Journal of Mathematics
Volume 2015 (2015), Article ID 460191, 6 pages
http://dx.doi.org/10.1155/2015/460191

Research Article

The Distortion Theorems for Harmonic Mappings with Analytic Parts Convex or Starlike Functions of Order

Mengkun Zhu and Xinzhong Huang

Department of Mathematics, Huaqiao University, Quanzhou, Fujian 362021, China

Received 12 July 2015; Accepted 27 September 2015

Academic Editor: Mario Ohlberger

Copyright © 2015 Mengkun Zhu and Xinzhong Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Some sharp estimates of coefficients, distortion, and growth for harmonic mappings with analytic parts convex or starlike functions of order are obtained. We also give area estimates and covering theorems. Our main results generalise those of Klimek and Michalski.

1. Introduction

Let denote the class of functions of the form that are analytic and univalent in the unit disk . An analytic function is said to be convex of order if

for which we write . An analytic function is said to be starlike of order if

for which we write , where . Moreover, when the positive constant vanishes in (1), (2), the function turns to be starlike or convex function. That is to say, when an analytic function only satisfies or , we call belongs to starlike or convex function, for which we write or for convenience.

A complex-valued harmonic function in the open unit disk has a canonical decomposition

where and are analytic in . Generally, we call the analytic part and the coanalytic part of . An elegant and complete account of the theory of planar harmonic mappings is given in Duren’s monograph [1].

Lewy [2] proved in 1936 that a necessary and sufficient condition for to be locally univalent and sense-preserving in is , where

To such a function , does not vanish in ; let

and then the second complex dilatation is analytic with .

Clunie and Sheil-Small introduced the class of all sense-preserving univalent harmonic mappings of with that is denoted by [3]. In [4], Hotta and Michalski denoted the class of all locally univalent and sense-preserving harmonic functions in the unit disk with . Obviously, . Every function is uniquely determined by coefficients of the following power series expansions:

where ,  .

In [5], the classes of starlike and convex functions of order were first introduced by Robertson. Then, such functions have been studied and used in [6–9], and so forth. In [10, 11], Klimek and Michalski studied the cases when the analytic part is the identity mapping (convex of order 1) and convex mapping (convex of order 0), respectively. In [4], Hotta and Michalski considered the case when the analytic part is a starlike analytic mapping (starlike of order 0). The main idea of this paper is to characterize the subclasses of when and the subclasses of when , where .

In order to establish our main results, we need the following lemma.

Lemma 1 (see [12]). If is analytic and on , then

2. Main Results and Their Proofs

In what follows, the harmonic mappings that we consider are all normalized locally univalent and sense-preserving.

Definition 2. For , let

By [3, Theorem  5.7], if with , then ; hence, the class is well-defined.

Definition 3. For , let

In particular, we establish a smaller subclass of ,

Lemma 4. If , then , where , , and , are related by , , .

Proof. By the definition of , . Using classical Alexander’s theorem [13, page 43], the function . Also, , , and . Let , ; then

Hence,

which implies that is a sense-preserving and locally univalent harmonic mapping in . By [11, Corollary  2.3], we obtain that .

Applying Lemma 1, we can prove the following theorem.

Theorem 5. If , then

Specially,

The estimate for is sharp; the extremal functions are

Proof. Assuming , , then by [7] we have (13). Let , where is the dilatation of . Since is analytic in , it has a power series expansion

where , , and . Recall that for all ; then, by Lemma 1, we have

Together with formulas (5), (6), and (17) we give

which leads to

Comparing coefficients, we obtain

Applying formulas (13) and (18) and by simple calculation, we have

In particular,

Next, we will prove the estimate is sharp. For , consider a function , such that

and suppose that the dilatation of satisfies

Applying formula (5), we obtain

which implies the estimate of (15) is sharp. Obviously, , , which means . Hence, the proof is completed.

Corollary 6. If , then , ,

Proof. If , then, by Lemma 4, the function , where , , . Let be expanded in the power series

Together with expansion (6) of and formula , we have ; then by Theorem 5 we can easily obtain the coefficient estimates of .
Specially, by comparing coefficients, we have , which easily leads to the estimate by the condition of Corollary 6.

Since the analytic part of belongs to , then, by [7], we have the following distortion estimate of :

Our next aim is to give the distortion estimate of the coanalytic part of .

Theorem 7. If , then

These inequalities are sharp. The equalities hold for the harmonic function which is defined in (16).

Proof. Let . Consider the function

which satisfies assumptions of the Schwarz lemma; then we have

It is equivalent to

Hence, applying the triangle inequalities to formula (34) we have

Finally, applying formula (29) together with (35) to the identity , we obtain (30) and (31). The function defined in (16) shows that inequalities (30) and (31) are sharp. The proof is completed.

Corollary 8. If , then

Proof. In [7], we know that if , then

Using inequality (35) to identity , then the corollary can be obtained immediately.

By [7], we have the following growth estimate of , where .

In the case ,

In the case ,

In the next results, we give the growth estimate of coanalytic part of .

Theorem 9. If , then

The inequality is sharp. The equality holds for the harmonic function which is defined in (16).

Proof. Let ; applying estimate (31) we have

where . The function defined (16) shows that inequality (40) is sharp.

For , by [7], we have

Now, we give the growth estimates of coanalytic part of .

Corollary 10. If , then

Using the distortion estimates in (29) and (35), we can easily deduce the following area estimates of .

Theorem 11. Let and ; if , then

where .

Proof. Observe that if , then does not vanish in . We can give the Jacobian of in the form

where is the dilatation of . Applying (29) and (35) to (45) we obtain

where ; this completes the proof.

Remark 12. To avoid the maximum of having no sense, we give the limiting condition in Theorem 11.

Corollary 13. Let and ; if , then

Theorem 14. If , then

Proof. For any point , let and denote

and then . Hence, there exists such that . Let , ; then and is a well-defined Jordan arc. Since , then we can obtain

By   with formulas (29) and (35), we have

Hence, we obtain

To prove (49) we simply use the inequality

By formulas (38), (39), and (40) with simple calculation we have (49); this completes the proof.

Corollary 15. If , then

Finally, the growth estimate of yields the following covering estimate.

Theorem 16. If , then

where

Proof. Let tend to in estimate (48); then Theorem 16 follows immediately from the argument principle for harmonic mappings.

Corollary 17. If , then

where

Proof. Let tend to in estimate (55); then Corollary 17 follows immediately from the argument principle for harmonic mappings.

Remark 18. The univalence problem of a locally univalent harmonic mapping with starlike analytic parts is an open problem. Though have stronger properties than , we cannot obtain the sharp value of such that is univalent. It has important sense to study. Moreover, case of was given a systematic study in [4, 10, 11].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

It is supported by NNSF of China (11101165), the Natural Science Foundation of Fujian Province of China (2014J01013), NCETFJ Fund (2012FJ-NCET-ZR05), and Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX110).

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