# Coefficient Estimates for Certain Subclasses of Biunivalent Functions Defined by Convolution.

1. Introduction

Let A denote the class of functions of the form

f(z) = z + [[infinity].summation over (j=2)] [a.sub.j][z.sup.j], ([a.sub.j] [greater than or equal to] 0) (1)

which are analytic in the open disc [DELTA] = {z : [absolute value of (z)] < 1} and normalized by f(0) = 0, f'(0) = 1. Let S be the subclass of A consisting of univalent functions f(z) of form (1). For f(z) defined by (1) and h(z) defined by

h(z) = z + [[infinity].summation over (j=2)] [h.sub.j][z.sup.j], ([h.sub.j] [greater than or equal to] 0), (2)

the Hadamard product (or convolution) of f and h is defined by

[mathematical expression not reproducible]. (3)

It is well known that every function f [member of] S has an inverse [f.sup.-1] defined by

[mathematical expression not reproducible]. (4)

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

[mathematical expression not reproducible]. (5)

A function f [member of] A is said to be biunivalent in [DELTA] if f(z) and [f.sup.-1](z) are univalent in [DELTA]. Let [SIGMA] denote the class of biunivalent functions in [DELTA] given by (1). In 1967, Lewin [1] investigated the biunivalent function class [SIGMA] and showed that [absolute value of ([a.sub.2])] < 1.51. Brannan and Clunie [2] conjectured that [absolute value of ([a.sub.2])] [less than or equal to] [square root of 2]. Netanyahu [3] introduced certain subclasses of biunivalent function class [SIGMA] similar to the familiar subclasses [S.sup.*]([alpha]) and k([alpha]) of starlike and convex functions of order a (0< [alpha] [less than or equal to] 1). Brannan and Taha [4] defined f [member of] A in the class [S.sub.[SIGMA]] ([alpha]) of strongly bistarlike functions of order [alpha] (0 < [alpha] [less than or equal to] 1) if each of the following conditions is satisfied:

[mathematical expression not reproducible]. (6)

where g is as defined by (5). They also introduced the class of all bistarlike functions of order [beta] defined as a function f [member of] A, which is said to be in the class [S.sup.*.sub.[SIGMA]]([beta]) if the following conditions are satisfied:

[mathematical expression not reproducible], (7)

where the function g is as defined in (5). The classes [S.sup.*.sub.[sigma]]([alpha]) and [K.sub.[SIGMA]]([alpha]) of bistarlike functions of order [beta] and biconvex functions of order ft, corresponding to the function classes [s.sup.*]([beta]) and K([beta]), were introduced analogously. For each of the function classes [S.sup.*.sub.[SIGMA]]([beta]) and [K.sup.*.sub.[SIGMA]]([beta]), they found nonsharp estimates on the first two Taylor-Maclaurin coefficients [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])] (see [2, 5]). Some examples of biunivalent functions are z/(1 - z), (1/2) log((1 + z)/(1 - z)), and - log(1 - z) (see [6]). The coefficient estimate problem for each of the following Taylor-Maclaurin coefficients, [absolute value of ([a.sub.n])] (n [member of] N, n [greater than or equal to] 3), is still open ([6]). Various subclasses of biunivalent function class [SIGMA] were introduced and nonsharp estimates on the first two coefficients [absolute value of ([a.sub.2])] and [absolute value of ([a.sub.3])] in the Taylor-Maclaurin series (1) were found in several investigations (see [7-11]).

In this present investigation, motivated by the works of Brannan and Taha [2] and Srivastava et al. [6], we introduce two new subclasses of biunivalent functions involving convolution. The first two initial coefficients of each of these two new subclasses are obtained. Further, we prove that Brannan and Clunie's conjecture is true for our subclasses.

In order to derive our main results, we have to recall the following lemma.

Lemma 1 (see [12]). If p [member of] P, then [absolute value of ([p.sub.k])] [less than or equal to] 2 for each k, where p is the family of all functions p(z) analytic in [DELTA] for which Re{p(z)} > 0;

p(z) = 1 + [p.sub.1]z + [p.sub.2][z.sup.2] + [p.sub.3][z.sup.3] + ... [for all]z [member of] [DELTA]. (8)

2. Coefficient Bounds for the Classes [S.sub.[SIGMA]](h, [alpha], [lambda]) and [S.sup.*.sub.[SIGMA]](h, [beta], [lambda])

Definition 2. A function f(z) given by (1) is said to be in the class [S.sub.[SIGMA]](h, [alpha], [lambda]), if the following conditions are satisfied:

[mathematical expression not reproducible], (9)

where the function h(z) is defined by (2) and [(f * h).sup.-1](w) is defined by

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible]. (11)

Clearly, [S.sub.[SIGMA]](z/(1 - z), [alpha], 0) = [S.sub.[SIGMA]]([alpha]), the class of all strong bistarlike functions of order a introduced by Brannan and Taha [2].

