Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric Points

Abstract

We introduce two subclasses of biunivalent functions and find estimates on the coefficients and for functions in these new subclasses. Also, consequences of the results are pointed out.

1. Introduction and Definitions

Let denote the class of analytic functions in the unit disk that have the form Further, by we will denote the class of all functions in which are univalent in .

The Koebe one-quarter theorem [1] states that the image of under every function from contains a disk of radius . Thus every such univalent function has an inverse which satisfies where

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 .

If the functions and are analytic in , then is said to be subordinate to , written as if there exists a Schwarz function , analytic in , with such that

Lewin [2] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient . Subsequently, Netanyahu [3] showed that if . Brannan and Clunie [4] conjectured that for . Brannan and Taha [5] introduced certain subclasses of the biunivalent function class similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike, and convex functions. They introduced bistarlike functions and obtained estimates on the initial coefficients. Bounds for the initial coefficients of several classes of functions were also investigated in [6–15].

Not much is known about the bounds on the general coefficient for . In the literature, there are only a few works determining the general coefficient bounds for the analytic biunivalent functions ([16–20]). The coefficient estimate problem for each of is still an open problem.

By and we denote the following classes of functions:

The classes and are the extensions of classical sets of starlike and convex functions and in such form were defined and studied by Ma and Minda [21].

In [22], Sakaguchi introduced the class of starlike functions with respect to symmetric points in , consisting of functions that satisfy the condition , . Similarly, in [23], Wang et al. introduced the class of convex functions with respect to symmetric points in , consisting of functions that satisfy the condition , . In the style of Ma and Minda, Ravichandran (see [24]) defined the classes and .

A function is in the class if and in the class if

In this paper, we introduce two new subclasses of biunivalent functions. Further, we find estimates on the coefficients and for functions in these subclasses.

2. Coefficient Estimates for the Function Class

Definition 1. Let the functions be so constrained that

Definition 2. A function is said to be in the class if the following conditions are satisfied: where .

Definition 3. One notes that, for , one gets the class which is defined as follows:

Theorem 4. Let given by (2) be in the class . Then

Proof. Let and be the analytic extension of to . It follows from (12) that where and satisfy the conditions of Definition 1. Furthermore, the functions and have the following Taylor-Maclaurin series expansions: respectively. From (15), we deduce From (18) and (20) we obtain By adding (19) to (21), we get Therefore, we find from (23) and (24) that
Subtracting (21) from (19) we have Then, upon substituting the value of from (23) and (24) into (26), it follows that We thus find that This completes the proof of Theorem 4.

Taking we get the following.

Corollary 5. If then

Corollary 6. If we let then inequalities (14) become

Corollary 7. If we let then inequalities (14) become

Remark 8. Corollaries 6 and 7 provide an improvement of the estimate obtained by Altınkaya and Yalçın [25].

Remark 9. The estimates on the coefficients and of Corollaries 6 and 7 are improvement of the estimates in [7].

3. Coefficient Estimates for the Function Class

Definition 10. A function is said to be if the following conditions are satisfied: where .

We note that, for , the class reduces to the class .

Definition 11. One notes that, for , one gets the class which is defined as follows:

Theorem 12. Let given by (2) be in the class . Then

Proof. Let and be the analytic extension of to . We have It follows from (34) that where and satisfy the conditions of Definition 1.
From (39), we deduce From (40) and (42) we obtain By adding (41) to (43), we get which gives us the desired estimate on as asserted in (36).
Subtracting (43) from (41) we have Then, in view of (45) and (46), it follows that as claimed. This completes the proof of Theorem 12.

Taking we get the following.

Corollary 13. If then

Corollary 14. If we let then inequalities (36) and (37) become

Corollary 15. If we let then inequalities (36) and (37) become

Remark 16. Corollaries 14 and 15 provide an improvement of the estimate obtained by Altınkaya and Yalçın [25].

Remark 17. The estimates on the coefficients and of Corollaries 14 and 15 are improvement of the estimates obtained in [7].

Conflict of Interests

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