ) cos α + ( 1 r ) sin α ] F ( cos α , r ) ,
(1.5)

where

F(cosα,r)= 1 ( 1 + r ) cos α ( cos α 1 ) ( 1 r ) cos α ( cos α + 1 )
(1.6)

for all |z|=r<1 and |α|<π/2.

Lemma 1.2 [6]

Let ϕ(z) be regular in the unit disc with ϕ(0)=0, then if |ϕ(z)| attains its maximum value on the circle |z|=r at the point z 1 , one has z 1 ϕ ( z 1 )=kϕ( z 1 ) for some k1.

Lemma 1.3 [7]

If s 1 (z) and s 2 (z) are regular in , s 1 (0)= s 2 (0), s 2 (z) maps onto a many-sheeted region which is starlike with respect to the origin, and s 1 (z)/ s 2 (z)P, then s 1 (z)/ s 2 (z)P.

Lemma 1.4 [8]

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
(1.7)

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 .

2 Main results

Theorem 2.1 s(z) S α if and only if

z s ( z ) s ( z ) 1 2 cos α e i α z 1 z
(2.1)

for all zD.

Proof Let s(z) in S α , then we have

e i α z s ( z ) s ( z ) =cosα 1 + ϕ ( z ) 1 ϕ ( z ) +isinα

or

e i α z s ( z ) s ( z ) = e i α 1 + ϕ ( z ) 1 ϕ ( z )

for some ϕ(z)Ω and all zD. Thus

z s ( z ) s ( z ) 1 = 1 + e 2 i α ϕ ( z ) 1 ϕ ( z ) 1 = 1 + ( cos 2 α i sin 2 α ) ϕ ( z ) 1 + ϕ ( z ) 1 ϕ ( z ) = 2 cos α e i α ϕ ( z ) 1 ϕ ( z )

for some ϕ(z)Ω and all zD. Since ϕ(z)Ω, we have that (2.1) is true. The sufficient part of the proof can be seen by following the above steps in the opposite direction by considering the subordination principle. □

Theorem 2.2 Let f=h(z)+ g ( z ) ¯ be an element of S H ( α ) . If w(z)= g ( z ) h ( z ) P, then g ( z ) h ( z ) P for all zD.

Proof A version of this theorem was proved by Sakaguchi for a univalent starlike function [7, 9]. Since f=h(z)+ g ( z ) ¯ S H ( α ) , then h(z) and g(z) are regular in and h(0)=g(0)=0. On the other hand, we have

w(z)= g ( z ) h ( z ) Pif and only if g ( z ) h ( z ) 1 + z 1 + z
(2.2)

for all zD. Geometrically, this means that g ( z ) h ( z ) maps inside the open disc centered on the real axis with diameter end points 1 r 1 + r and 1 + r 1 r . Now we define a function ϕ(z) by

g ( z ) h ( z ) = 1 + ϕ ( z ) 1 ϕ ( z ) (zD).
(2.3)

Then ϕ(z) is analytic in , and ϕ(0)=0. On the other hand,

w(z)= g ( z ) h ( z ) = 2 z ϕ ( z ) 1 ϕ ( z ) 1 e i α ( 1 + e 2 i α ) ϕ ( z ) + 1 + ϕ ( z ) 1 ϕ ( z ) (zD).
(2.4)

Now, it is easy to realize that the subordination (2.2) is equivalent to |ϕ(z)|<1 in (2.3) for all zD. Indeed, assume to the contrary that there exists z 1 D such that |ϕ( z 1 )|=1. Then by Jack’s lemma (Lemma 1.2), z 1 ϕ ( z 1 )=kϕ( z 1 ), k1, for such z 1 we have

w( z 1 )= g ( z 1 ) h ( z 1 ) = 2 k ϕ ( z 1 ) 1 ϕ ( z 1 ) 1 e i α ( 1 + e 2 i α ) ϕ ( z 1 ) + 1 + ϕ ( z 1 ) 1 ϕ ( z 1 ) =w ( ϕ ( z 1 ) ) w(D),

since |ϕ( z 1 )|=1 and k1. But this is a contradiction to the condition w(z)= g ( z ) h ( z ) 1 + z 1 z , and so the assumption is wrong, i.e., |ϕ(z)|<1 for all zD. □

Remark 2.3 Theorem 2.2 is an extension of Lemma 1.3 to the harmonic mappings.

Corollary 2.4 Let f=h(z)+ g ( z ) ¯ be an element of S H ( α ) , then

r ( 1 r ) 1 + r F(cosα,r) | g ( z ) | r ( 1 + r ) 1 r F(cosα,r)
(2.5)

and

[ ( 1 r ) cos α ( 1 + r ) sin α ] 1 r 1 + r F ( cos α , r ) | g ( z ) | [ ( 1 + r ) cos α + ( 1 r ) sin α ] 1 + r 1 r F ( cos α , r ) ,
(2.6)

where F(cosα,r) is given by (1.6) for all |z|=r<1 and |α|<π/2.

Proof The proof of this theorem is a simple consequence of Theorem 1.1 and Theorem 2.2 since

Rew(z)=Re g ( z ) h ( z ) g ( z ) h ( z ) 1 + z 1 z 1 r 1 + r | g ( z ) h ( z ) | 1 + r 1 r ,

then

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

and

Re g ( z ) h ( z ) >0 g ( z ) h ( z ) 1 + z 1 z 1 r 1 + r | g ( z ) h ( z ) | 1 + r 1 r ,

then

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

 □

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

[ ( 1 r ) cos α ( 1 + r ) sin α ] 2 ( F ( cos α , r ) ) 2 ( 1 r 2 ) ( 1 | b 1 | 2 ) ( 1 + | b 1 | r ) 2 J f ( z ) [ ( 1 + r ) cos α + ( 1 r ) sin α ] 2 ( F ( cos α , r ) ) 2 ( 1 r 2 ) ( 1 | b 1 | 2 ) ( 1 | b 1 | r ) 2 ,

where F(cosα,r) is given by (1.6) for all |z|=r<1, |α|<π/2 and J f ( z ) is the Jacobian of f defined by J f ( z ) = | h ( z ) | 2 | g ( z ) | 2 for all zD.

Proof The proof of this corollary is a simple consequence of Lemma 1.1 and Lemma 1.4. □

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Correspondence to Emel Yavuz Duman.

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Yavuz Duman, E., Polatoğlu, Y. & Kahramaner, Y. An investigation on a new class of harmonic mappings. J Inequal Appl 2013, 478 (2013). https://doi.org/10.1186/1029-242X-2013-478

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Keywords

  • harmonic mappings
  • distortion theorem
  • grow theorem