The Distortion Theorems for Harmonic Mappings with Analytic Parts Convex or Starlike Functions of Order
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.
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 iffor which we write . An analytic function is said to be starlike of order iffor 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 decompositionwhere 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 .
Lewy  proved in 1936 that a necessary and sufficient condition for to be locally univalent and sense-preserving in is , whereTo such a function , does not vanish in ; letand 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 . In , 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 , 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 , 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 ). 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 , letIn 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 , ; thenHence,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 , thenSpecially,The estimate for is sharp; the extremal functions are
Proof. Assuming , , then by  we have (13). Let , where is the dilatation of . Since is analytic in , it has a power series expansionwhere , , and . Recall that for all ; then, by Lemma 1, we haveTogether with formulas (5), (6), and (17) we givewhich leads toComparing coefficients, we obtainApplying formulas (13) and (18) and by simple calculation, we haveIn particular,Next, we will prove the estimate is sharp. For , consider a function , such thatand suppose that the dilatation of satisfiesApplying formula (5), we obtainwhich 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 seriesTogether 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 , we have the following distortion estimate of :
Our next aim is to give the distortion estimate of the coanalytic part of .
Theorem 7. If , thenThese inequalities are sharp. The equalities hold for the harmonic function which is defined in (16).
Proof. Let . Consider the functionwhich satisfies assumptions of the Schwarz lemma; then we haveIt is equivalent toHence, applying the triangle inequalities to formula (34) we haveFinally, 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 , we know that if , thenUsing inequality (35) to identity , then the corollary can be obtained immediately.
By , 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 , thenThe inequality is sharp. The equality holds for the harmonic function which is defined in (16).
Proof. Let ; applying estimate (31) we havewhere . The function defined (16) shows that inequality (40) is sharp.
For , by , we haveNow, 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 , thenwhere .
Proof. Observe that if , then does not vanish in . We can give the Jacobian of in the formwhere 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 denoteand then . Hence, there exists such that . Let , ; then and is a well-defined Jordan arc. Since , then we can obtainBy with formulas (29) and (35), we haveHence, we obtainTo prove (49) we simply use the inequalityBy 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 , thenwhere
Proof. Let tend to in estimate (48); then Theorem 16 follows immediately from the argument principle for harmonic mappings.
Corollary 17. If , thenwhere
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.
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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