Coefficient Estimate of Biunivalent Functions of Complex Order Associated with the Hohlov Operator

We introduce and investigate a new subclass of the function class Σ of biunivalent functions of complex order defined in the open unit disk, which are associated with the Hohlov operator, satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients |a 2 | and |a 3 | for functions in this new subclass. Several, known or new, consequences of the results are also pointed out.

Example 2. For  = 1 and  ∈ C \ {0}, a function  ∈ Σ, given by (1), is said to be in the class S ,; Σ (, ), if the following conditions are satisfied: where ,  ∈ U and the function  is given by (4).
Example 3.For  = 0 and  ∈ C \ {0}, a function  ∈ Σ, given by (1), is said to be in the class G ,; Σ (, ), if the following conditions are satisfied: where ,  ∈ U and the function  is given by ( 4).
It is of interest to note that, for  =  and  = 1, the class S ,; Σ (, , ) reduces to the following new subclasses.
Example 4. For  = 1 and  ∈ C \ {0}, a function  ∈ Σ, given by ( 1), is said to be in the class S * Σ (, ), if the following conditions are satisfied: where ,  ∈ U and the function  is given by ( 4).
Example 5.For  = 0 and  ∈ C \ {0}, a function  ∈ Σ, given by ( 1), is said to be in the class H * Σ (, ), if the following conditions are satisfied: where ,  ∈ U and the function  is given by ( 4).
In the following section, we find estimates on the coefficients | 2 | and | 3 | for functions in the above-defined subclasses S ,; Σ (, , ) of the function class Σ by employing the technique which is different from that used by earlier authors.Earlier authors investigated the coefficients of biunivalent functions mainly by using the following lemma.
Next, in order to find the bound on | 3 |, by subtracting (28) from (26), we get It follows from ( 20 This completes the proof of Theorem 7.
By putting  = 1 in Theorem 7, we have the following corollary.
Recently, there has been triggering interest to study biunivalent function class Σ and obtained nonsharp coefficient estimates on the first two coefficients | 2 | and | 3 | of (1).
, in the present paper, we introduce new subclasses of the function class Σ of complex order  ∈ C \ {0}, involving Hohlov operator I ,; , and find estimates on the coefficients | 2 | and | 3 | for functions in the new subclasses of function class Σ.Several related classes are also considered, and connection to earlier known results are made.