JCA Journal of Complex Analysis 2314-4971 2314-4963 Hindawi Publishing Corporation 474231 10.1155/2013/474231 474231 Research Article Coefficient Estimate Problem for a New Subclass of Biunivalent Functions Magesh N. 1 Rosy T. 2 Varma S. 2 Heittokangas Janne 1 Postgraduate and Research Department of Mathematics Government Arts College for Men Krishnagiri Tamil Nadu 635001 India 2 Department of Mathematics Madras Christian College Thambaram Chennai Tamil Nadu 600 059 India mcc.edu.in 2013 28 10 2013 2013 25 05 2013 19 09 2013 2013 Copyright © 2013 N. Magesh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce a unified subclass of the function class Σ of biunivalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients |a2| and |a3| for functions in this subclass. In addition, many relevant connections with known or new results are pointed out.

1. Introduction

Let 𝒜 denote the class of functions of the form (1)f(z)=z+n=2anzn which are analytic in the open unit disc 𝕌={z:zand|z|<1}. Further, by 𝒮, we will denote the class of all functions in 𝒜 which are univalent in 𝕌.

Some of the important and well-investigated subclasses of the univalent function class 𝒮 include, for example, the class 𝒮*(α) of starlike functions of order α in 𝕌 and the class 𝒦(α) of convex functions of order α in 𝕌.

It is well known that every function f𝒮 has an inverse f-1, defined by (2)f-1(f(z))=z(z𝕌),f(f-1(w))=w(|w|<r0(f);  r0(f)14), where (3)f-1(w)=w-a2w2+(2a22-a3)w3-(5a23-5a2a3+a4)w4+.

A function f𝒜 is said to be biunivalent in 𝕌 if both f(z) and f-1(z) are univalent in 𝕌. Let Σ denote the class of biunivalent functions in 𝕌 given by (1).

In 1967, Lewin  investigated the biunivalent function class Σ and showed that |a2|<1.51; on the other hand Brannan and Clunie  (see also ) and Netanyahu  made an attempt to introduce various subclasses of biunivalent function class Σ and obtained nonsharp coefficient estimates on the first two coefficients |a2| and |a3| of (1). But the coefficient problem for each of the following Taylor-Maclaurin coefficients |an| for n{1,2}; :={1,2,3,} is still an open problem. In this line, following Brannan and Taha , recently, many researchers have introduced and investigated several interesting subclasses of biunivalent function class Σ and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|; for details, one can refer to the works of .

Now, we define Σ(α,λ) of function f𝒜 satisfying the following conditions: (4)fΣ,|arg(z1-λf(z)(f(z))1-λ)|<απ2,|arg(w1-λg(w)(g(w))1-λ)|<απ2(z,w𝕌;  λ0) for some α(0<α1), where g(w) is the extension of f-1(w) to 𝕌. Similarly, we say that a function f𝒜 belongs to the class Σ(β,λ) if f(z) satisfies the following inequalities: (5)fΣ,(z1-λf(z)(f(z))1-λ)>β,(w1-λg(w)(g(w))1-λ)>β(z,w𝕌;  λ0), for some β(0β<1), where g(w) is the extension of f-1(w) to 𝕌. The classes Σ(α,λ) and Σ(β,λ) were introduced by Prema and Keerthi ; furthermore, for these classes, they have found the following estimates on the first two Taylor-Maclaurin coefficients in (1).

Theorem 1.

If fΣ(α,λ), 0<α1, and λ0, then (6)|a2|2α(α+1+λ)(1+λ),|a3|4α2(1+λ)2+2α2+λ.

Theorem 2.

If fΣ(β,λ), 0β<1, and λ0, then (7)|a2|2(1-β)1+λ,|a3|4(1-β)2(1+λ)2+2(1-β)2+λ.

Motivated by the works of Xu et al. [12, 13], we introduce the following generalized subclass Σ(φ,ψ,λ) of the analytic function class 𝒜.

Definition 3.

Let f𝒜, and let the functions φ,ψ:𝕌 be so constrained that (8)min{(φ(z)),(ψ(z))}>0(z𝕌),φ(0)=ψ(0)=1. We say that fΣ(φ,ψ,λ) if the following conditions are satisfied: (9)fΣ,z1-λf(z)(f(z))1-λφ(𝕌),w1-λg(w)(g(w))1-λψ(𝕌)(z,w𝕌), where λ0 and the function g(w) is the extension of f-1(w) to 𝕌.

