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Coefficient Estimates for Starlike and Convex Classes of -fold Symmetric Bi-univalent Functions

S. Sivasubramanian, P. Gurusamy*

Department of Mathematics, Velammal Engineering College, Surapet, Chennai, India

Abstract: In an article of Pommerenke [10] he remarked that, for an -fold symmetric functions in the class , the well known lemma stated by Caratheodary for a one fold symmetric functions in still holds good. Exploiting this concept, we introduce certain new subclasses of the bi-univalent function class in which both and are -fold symmetric analytic with their derivatives in the class of analytic functions. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for and We remark here that the concept of -fold symmetric bi-univalent is not in the literature and the authors hope it will make the researchers interested in these type of investigations in the forseeable future. By the working procedure and the difficulty involved in these procedures, one can clearly conclude that there lies an unpredictability in finding the coefficients of a -fold symmetric bi-univalent functions.

Keywords: analytic functions, univalent functions, bi-univalent functions, -fold symmetric functions, subordination

Received March 31, 2015; Revised May 31, 2015; Accepted September 14, 2015

Citation: S. Sivasubramanian, P. Gurusamy, Coefficient Estimates for Starlike and Convex Classes of -fold Symmetric Bi-univalent Functions. Turkish Journal of Analysis and Number Theory, 3 (3). 83-86. January 2015. doi:10.12691/tjant-3-3-3

1. Introduction

Let denote the class of functions of the form:

[picture] | (1)

which are analytic in the open unit disk Further, by , we mean the class of all functions in which are univalent in . For more details on univalent functions, see [3]. It is well known that every function has an inverse , defined by

[picture] | (2)

and

[picture] | (3)

Indeed, the inverse function may have an analytic continuation to , with

[picture] | (4)

A function is said to be bi-univalent in if both and are univalent in . Let denote the class of bi-univalent functions in , given by equation (1). An analytic function is subordinate to an analytic function , written , provided there is an analytic function defined on with and satisfying Lewin [8] investigated the class of bi-univalent functions and obtained a bound . Motivated by the work of Lewin [8], Brannan and Clunie [1] conjectured that Some examples of bi-univalent functions are and (see also the work of Srivastava et al. [11]). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients: is still open([11]). In recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al. [11]. Motivated by this, many researchers (see [4,11,12,13,14,15,17]) recently investigated several interesting subclasses of the class and found non-sharp estimates on the first two Taylor-Maclaurin coefficients. For each function in , the function is univalent and maps the unit disk into a region with -fold symmetry. A function is -fold symmetric (see [10]) if it has the normalized form

[picture] | (5)

and we denote the class of -fold symmetric univalent functions by , which are normalized by the above series expansion. In fact, the functions in the class are one fold symmetric. Analogous to the concept of -fold symmetric univalent functions, one can think of the concept of -fold symmetric bi-univalent functions in a natural way. Each function in the class in , generates an -fold symmetric bi-univalent function for each integer . The normalized form of is given as in (5) and is given as follows.

[picture] | (6)

where We denote the class of -fold symmetric bi-univalent functions by For , the formula (6) coincides with the formula (4) of the class . Denote also, by , the class of analytic functions of the form such that in . In view of Pommerenke [10], the -fold symmetric function in the class is of the form

[picture] | (7)

It is assumed that is an analytic functions with positive real part in the unit disk , with and is symmetric with respect to the real axis. Such a function has a series expansion of the form

[picture] | (8)

Suppose that and are analytic functions in the unit disk with and and suppose that

[picture] | (9)

and

[picture] | (10)

We observe that

[picture] | (11)

By a simple computations, we have

[picture] | (12)

and

[picture] | (13)

Motivated essentially by the work of Ma and Minda [9], we introduce some new subclasses of -fold symmetric bi-univalent functions and obtain coefficient bounds of and for functions in these classes. The results presented in this paper improve the earlier results of Frasin and Aouf [4], Srivastava et al. [11] for the case of one fold symmetric functions.

