New Subclasses of Biunivalent Functions Involving Dziok-Srivastava Operator

We introduce two new subclasses of biunivalent functions which are defined by using the Dziok-Srivastava operator. Furthermore, we find estimates on the coefficients and for functions in these new subclasses.

It is well known that every function  ∈  has an inverse  −1 , defined by where A function  ∈  is said to be bi-univalent in  if both () and  −1 () are univalent in .Let Σ denote the class of bi-univalent functions in  given by (1).For a brief history and interesting examples in the class Σ see [5].
For function  given by (1) and  given by the Hadamard product (or convolution) of  and  is defined by For complex parameters  1 , . . .,   and  1 , . . .,   ( ∉ Z − 0 = {0, −1, −2, . ..};  = 1, . . ., ), the generalized hypergeometric function    is defined by the following infinite series: where ()  is the Pochhammer symbol (or shift factorial) defined, in terms of the Gamma function Γ, by Correspondingly a function ℎ( Dziok and Srivastava [9] (see also [10]) considered a linear operator defined by the following Hadamard product: If  ∈  is given by ( 1), then we have where To make the notation simple, we write The linear operator  , [ 1 ;  1 ; ] is a generalization of many other linear operators considered earlier.
The object of the present paper is to introduce two new subclasses of the bi-univalent functions which are defined by using the Dziok-Srivastava operator and find estimates on the coefficients | 2 | and | 3 | for functions in these new subclasses of the function class Σ employing the techniques used earlier by Srivastava et al. [5] (see also [11]).
In order to derive our main results, we have to recall here the following lemma [12].

Theorem 4. Letting 𝑓(𝑧) given by (1) be in the class 𝑇
Proof.It follows from ( 18) that where () and () in  have the forms Now, equating the coefficients in (22), we get From ( 25) and ( 27), we get Now from ( 26), (28), and (30), we obtain (31) Therefore, we have Applying Lemma 1 for the coefficients  2 and  2 , we immediately have This gives the bound on | 2 | as asserted in (20).Next, in order to find the bound on | 3 |, by subtracting (28) from (26) and using (29), we get It follows from (30) and (34) that And, then, Applying Lemma 1 once again for the coefficients  1 ,  2 ,  1 , and  2 , we readily get This completes the proof of Theorem 4.
where the function  is defined by (19).