Coefficient estimates for families of bi-univalent functions defined by Ruscheweyh derivative operator

The main purpose of this manuscript is to find upper bounds for the second and third Taylor-Maclaurin coefficients for two families of holomorphic and bi-univalent functions associated with Ruscheweyh derivative operator. Further, we point out certain special cases for our results.


Introduction
Indicate by A the collection of all holomorphic functions in the open unit disk U = {z ∈ C : |z| < 1} , that have the form (1) f (z) = z + ∞ n=2 a n z n .
Further, assume that S stands for the sub-collection of the set A containing of functions in U satisfying (1) which are univalent in U .

Lemma 1 ([6]
). If h ∈ P, then |c k | ≤ 2 for each k ∈ N, where P is the family of all functions h holomorphic in U for which where h(z) = 1 + c 1 z + c 2 z 2 + · · · (z ∈ U ).

Definition 2. A function f ∈ Σ given by
where z, w ∈ U and g = f −1 is given by (2).

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Coefficient estimates for families of bi-univalent functions. . .
Adding (23) and (25), we obtain Hence, we find from (26) and (27) that respectively. By applying Lemma 1 for the coefficients p 2 and q 2 , we deduce that respectively. To determinate the bound on |a 3 |, by subtracting (25) from (23), we get or equivalently .

Conclusion
In this investigation, we have introduced and defined two a certain families of holomorphic and bi-univalent functions in the open unit disk U associated with Ruscheweyh derivative operator. We have then derived the initial coefficient estimations for functions belonging to these families. Further by specializing the parameters, several consequences of these families are mentioned.