Open Access

An investigation on a new class of harmonic mappings

  • Emel Yavuz Duman1Email author,
  • Yaşar Polatoğlu1 and
  • Yasemin Kahramaner2
Journal of Inequalities and Applications20132013:478

https://doi.org/10.1186/1029-242X-2013-478

Received: 28 December 2012

Accepted: 28 August 2013

Published: 7 November 2013

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.

Keywords

harmonic mappings distortion theorem grow theorem

1 Introduction

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

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 z D .

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 , z D
(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 z D , 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 z D .

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
r F ( cos α , r ) | s ( z ) | r F ( cos α , 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 ( cos α , 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 ) = k ϕ ( z 1 ) for some k 1 .

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 z D .

Proof Let s ( z ) in S α , then we have
e i α z s ( z ) s ( z ) = cos α 1 + ϕ ( z ) 1 ϕ ( z ) + i sin α
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 z D . 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 z D . 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 z D .

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 ) P if and only if g ( z ) h ( z ) 1 + z 1 + z
(2.2)
for all z D . 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 ) ( z D ) .
(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 ) ( z D ) .
(2.4)
Now, it is easy to realize that the subordination (2.2) is equivalent to | ϕ ( z ) | < 1 in (2.3) for all z D . 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 ) = k ϕ ( z 1 ) , k 1 , 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 k 1 . 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 z D . □

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 ( cos α , r ) | g ( z ) | r ( 1 + r ) 1 r F ( cos α , 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 ( cos α , 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
Re w ( 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 ( cos α , 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 z D .

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

Declarations

Authors’ Affiliations

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

References

  1. Duren P: Harmonic Mappings in the Plane. Cambridge University Press, Cambridge; 2004.View ArticleGoogle Scholar
  2. Goodman AW I. In Univalent Functions. Mariner Pub. Comp. Inc., Tampa; 1983.Google Scholar
  3. Goodman AW II. In Univalent Functions. Mariner Pub. Comp. Inc., Tampa; 1983.Google Scholar
  4. S̆pac̆ek L: Contribution à la théorie des fonctions univalentes. Čas. Pěst. Mat. 1932, 62: 12-19.Google Scholar
  5. Polatoğlu Y: Grow and distortion theorem for Janowski α -spirallike functions in the unit disc. Stud. Univ. Babeş-Bolyai, Math. 2012, 57(2):255-259.Google Scholar
  6. Jack IS: Functions starlike and convex of order α . J. Lond. Math. Soc. 1971, 3: 469-474.MathSciNetView ArticleGoogle Scholar
  7. Libera RJ: Some classes of regular univalent functions. Proc. Am. Math. Soc. 1965, 16: 755-758. 10.1090/S0002-9939-1965-0178131-2MathSciNetView ArticleGoogle Scholar
  8. Aydoğan M, Yavuz Duman E, Polatoğlu Y, Kahramaner Y: Harmonic function for which the second dilatation is α -spiral. J. Inequal. Appl. 2012., 2012: Article ID 262Google Scholar
  9. Sakaguchi K: On a certain univalent mappings. J. Math. Soc. Jpn. 1959, 11: 72-75. 10.2969/jmsj/01110072MathSciNetView ArticleGoogle Scholar

Copyright

© Yavuz Duman et al.; licensee Springer. 2013

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.