Coefficient Bounds for Subclasses of Biunivalent Functions Associated with the Chebyshev Polynomials
We introduce and investigate new subclasses of biunivalent functions defined in the open unit disk, involving Sălăgean operator associated with Chebyshev polynomials. Furthermore, we find estimates of the first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we give Fekete-Szegö inequalities for these function classes. Several consequences of the results are also pointed out.
Let denote the class of analytic functions of the form normalized by the conditions defined in the open unit disk Let be the subclass of consisting of functions of form (1) which are also univalent in Let and denote the well-known subclasses of , consisting of starlike and convex functions of order , respectively.
The Koebe one-quarter theorem  ensures that the image of under every univalent function contains a disk of radius . Thus every univalent function has an inverse satisfying A function is said to be biunivalent in if both and are univalent in . Let denote the class of biunivalent functions defined in the unit disk . Since has the Maclaurin series given by (1), a computation shows that its inverse has the expansion
An analytic function is subordinate to an analytic function , written as , provided there is an analytic function defined on with and satisfying .
Chebyshev polynomials, which are used by us in this paper, play a considerable role in numerical analysis. We know that the Chebyshev polynomials are four kinds. The most of books and research articles related to specific orthogonal polynomials of Chebyshev family contain essentially results of Chebyshev polynomials of first and second kinds and and their numerous uses in different applications; see Doha  and Mason .
The well-known kinds of the Chebyshev polynomials are the first and second kinds. In the case of real variable on , the first and second kinds are defined by where the subscript denotes the polynomial degree and . We consider the function We note that if , , then for all Thus, we write where , for , are the second kind of the Chebyshev polynomials. Also, it is known that The Chebyshev polynomials , , of the first kind have the generating function of the form
All the same, the Chebyshev polynomials of the first kind and the second kind are well connected by the following relationship:
Several authors have introduced and investigated subclasses of biunivalent functions and obtained bounds for the initial coefficients (see [4–10]). In , making use of the Sălăgean  differential operator, defined by and further for functions of the form (4) Vijaya et al.  (also see ) defined and introduced two new subclasses of biunivalent functions. In this paper, we use Chebyshev polynomials to obtain the estimates on the coefficients and .
2. Biunivalent Function Classes and
Motivated by recent works of Altinkaya and Yalcin  (also see ) and recent studies on biunivalent functions involving Sălăgean operator [11, 13], in this section, we introduce two new subclasses of Σ associated with Chebyshev polynomials and obtain the initial Taylor coefficients and for the function classes by subordination.
We note that by specializing the parameters and suitably fixing the values for in Definition 1, we introduce (had not been studied so far) the following new subclasses of as listed below.
Remark 2. Supposing and , then we denote(1),(2),(3),(4)
In Definition 3, by specializing the parameters and suitably fixing the values for (had not been studied so far) the following new subclasses of are as listed below.
Remark 4. Supposing and , then we denote (1),(2),(3),(4)
In the following theorems we determine the initial Taylor coefficients and for the function classes and .
Theorem 5. Let given by (1) be in the class and Then where and
Proof. Let and Considering (17), we have Define the functions and by which are analytic in with and , , for all . It is well known that and then Using (22) and (23) in (20) and (21), respectively, we have In light of (1), (4), (10), (15), and (16) and from (26), we have This yields the following relations:From (28) and (30) it follows that Adding (29) to (31) and using (33), we obtain Applying (25) to the coefficients and and using (10) we have By subtracting (31) from (29) and using (32) and (33), we get Using (10), once again applying (25) to the coefficients , , , and , we get
By taking or and , one can easily state the estimates and for the function classes and , respectively.
Remark 6. Let given by (1) be in the class Then
Remark 7. Let given by (1) be in the class Then
For , Theorem 5 yields the following corollary.
Corollary 8. Let given by (1) be in the class Then where and
By taking in the above remarks we get the estimates and for the function classes and
Remark 9. Let given by (1) be in the class Then
Remark 10. Let given by (1) be in the class Then, for ,
Theorem 11. Let given by (1) be in the class and Then
Proof. Proceeding as in the proof of Theorem 5 we can arrive at the following relations:From (46) and (48) it follows that From (47), (49), and (51), we obtain Using (10) and (25) for the coefficients and , we immediately get the desired estimate on as asserted in (44).
By subtracting (49) from (47) and using (50) and (51), we get Again using (10) and (25) for the coefficients , , , and , we get the desired estimate on as asserted in (45).
Remark 12. Let given by (1) be in the class . Then
Remark 13. Let given by (1) be in the class . Then
By taking we deduce the following results.
Remark 14. Let given by (1) be in the class . Then
Remark 15. Let given by (1) be in the class Then
Remark 16. Let given by (1) be in the class Then
3. Fekete-Szegö Inequality for the Function Classes and
Due to Zaprawa , in this section we obtain the Fekete-Szegö inequality for the function classes and
Theorem 17. Let given by (1) be in the class and . Then one has
Corollary 18. If , then
Corollary 19. Let given by (1) be in the class and . Then one has Particularly, for if one obtains
Corollary 20. Let given by (1) be in the class and . Then one hasParticularly, for if one obtains
Theorem 21. Let given by (1) be in the class and . Then one has
Corollary 22. If , then
Corollary 23. Let given by (1) be in the class and . Then one has Particularly, for if one obtains
Corollary 24. Let given by (1) be in the class and . Then one has Particularly, for if one obtains
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
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