2. Coefficient Estimates for the Function Class

Definition 2.1 A function , given by (5), is said to be in the class , if the following conditions are satisfied:

where the function is defined by (6).

For the special choices of the function and for the choice of our class reduces to the following.

1. For and , the class of strongly bi-starlike functions of order studied by Brannan and Taha [2].

2. For and , the class of bi-starlike functions of order studied by Brannan and Taha [2].

We first state and prove the following theorem.

Theorem 2.1 Let given by (5), be in the class

Then

[picture] | (14)

and

[picture] | (15)

Proof. Let and . Then there are analytic functions , with satisfying

[picture] | (16)

Since

and

it follows from (12), (13) and (16) that

[picture] | (17)

[picture] | (18)

[picture] | (19)

and

[picture] | (20)

From (17) and (19), we get

[picture] | (21)

By adding (18) and (20) and in view of the computations using (17) and (21), we get

[picture] | (22)

Further, (21), (22), together with (11), gives

[picture] | (23)

Now from (17) and (23), we get

as asserted in (14).

By simple calculations from (18) and (20) using with the equations (17) and (21), we get

[picture] | (24)

Then using the equation (11) in (24), we get

[picture] | (25)

Since

[picture] | (26)

substituting (26) in (25), we get

as asserted in (15). This completes the proof of Theorem 2.1.

For the case of one fold symmetric functions, Theorem 2.1 reduces to the coefficient estimates for Ma-Minda bi-starlike functions in Srivastava et al [11].

Corollary 2.1 Let given by (5), be in the class Then

[picture] | (27)

and

[picture] | (28)

For the case of one fold symmetric functions and for the class of strongly starlike functions , the function is given by

[picture] | (29)

which gives and Hence Theorem 2.1 reduce to the result in Brannan and Taha [2].

Corollary 2.2 [2] Let given by (5), be in the class Then

[picture] | (30)

and

[picture] | (31)

For the case of one fold symmetric functions and for the class of strongly starlike functions , the function is given by

then and the Theorem 2.1 reduce to the result in Brannan and Taha [2].

Corollary 2.3 [2] Let given by (5), be in the class . Then

[picture] | (32)

and

[picture] | (33)

3. Coefficient Bound for the Function Class

Definition 3.1 A function given by (5), is said to be in the class if the following conditions are satisfied:

and

where the function is defined by (6).

For one fold symmetric, a function in the class is called bi-Mocanu-convex function of Ma-Minda type. For the special choices of the function and for the choice of , our class reduces to the following.

1. For and the class of strongly bi-convex functions of order studied by Brannan and Taha [2].

2. For and the class of bi-convex functions of order studied by Brannan and Taha [2].

Theorem 3.1 Let given by (5), be in the class Then

[picture] | (34)

and

[picture] | (35)

Proof. Let Then there are analytic functions with satisfying

[picture] | (36)

Since

and

from (12), (13), and (36), we get

[picture] | (37)

[picture] | (38)

[picture] | (39)

and

[picture] | (40)

From (37) and (39), we get

[picture] | (41)

By adding the equations (38) and (40), in view of computations using (37) and (41), we get

[picture] | (42)

Further, from the equations (41), (42), together with (11), we have

[picture] | (43)

Now from (37) and (43), we get

as asserted in (34). By simple calculations from (38) and (40) using with the equations (37) and (41), we get

[picture] | (44)

Then using the equation (11) in (44), we get

[picture] | (45)

Since

[picture] | (46)

substituting (46) in (45), we get

as asserted in (35).

For one fold symmetric functions then, Theorem 3.1 gives the coefficient for Ma-Minda bi-convex functions in Brannan and Taha [2]

Corollary 3.1 [2] Let given by (5), be in the class Then

and

For the case of one fold symmetric functions and for the class of strongly starlike functions, the function is given by

then and the Theorem 3.1 reduce to the result in Brannan and Taha[2].

Corollary 3.2 [2] Let given by (5), be in the class . Then

and

References

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