Abstract

An analytic function defined on the open unit disk is biunivalent if the function and its inverse are univalent in . Estimates for the initial coefficients of biunivalent functions are investigated when and , respectively, belong to some subclasses of univalent functions. Some earlier results are shown to be special cases of our results.

1. Introduction

Let be the class of all univalent analytic functions in the open unit disk and normalized by the conditions and . For , it is well known that the th coefficient is bounded by . The bounds for the coefficients give information about the geometric properties of these functions. Indeed, the bound for the second coefficient of functions in the class gives rise to the growth, distortion and covering theorems for univalent functions. In view of the influence of the second coefficient in the geometric properties of univalent functions, it is important to know the bounds for the (initial) coefficients of functions belonging to various subclasses of univalent functions. In this paper, we investigate this coefficient problem for certain subclasses of biunivalent functions.

Recall that the Koebe one-quarter theorem [1] ensures that the image of under every univalent function contains a disk of radius 1/4. Thus, every univalent function has an inverse satisfying , , and A function is biunivalent in if both and are univalent in . Let denote the class of biunivalent functions defined in the unit disk . Lewin [2] investigated this class and obtained the bound for the second coefficient of the biunivalent functions. Several authors subsequently studied similar problems in this direction (see [3, 4]). A function is bistarlike or strongly bistarlike or biconvex of order if and are both starlike, strongly starlike, or convex of order , respectively. Brannan and Taha [5] obtained estimates for the initial coefficients of bistarlike, strongly bistarlike, and biconvex functions. Bounds for the initial coefficients of several classes of functions were also investigated in [6–24].

An analytic function is subordinate to an analytic function , written , if there is an analytic function with satisfying . Ma and Minda [25] unified various subclasses of starlike () and convex functions () by requiring that either the quantity or is subordinate to a more general superordinate function with positive real part in the unit disk , , , maps onto a region starlike with respect to and symmetric with respect to the real axis. The class of Ma-Minda starlike functions with respect to consists of functions satisfying the subordination . Similarly, the class of Ma-Minda convex functions consists of functions satisfying the subordination . Ma and Minda investigated growth and distortion properties of functions in and as well as Fekete-Szegö inequalities for and . Their proof of Fekete-Szegö inequalities requires the univalence of . Ali et al. [7] investigated Fekete-Szegö problems for various other classes and their proof does not require the univalence or starlikeness of . In particular, their results are valid even if one just assumes the function to have a series expansion of the form ,  . So, in this paper, we assume that has series expansion , , are real, and . A function is Ma-Minda bistarlike or Ma-Minda biconvex if both and are, respectively, Ma-Minda starlike or convex. Motivated by the Fekete-Szegö problem for the classes of Ma-Minda starlike and Ma-Minda convex functions [25], Ali et al. [26] recently obtained estimates of the initial coefficients for biunivalent Ma-Minda starlike and Ma-Minda convex functions.

The present work is motivated by the results of Kędzierawski [27] who considered functions belonging to certain subclasses of univalent functions while their inverses belong to some other subclasses of univalent functions. Among other results, he obtained the following coefficient estimates.

Theorem 1 (see [27]). Let with Taylor series and . Then,

We need the following classes investigated in [6, 7, 26].

Definition 2. Let be analytic and with and . For , let

In this paper, we obtain the estimates for the second and third coefficients of functions when(i) and , or , or ,(ii) and , or ,(iii) and .

2. Coefficient Estimates

In the sequel, it is assumed that and are analytic functions of the form

Theorem 3. Let and . If , and is of the form then where .

Proof. Since and , , then there exist analytic functions , with , satisfying Define the functions and by or, equivalently, Then, and are analytic in with . Since , the functions and have positive real part in , and and . In view of (8) and (10), it is clear that Using (10) together with (4), it is evident that Since has the Maclaurin series given by (5), a computation shows that its inverse has the expansion Since it follows from (11) and (12) that It follows from (15) and (17) that Equations (15), (16), (18), and (19) lead to where , which, in view of and , gives us the desired estimate on as asserted in (6).
By using (16), (18), and (19), we get and this yields the estimate given in (7).

Remark 4. When and , , then (6) reduces to Theorem 1. When and , Theorem 3 reduces to [26, Theorem 2.2].

Theorem 5. Let and . If and , then where .

Proof. Let and , . Then, there exist analytic functions , with , such that Since (12) and (24) yield It follows from (26) and (28) that Hence, (26), (27), (29), and (30) lead to which gives us the desired estimate on as asserted in (22) when and .
Further, (27), (29), and (30) give and this yields the estimate given in (23).

Theorem 6. Let and . If and , then where .

Proof. Let and , . Then, there are analytic functions , with , satisfying Using and (12) and (34) will yield Further implication of (36) and applying the fact that and give the estimates in (33).

Theorem 7. Let and . If , , then where .

Proof. For and , , there exist analytic functions , with , satisfying Since then (12) and (38) yield Further implication of (40) and applying the fact that and give the estimates in (37).

Remark 8. When and , Theorem 7 reduces to [26, Theorem 2.3].

The following theorems give the estimates for the second and third coefficients of functions when (i) and and (ii) and . The proofs are similar as for the theorems above; hence, they are omitted here.

Theorem 9. Let and . If and , then where .

Theorem 10. Let and . If and , then where .

Remark 11. When and , Theorem 10 reduces to [26, Theorem 2.4].

Conflict of Interests

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

Acknowledgments

The research of the first and last authors is supported, respectively, by FRGS Grant and MyBrain MyPhD Programme of the Ministry of Higher Education, Malaysia.