Applications of Borel distribution series on holomorphic and bi-univalent functions

In present manuscript, we introduce and study two families BΣ(λ, δ;α) and B∗ Σ(λ, δ;β) of holomorphic and bi-univalent functions which involve the Borel distribution series. We establish upper bounds for the initial Taylor-Maclaurin coefficients |a2| and |a3| for functions in each of these families. We also point out special cases and consequences of our results.


Introduction
We indicate by A the family of functions which are holomorphic in the open unit disk U = {z : z ∈ C and |z| < 1} and have the following normalized type: (1) f (z) = z + ∞ k=2 a k z k .
We also indicate by S the subclass of A consisting of functions which are also univalent in U. According to the Koebe one-quarter theorem [8], every function f ∈ S has an inverse f −1 defined by , and f f −1 (w) = w, quad |w| < r 0 (f ); r 0 (f ) 1 4 ), where (2) g(w) = f −1 (w) = w − a 2 w 2 + 2a 2 2 − a 3 w 3 − 5a 3 2 − 5a 2 a 3 + a 4 w 4 + · · · . A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U. Let Σ stand for the class of normalized bi-univalent functions in U given by (1). For a brief historical account and for several interesting examples of functions in the class Σ, see the pioneering work on this subject by Srivastava et al. [18], which actually revived the study of bi-univalent functions in recent years. From the work of Srivastava et al. [18], we choose to recall here the following examples of functions in the class Σ : We notice that the class Σ is not empty. However, the Koebe function is not a member of Σ.
In a considerably large number of sequels to the aforementioned work of Srivastava et al. [18], several different subclasses of the bi-univalent function class Σ were introduced and studied analogously by the many authors (see, for example, [1-7, 9-11, 13, 14, 16, 17, 19-28, 30, 31]), but only non-sharp estimates on the initial coefficients |a 2 | and |a 3 | in the Taylor-Maclaurin expansion (1) were obtained in many of these recent papers. The problem to find the general coefficient bounds on the Taylor-Maclaurin coefficients for functions f ∈ Σ is still not completely addressed for many of the subclasses of the bi-univalent function class Σ (see, for example, [14,19,21]).
Recently, Srivastava [12] in his survey-cum-expository review article, explored the mathematical application of q-calculus, fractional q-calculus and fractional q-differential operators in Geometric Function Theory.

Wanas and Khuttar
We note by the familiar Ratio Test that the radius of convergence of the above series is infinity. Now, we considered the linear operator B δ : A −→ A which is defined as follows: where ( * ) indicate the Hadamard product (or convolution) of two series. Very recently, Srivastava and El-Deeb [15] have introduced some applications of the Borel distribution.
We now recall the following lemma that will be used to prove our main results.
Lemma 1 (see [8]). If h ∈ P, then where P is the family of all functions h, holomorphic in U, for which

Coefficient estimates for the bi-univalent function class
In this section, we first define the bi-univalent function class B Σ (λ, δ; α).
In particular, if we choose λ = 1 in Definition 1, the family B Σ (λ, δ; α) reduces to the family S Σ (δ; α) of bi-starlike functions which satisfying the following conditions If we choose λ = 0 in Definition 1, the family B Σ (λ, δ; α) reduces to the family K Σ (δ; α) of bi-convex functions which satisfying the following conditions: Applications of Borel distribution series on holomorphic. . . and Our first main result is asserted by Theorem 1 below.
By comparing the corresponding coefficients of (5) and (6), we find that Thus, by using (9) and (11), we conclude that If we add (10) to (12), we obtain Substituting the value of p 2 1 + q 2 1 from (14) into the right-hand side of (15), and after some computations, we deduce that By taking the moduli of both sides of (16) and applying the Lemma 1 for the coefficients p 2 and q 2 , we have Next, in order to determinate the bound on |a 3 |, by subtracting (12) from (10), we get Now, upon substituting the value of a 2 2 from (14) into (17) and using (13), we deduce that (18) Finally, by taking the moduli on both sides of (18) and applying the Lemma 1 once again for the coefficients p 1 , p 2 , q 1 and q 2 , it follows that .
This completes the proof of Theorem 1.
In particular, if we choose λ = 1 in Definition 2, the family B * Σ (λ, δ; β) reduces to the family S * Σ (δ; β) of bi-starlike functions which satisfying the following conditions Also, if we choose λ = 0 in Definition 2, the family B * Σ (λ, δ; β) reduces to the family K * Σ (δ; β) of bi-convex functions which satisfying the following conditions Our second main result is asserted by Theorem 2 below.
We have thus completed the proof of Theorem 2.