We note that by specializing λ, φ, and ψ, we get the following interesting subclasses:

Σ(φ,ψ,1)=Σφ,ψ; see ,

Σ(((1+z)/(1-z))α,  ((1+z)/(1-z))α,λ)=Σ(α,λ) (0<α1; λ0) and Σ((1+(1-2β)z)/(1-z),  (1+(1-2β)z)/(1-z),λ)=Σ(β,λ) (0β<1; λ0); see ,

Σ(((1+z)/(1-z))α,((1+z)/(1-z))α,1)=Σα (0<α1) and Σ((1+(1-2β)z)/(1-z),(1+(1-2β)z)/(1-z),1)=Σβ (0β<1); see .

The objective of the present paper is to introduce a new subclass Σ(φ,ψ,λ) and to obtain the estimates on the coefficients |a2| and |a3| for the functions in theaforementioned class, employing the techniques used earlier by Xu et al. [12, 13].

2. Main Result

In this section, we find the estimates on the coefficients |a2| and |a3| for the functions in the class Σ(φ,ψ,λ).

Theorem 4.

Let f(z) be of the form (1). If fΣ(φ,ψ,λ), then (10)|a2||φ′′(0)|+|ψ′′(0)|8+4λ,(11)|a3||φ′′(0)|4+2λ.

Proof.

Since fΣ(φ,ψ,λ), from (9), we have, (12)z1-λf(z)(f(z))1-λ=φ(z)(z𝕌),w1-λg(w)(g(w))1-λ=ψ(w)(w𝕌), where (13)φ(z)=1+φ1z+φ2z2+,ψ(z)=1+ψ1z+ψ2z2+ satisfy the conditions of Definition 3. Now, equating the coefficients in (12), we get (14)(1+λ)a2=φ1,(15)(2+λ)a3=φ2,(16)-(1+λ)a2=ψ1,(17)(2+λ)(2a22-a3)=ψ2. From (14) and (16), we get (18)φ1=-ψ1,2(1+λ)2a22=φ12+ψ12. From (15) and (17), we obtain (19)a22=φ2+ψ22(2+λ). Since φ(z)φ(𝕌) and ψ(z)ψ(𝕌), we immediately have (20)|a2||φ′′(0)|+|ψ′′(0)|8+4λ. This gives the bound on |a2| as asserted in (10).

Next, in order to find the bound on |a3|, by subtracting (17) from (15), we get (21)2(2+λ)a3-2(2+λ)a22=φ2-ψ2. It follows from (19) and (21) that (22)a3=φ22+λ. Since φ(z)φ(𝕌) and ψ(z)ψ(𝕌), we readily get |a3||φ′′(0)|/(4+2λ) as asserted in (11). This completes the proof of Theorem 4.

By setting φ(z)=ψ(z)=((1+Az)/(1+Bz))α, where -1B<A1 and 0<α1, in Theorem 4, we get the following corollary.

Corollary 5.

Let f(z) be of the form (1) and in the class Σ(A,B,α,λ). Then, (23)|a2|α2(A-B)2-α(A2-B2)4+2λ,|a3|α2(A-B)2-α(A2-B2)4+2λ.

If we choose A=1 and B=-1 in Corollary 5, we have the following corollary.

Corollary 6.

Let f(z) be of the form (1) and in the class Σ(α,λ), 0<α1 and λ0. Then, (24)|a2|α22+λ,|a3|2α22+λ.

Remark 7.

The estimates found in Corollary 6 would improve the estimates obtained in [14, Theorem 2.2].

If we set A=1-2β, B=-1, where 0β<1 and α=1 in Corollary 5, we readily have the following corollary.

Corollary 8.

Let f(z) be of the form (1) and in the class Σ(β,λ), 0β<1 and λ0. Then (25)|a2|2(1-β)2+λ,|a3|2(1-β)2+λ.

Remark 9.

The estimates found in Corollary 8 would improve the estimates obtained in [14, Theorem 3.2].

Remark 10.

For λ=1, the bounds obtained in Theorem 4 are coincident with the outcome of Xu et al. . Taking λ=0 in Corollaries 6 and 8, the estimates on the coefficients |a2| and |a3|, are the improvement of the estimates on the first two Taylorû Maclaurin coefficients obtained in [10, Corollaries 2.3 and 3.3]. Also, for the choices of λ=1, the results stated in Corollaries 6 and 8 would improve the bounds stated in [11, Theorems 1 and 2], respectively. Furthermore, various other interesting corollaries and consequences of our main result could be derived similarly by specializing φ and ψ.

Acknowledgment

The authors would like to thank the referee for his valuable suggestions.

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