Definition 3. A function f(z) given by (1) is said to be in the class [S.sup.*.sub.[SIGMA]](h, [beta], [lambda]), if the following conditions are satisfied:

[mathematical expression not reproducible], (12)

where h(z) and [(f * h).sup.-1](w) are defined, respectively, as in (2) and (10).

Clearly, [S.sub.[SIGMA]](z/(1 - z), [beta], 0) [equivalent to] [S.sup.*.sub.[SIGMA]]([beta]), the class of all strong bistarlike functions of order [beta] introduced by Brannan and Taha [2].

Theorem 4. Let f(z) given by (1) be in the class [S.sub.[SIGMA]](h, [alpha], [lambda]), 0 < [alpha] [less than or equal to] 1 and [lambda] [greater than or equal to] 0. Then

[mathematical expression not reproducible]. (13)

Further, for the choice of h(z) = z/[(1-z).sup.2] = z + [[SIGMA].sup.[infinity].sub.n=2] n[z.sup.n], one gets

[mathematical expression not reproducible]. (14)

Proof. It follows from (9) that

[mathematical expression not reproducible], (15)

where p(z) and q(w) satisfy the following inequalities:

Re {p(z)} > 0 (z [member of] [DELTA]), Re {q(w)} > 0 (w [member of] [DELTA]). (16)

Furthermore, the functions p(z) and q(w) have the forms

p(z) = 1 + [p.sub.1]z + [p.sub.2][z.sup.2] + [p.sub.3][z.sup.3] + ..., (17)

q(w) = 1 + [q.sub.1]w + [q.sub.2][w.sup.2] + [q.sub.3][w.sup.3] + .... (18)

Now, equating the coefficients in (15), we get

(1 + 2X)[a.sub.2][h.sub.2] = [alpha][p.sub.1] (19)

[mathematical expression not reproducible] (20)

- (1 + 2[lambda]) [a.sub.2][h.sub.2] = [alpha][q.sub.1], (21)

[mathematical expression not reproducible]. (22)

From (19) and (21), we get

[p.sub.1] = -[q.sub.1], (23)

2[(1+2[lambda]).sup.2] [a.sup.2.sub.2][h.sup.2.sub.2] = [[alpha].sup.2] ([p.sup.2.sub.1] + [q.sup.2.sub.1]). (24)

Now, from (20), (22), and (24), we obtain

[mathematical expression not reproducible]. (25)

Applying Lemma 1 for the coefficients [p.sub.2] and [q.sub.2], we immediately have

[mathematical expression not reproducible]. (26)

This gives the bound on [absolute value of ([a.sub.2])].

Next, in order to find the bound on [absolute value of ([a.sub.3])], by subtracting (20) from (22), we get

[mathematical expression not reproducible]. (27)

Upon substituting the value of [a.sup.2.sub.2] from (24) and observing that [p.sup.2.sub.1] = [q.sup.2.sub.1], it follows that

[mathematical expression not reproducible]. (28)

Applying Lemma 1 once again for the coefficients [p.sub.1], [p.sub.2], [q.sub.1], and [q.sub.2], we get

[absolute value of ([a.sub.3])] [less than or equal to] 4[[alpha].sup.2]/[h.sub.3][(1 + 2[lambda]).sup.2] + [alpha]/[h.sub.3](1 + 3[lambda]). (29)

This completes the proof.

Remark 5. When h(z) = z/(1 - z) and [lambda] = 0, in (13), we get the results obtained due to [4].

Remark 6. When [lambda] = 0, [alpha] = 1, and [h.sub.2] = 1, we obtain Brannan and Clunie's [2] conjecture [absolute value of ([a.sub.2])] [less than or equal to] [square root of 2].

