Certain Subclasses of Bistarlike and Biconvex Functions Based on Quasi-Subordination

Abstract

We introduce the unified biunivalent function class defined based on quasi-subordination and obtained the coefficient estimates for Taylor-Maclaurin coefficients and . Several related classes of functions are also considered and connections to earlier known and new results are established.

1. Introduction

Let denote the class of functions of the formwhich are analytic in the open unit disc . Further, by we denote the family of all functions in which are univalent in . Let be an analytic function in and , such thatwhere all coefficients are real. Also, let be an analytic and univalent function with positive real part in with , and maps the unit disk onto a region starlike with respect to 1 and symmetric with respect to the real axis. Taylor’s series expansion of such function is of the formwhere all coefficients are real and . Throughout this paper we assume that the functions and satisfy the above conditions one or otherwise stated.

For two functions and are analytic in , we say that the function is subordinate to in and write if there exists a Schwarz function , analytic in , with such that In particular, if the function is univalent in , the above subordination is equivalent to

For two analytic functions and , the function is quasi-subordinate to in the open unit disc if there exist analytic functions and , with , , and , such that is analytic in and written as We also denote the above expression by and this is equivalent to

Observe that if , then , so that in . Also notice that if , then and it is said that is majorized by and written by in . Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization (see [1]).

In [2] Ma and Minda introduced the unified classes and given below:

For the choiceorthe classes and consist of functions known as the starlike (resp., convex) functions of order or strongly starlike (resp., convex) functions of order , respectively.

Recently, El-Ashwah and Kanas [3] introduced and studied the following two subclasses:We note that when , the classes and reduce, respectively, to the familiar classes and of Ma-Minda starlike and convex functions of complex order () in (see [4]). For , the classes and reduce to the classes and , respectively, that are analogous to Ma-Minda starlike and convex functions, introduced by Mohd and Darus [5].

It is well known that every function has an inverse , defined by wherewhereand is the th order determinant whose entries are defined in terms of the coefficients of by the following:For initial values of , we haveand so on. A function is said to be biunivalent in if both and are univalent in . Let denote the class of biunivalent functions in given by (1). For a brief history and interesting examples of functions which are in (or which are not in) the class , together with various other properties of the biunivalent function class , one can refer to the work of Srivastava et al. [6] and references therein. Recently, various subclasses of the biunivalent function class were introduced and nonsharp estimates on the first two coefficients and in the Taylor–Maclaurin series expansion (1) were found in several recent investigations (see, e.g., [7–17]). But the problem of finding the coefficient bounds on () for functions is still an open problem.

Motivated by the above mentioned works, we define the following subclass of function class .

A function given by (1) is said to be in the class , , , if the following quasi-subordination conditions are satisfied:whereand the function is the extension of to .

It is interesting to note that the special values of , , , and and the class unify the following known and new classes.

Remark 1. Setting in the above class, we have In particular, for , we have which was introduced and studied by Goyal and Kumar [18, Definition , p. ]. Also, we note that for the class was introduced and studied by Ali et al. [7] (see also [19]). If we take by (12) in the class , we are led to the class which we denote, for convenience, by , introduced and studied by Li and Wang [12, Definition ., p. ], and upon replacing by (13) in the class , we have ; this class was introduced and studied by Li and Wang [12, Definition ., p. ].

Remark 2. Taking and in the class , we have In particular, for , we have The class is particular case of the class , when and it was introduced and studied by Goyal and Kumar [18, Definition , p. ]. We note that, for , the class was introduced and studied by Deniz [10]. Further, for , the class was introduced by Ali et al. [7] and Srivastava et al. [16]. For given by (12), the class was introduced by Brannan and Taha [20] and studied by Bulut [8], Çaglar et al. [9], Li and Wang [12], and others.

Remark 3. Setting and in the class , we have In particular, for , we get The class is particular case of the class , when and it was introduced and studied by Goyal and Kumar [18, Definition , p. ]. We note that, for , the class was introduced and studied by Deniz [10]. Further, for , the class was considered by Ali et al. [7]. For given by (12), we get the class , introduced by Brannan and Taha [20] and studied by Li and Wang [12] and others.

Remark 4. Taking , we have the class as defined below.
A function is said to be in the class , , , if the following quasi-subordinations hold:where . A function in the class is called both bi--convex functions and bi--starlike functions of complex order of Ma-Minda type. For , the class was introduced and studied by Deniz [10].

Remark 5. Putting , we have the class as defined below.
A function is said to be in the class , , , if the following quasi-subordinations hold:where .

Remark 6. For , the class was introduced in [21].

In this paper we introduce the unified biunivalent function class as defined above and obtain the coefficient estimates for Taylor-Maclaurin coefficients and for functions belonging to . Some interesting applications of the results presented here are also discussed.

In order to derive our results, we need the following lemma.

Lemma 7 (see [22]). If , then for each , where is the family of all functions , analytic in , for which where

2. Coefficient Estimates for the Class

Theorem 8. Let given by (1) be in the class , , , and . Then

Proof. Since , there exist two analytic functions , with , such thatDefine the functions and byor equivalentlyUsing (36) in (34), we haveAgain using (36) along with (3), it is evident thatIt follows from (37) and (38) thatFrom (39) and (41), we find thatit follows thatAdding (40) and (42), we haveSubstituting (43) and (44) into (46), we getApplying Lemma 7 in (47), we get desired inequality (32). Subtracting (40) from (42) and a computation using (44) finally lead toAgain applying Lemma 7, (48) yields desired inequality (33). This completes the proof of Theorem 8.

In light of Remarks 1–5, we have following corollaries.

Corollary 9. If , , then

Remark 10. Corollary 9 reduces to [23, Corollary , p. ].

Corollary 11. If , , then

Corollary 12. If , , and , then

Corollary 13. If , , and , then

Corollary 14. If , , and , then

Remark 15. Taking in Corollary 9, we get estimates in [10, Corollary , p. ] and setting in Corollary 11 we have bounds in [10, Corollary , p. ]. For and , the inequalities obtained in Corollary 11 coincide with [7, Corollary , p. ]. For and , the estimates in Corollary 12 reduce to a known result in [7, Theorem , p. ]. Further, for , , and given by (12) the inequalities in Corollary 12 reduce to a result proven earlier by [12, Theorem , p. ] and for , , and given by (13) the inequalities in Corollary 12 would reduce to known result in [12, Theorem , p. ]. Also, for , the estimates in Corollary 13 provide improvement over the estimates derived by Deniz [10, Theorem , p. ]. For , the results obtained in this paper coincide with results in [21]. Furthermore, various other interesting corollaries and consequences of our results could be derived similarly by specializing .

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.