SUBCLASSES OF BI-UNIVALENT FUNCTIONS BASED ON HOHLOV OPERATOR

In this paper, we introduce two new subclasses of the function class Σ of bi-univalent functions defined in the open unit disc based on Hohlov Operator. Furthermore, we find estimates on the coefficients |a2| and |a3| for functions in these new subclasses. Also consequences of the results are pointed out. AMS Subject Classification: 30C45


Introduction and Definitions
Let A denote the class of functions of the form which are analytic in the open unit disc U = {z : z ∈ C and |z| < 1}.Further, by S we shall denote the class of all functions in A which are univalent in U.It is well known that every function f ∈ S has an inverse f −1 , defined by where A function f ∈ A is said to be bi-univalent in U if both f (z) and f −1 (z) are univalent in U.
In 1967, Lewin [5] investigated the bi-univalent function class Σ and showed that |a 2 | < 1.51.Subsequently, Brannan and Clunie [1] conjectured that |a 2 | ≤ √ 2. Netanyahu [7], on the other hand, showed that max Brannan and Taha [2] introduced certain subclasses of the bi-univalent function class Σ.A function f ∈ A is in the class S * Σ (α) of strongly bistarlike of order α (0 ≤ α < 1), if each of the following condition is satisfied: 2 , and arg wg ′ (w) < απ 2 , w ∈ U; and the class < απ 2 , and arg 1 + wg ′′ (w) where the function g is given and g is the extension of f −1 to U and found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 | (for details see [2]).
The study of operators plays an important role in the geometric function theory and its related fields.Many differential and integral operators can be written in terms of convolution of certain analytic functions.It is observed that this formalism brings an ease in further mathematical exploration and also helps to understand the geometric properties of such operators better.We recall a sufficiently adequate special case of the generalized convolution operator I a,b,c , due to Hohlov [4] in terms of the Hadamard product (or convolution).
The convolution or Hadamard product of two functions f, h ∈ A is denoted by f * h and is defined as where f (z) is given by (1.1) and h(z For the complex parameters a, b and where (α) n is the Pochhammer symbol (or the shifted factorial) defined as (1.6)By using the Gaussian hypergeometric function given by (1.5), Hohlov [4] introduced a generalized convolution operator I a,b,c as where and discussed some interesting geometrical properties exhibited by this operator.The three parameter family of operators I a,b;c contains as a special cases most of the known linear integral or differential operators.In particular, if b = 1 in (1.7), then I a,1;c reduces to the Carlson-Shaffer operator a special case of Hohlov operator.Similarly it is straightforward to show that Hohlov operator is also a generalization of Ruscheweyh and Bernardi operators.
The object of the present paper is to introduce two new subclasses of the function class Σ and find estimate on the coefficients |a 2 | and |a 3 | for functions in these new subclasses of the function class Σ employing the techniques used earlier by Srivastava et al. [11] and others (see [3,8,10,12,13]).Definition 1.1.For 0 < α ≤ 1; 0 ≤ λ ≤ 1, a function f (z) given by (1.1) is said to be in the class P a,b;c Σ (α, λ) if the following conditions are satisfied: and respectively,where the function g is given by (1.3)and z, w ∈ U.
Further it is of interest to note that by taking a = b and c = 1 in the Definitions 1.1and 1.2 we deduce the new subclasses of Σ defined and discussed by authors in [6] In order to derive our main results, we have to recall here the following lemma [9].
In the following section, we obtain the estimates on the coefficients |a 2 | and |a 3 | for f ∈ P a,b;c Σ (α, λ). and f ∈ Q a,b;c Σ (β, λ).

Coefficient Bounds for the
and Proof.It follows from (1.9) and (1.10) that where p(z) and q(w) in ℘ and have the forms and q(w) = 1 + q 1 w + q 2 w 2 + . . . .
Now, equating the coefficients in (2.3) and (2.4), we get From (2.7) and (2.9), we get From (2.8), (2.10) and (2.12), we obtain Applying Lemma 1.3 for the coefficients p 2 and q 2 , we immediately have This gives the bound on |a 2 | as asserted in (2.1).Next, in order to find the bound on |a 3 |, by subtracting (2.10) from (2.8), we get It follows from (2.11), (2.12) and (2.13) that (2.14) Applying Lemma 1.3 once again for the coefficients p 1 , p 2 , q 1 and q 2 , we readily get This completes the proof of Theorem 2.1.

Coefficient Bounds for the Function Class
and Proof.It follows from (1.11) and (1.12) that there exists p, q ∈ ℘ such that where p(z) and q(w) have the forms (2.5) and (2.6), respectively.Equating coefficients in (3.3) and (3.4), we get From (3.5) and (3.7), we get .
Applying Lemma 1.3 for the coefficients p 2 and q 2 , we immediately have This gives the bound on |a 2 | as asserted in (3.1).
Next, in order to find the bound on (3.12) Applying Lemma 1.3 once again for the coefficients p 1 , p 2 , q 1 and q 2 , we readily get This completes the proof of Theorem 3.1.

Concluding Remarks
Various other interesting corollaries and consequences of our main results (which are asserted by Theorems 2.1 3.1and above) can be derived similarly for the new classes defined in Example 1 and Example2.Further by taking a = b and c = 1 in Theorems 2.1 3.1we get the results discussed by authors in [6].