Coefficient Bounds for Certain Subclasses of -Fold Symmetric Biunivalent Functions

Abstract

We consider two new subclasses and of consisting of analytic and -fold symmetric biunivalent functions in the open unit disk . Furthermore, we establish bounds for the coefficients for these subclasses and several related classes are also considered and connections to earlier known results are made.

1. Introduction

Let denote the class of functions of the formwhich are analytic in the open unit disk , and let be the subclass of consisting of form (1) which is also 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 inverse which satisfieswhere

Function is said to be biunivalent in if both and are univalent in . Let denote the class of biunivalent functions defined in unit disk .

For a brief history and interesting examples in class , see [2]. Examples of functions in class areand so on. However, the familiar Koebe function is not a member of . Other common examples of functions in such asare also not members of (see [2]).

For each function , functionis univalent and maps unit disk into a region with -fold symmetry. A function is said to be -fold symmetric (see [3, 4]) if it has the following normalized form:

We denote by the class of -fold symmetric univalent functions in , which are normalized by the series expansion (7). In fact, the functions in class are one-fold symmetric.

Analogous to the concept of -fold symmetric univalent functions, we here introduced the concept of -fold symmetric biunivalent functions. Each function generates an -fold symmetric biunivalent function for each integer . The normalized form of is given as in (7) and the series expansion for , which has been recently proven by Srivastava et al. [5], is given as follows:where . We denote by the class of -fold symmetric biunivalent functions in . For , formula (8) coincides with formula (3) of class . Some examples of -fold symmetric biunivalent functions are given as follows:

Lewin [6] studied the class of biunivalent functions, obtaining the bound 1.51 for modulus of the second coefficient . Subsequently, Brannan and Clunie [7] conjectured that for . Later, Netanyahu [8] showed that if . Brannan and Taha [9] introduced certain subclasses of biunivalent function class similar to the familiar subclasses. and are of starlike and convex function of order , respectively (see [8]). Classes and of bistarlike functions of order and biconvex functions of order , corresponding to function classes and , were also introduced analogously. For each of function classes and , they found nonsharp estimates on the initial coefficients. In fact, the aforecited work of Srivastava et al. [2] essentially revived the investigation of various subclasses of biunivalent function class in recent years. Recently, many authors investigated bounds for various subclasses of biunivalent functions (see [2, 10–15]). Not much is known about the bounds on general coefficient for . In the literature, only few works determine general coefficient bounds for the analytic biunivalent functions (see [16–18]). The coefficient estimate problem for each of is still an open problem.

The aim of the this paper is to introduce two new subclasses of function class and derive estimates on initial coefficients and for functions in these new subclasses. We have to remember the following lemma here so as to derive our basic results.

Lemma 1 (see [4]). If is an analytic function in with positive real part, then

2. Coefficient Bounds for Function Class

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

Theorem 3. Let given by (7) be in class , . Then,

Proof. Let . Then,where   and in have the following forms:Now, equating the coefficients in (13), we getFrom (15) and (17), we obtainAlso from (16), (18), and (20), we haveTherefore, we haveApplying Lemma 1 for coefficients and , we obtainNext, in order to find the bound on , by subtracting (18) from (16), we obtainThen, in view of (19) and (20) and applying Lemma 1 for coefficients and , we haveThis completes the proof of Theorem 3.

3. Coefficient Bounds for Function Class

Definition 4. Function given by (7) is said to be in class if the following conditions are satisfied:where function .

Theorem 5. Let given by (7) be in class , . Then,

Proof. Let . Then,where and .
It follows from (28) thatFrom (29) and (31), we obtainAdding (30) and (32), we have Therefore, we obtainApplying Lemma 1 for coefficients and , we obtainNext, in order to find the bound on , by subtracting (32) from (30), we obtainThen, in view of (33), applying Lemma 1 for coefficients and , we haveThis completes the proof of Theorem 5.

If we set in Theorems 3 and 5, then classes and reduce to classes and and thus we obtain the following corollaries.

Corollary 6. Let given by (7) be in class . Then,

Corollary 7. Let given by (7) be in class . Then,

Classes and are, respectively, defined as follows.

Definition 8. Function given by (7) is said to be in class if the following conditions are satisfied:where function .

Definition 9. Function given by (7) is said to be in class if the following conditions are satisfied:where function .

For one-fold symmetric biunivalent functions and , Theorems 3 and 5 reduce to Corollaries 10 and 11, respectively, which were proven earlier by Murugusundaramoorty et al. [19].

Corollary 10. Let given by (7) be in class . Then,

Corollary 11. Let given by (7) be in class . Then,

Conflict of Interests

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