Theorem 7. Let f(z) given by (1) be in the class [S.sup.*.sub.[SIGMA]] (h, [beta], [lambda]), 0 [less than or equal to] [beta] < 1 and [lambda] [greater than or equal to] 0. Then

[mathematical expression not reproducible]. (30)

Further, for the choice of h(z) = z/[(1-z).sup.2] = z + [[summation].sup.[infinity].sub.n=2] n[z.sup.n], we get

[mathematical expression not reproducible], (32)

[mathematical expression not reproducible]. (32)

Proof. It follows from (12) that there exist p(z) and q(w), such that

[mathematical expression not reproducible] (33)

where p(z) and q(w) have forms (17) and (18), respectively. Equating coefficients in (33) we obtain

(1+2[lambda])[a.sub.2][h.sub.2] = [p.sub.1] (1 - [beta]) (34)

2(1 + 3[lambda])[a.sub.3][h.sub.3], = [p.sub.2] (1-[beta]) + [p.sup.2.sub.1][(1 - [beta]).sup.2] (35)

-(1 + 2[lambda])[a.sub.2][h.sub.2] = [q.sub.1](1-[beta]), (36)

2 (1 + 3[lambda]) (2[a.sup.2.sub.2][h.sup.2.sub.2] - [a.sub.3][h.sub.3]) = [q.sub.2] (1 - [beta]) + [q.sup.2.sub.1][(1 - [beta]).sup.2]/(1 + 2[lambda]). (37)

From (34) and (36), we get

[P.sub.1] = -[q.sub.1], (38)

2[(1 + 2[lambda]).sup.2] [a.sup.2.sub.2][h.sup.2.sub.2] = [(1- [beta]).sup.2] ([p.sup.2.sub.1] + [q.sup.2.sub.1]). (39)

Now from (35), (37), and (39), we obtain

[mathematical expression not reproducible] (40)

Therefore, we have

[a.sup.2.sub.2] = (1 - [beta])([p.sub.2] + [q.sub.2])/2[h.sup.2.sub.2] (1 + 4[lambda]). (41)

Applying Lemma 1, for the coefficients [p.sub.2] and [q.sub.2], we immediately have

[absolute value of ([a.sub.2])] [less than or equal to] 1/[h.sub.2] (42)

Next, in order to find the bound on [absolute value of ([a.sub.3])], by subtracting (35) from (37), we get

[mathematical expression not reproducible]. (43)

Applying Lemma 1 for the coefficients [p.sub.1], [p.sub.2], [q.sub.1], and [q.sub.2], we readily get

[mathematical expression not reproducible]. (44)

Remark 8. When h(z) = z/(1 - z) and [lambda] = 0 in (30), we have the following result due to [4]. The bounds are

[mathematical expression not reproducible]. (45)

Remark 9. When h(z) = z/(1 - z) and [lambda] = 0 in (30), we have the following result due to [4]. The bounds are

[mathematical expression not reproducible]. (46)

Remark 10. When [lambda] = 0, [beta] =0, and [h.sub.2] = 1 we obtain Brannan and Clunie's [2] conjecture [absolute value of ([a.sub.2] [less than or equal to] [square root of 2].

http://dx.doi.org/10.1155/2016/6958098

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The work of the third author is supported by a grant from Department of Science and Technology, Government of India; vide Ref SR/FTP/MS-022/2012 under fast track scheme.

References

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[4] D. A. Brannan and T. S. Taha, "On some classes of bi-univalent," Studia Universitatis Babes-Bolyai Mathematica, vol 31, no. 2, pp. 70-77, 1986.

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[8] B. A. Frasin and M. K. Aouf, "New subclasses of bi-univalent functions," Applied Mathematics Letters, vol. 24, no. 9, pp. 1569-1573, 2011.

[9] T. Hayami and S. Owa, "Coefficient bounds for bi-univalent functions," Panamerican Mathematical Journal, vol. 22, no. 4, pp. 15-26, 2012.

[10] Q.-H. Xu, Y.-C. Gui, and H. M. Srivastava, "Coefficient-estimates for a certain subclass of analytic and bi-univalent functions," Applied Mathematics Letters, vol. 25, no. 6, pp. 990-994, 2012.

[11] Q.-H. Xu, H.-G. Xiao, and H. M. Srivastava, "A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems," Applied Mathematics and Computation, vol. 218, no. 23, pp. 11461-11465, 2012.

[12] C. Pommerenke, Univalent Functions, Vandenheock and Ruprecht, Gottingen, Germany, 1975.

R. Vijaya, (1) T. V. Sudharsan, (2) and S. Sivasubramanian (3)

(1) Department of Mathematics, SDNB Vaishnav College for Women, Chromepet, Chennai, Tamil Nadu 600044, India

(2) Department of Mathematics, SIVET College, Gowrivakkam, Chennai, Tamil Nadu 600073, India

(3) Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam, Tamil Nadu 604001, India

Correspondence should be addressed to S. Sivasubramanian; sivasaisastha@rediffmail.com

Received 7 July 2016; Accepted 25 October 2016