A certain subclass of bi-univalent functions associated with Bell numbers and q − Srivastava Attiya operator

: In the present study, we introduced general a subclass of bi-univalent functions by using the Bell numbers and q − Srivastava Attiya operator. Also, we investigate coe ﬃ cient estimates and famous Fekete-Szeg¨o inequality for functions belonging to this interesting class.


Introduction and preliminaries
Let A be the class of all analytic functions of the form f (z) = z + ∞ k=2 a k z k (1.1) in the open unit disk D = {z ∈ C : |z| < 1} normalized by the conditions f (0) = 0 and f (0) = 1. The well-known Koebe one-quarter theorem [8] ensures that the image of D under every univalent function f ∈ A contains a disk of radius 1/4. Thus, every univalent function f has an inverse f −1 satisfying f −1 ( f (z)) = z, (z ∈ D) and (1. 2) A function f ∈ A is said to be bi-univalent in D if both f and g to D are univalent in D, where g is the analytic continuation of f −1 to the unit disk D. Let Σ denote the class of bi-univalent functions defined in the unit disk D given by 1.1. For a brief history of functions in the class Σ, see [3,4,16,19]. Later, Srivastava et al.'s [24,[26][27][28] gave very important contributions to this theory. Recently, for coefficient estimates of the functions in some particular subclasses of bi-univalent functions, one may see [6,7,10,15,20,25,29,30].
For analytic functions f and g in D, f is said to be subordinate to g if there exists an analytic function w such that w (0) = 0, |w (z)| < 1 and f (z) = g (w (z)). This subordination is denote by f (z) ≺ g (z). In particular, when g is univalent in D, f (z) ≺ g (z) ⇐⇒ f (0) = g (0) and f (D) ⊂ g (D) (z ∈ D) .
The q−difference operator, which was introduced by Jackson [13], is defined by , where f is the ordinary derivative of the function. For more properties of ∂ q see [9,11,12].
Thus, for function f ∈ A we have where [k] q is given by and the q− factorial is defined by As q → 1 − , then we get [k] q → k. Thus, if we choose the function g (z) = z k , while q → 1, then we have where g is the ordinary derivative. In order to derive our main results, we have to recall here the following lemmas.
If p ∈ P then |p k | ≤ 2 for each k, where P is the family of all functions p analytic in D for which Re p(z) > 0, 17]) If the function p ∈ P is given by the series 1.8 , then for some x, z with |x| ≤ 1 and |z| ≤ 1.
For a fixed non-negative integer n, the Bell numbers B n count the possible disjoint partitions of a set with n elements into non-empty subsets or, equivalently, the number of equivalence relations on it. The numbers B n are named the Bell numbers after Eric Temple Bell (1883 − 1960) (see [1,2]) who called then the "exponential numbers". The Bell numbers B n (n 0) are generated by the function e e z −1 as follows: e e z −1 = ∞ n=0 B n (z n /n!) (z ∈ R) . The Bell numbers B n satisfy the following recurrence relation involving binomial coefficients: B n+1 = n k=0 n k B k . Clearly, we have B 0 = B 1 = 1, B 2 = 2, B 3 = 5, B 4 = 15, B 5 = 52 and B 6 = 203. We now consider the function ϕ (z) := e e z −1 with its domain of definition as the open unit disk D. Recently Srivastava and co-auhors studied geometric properties and coefficients bounds for starlike functions related to the Bell numbers (see [5,14]).
On the other hand, Shah and Noor [21] introduced the q−analogue of the Hurwitz Lerch zeta function by the following series: (1.9) where b ∈ C \ Z − 0 , s ∈ C when |z| < 1, and Re (s) > 1 when |z| = 1. The a normalized form of 1.9 as follows: where * denotes convolution (or the Hadamard product). We note that: (i) If q → 1 − , then the function φ q (s, b; z) reduces to the Hurwitz-Lerch zeta function and the operator J s q,b coincides with the Srivastava-Attiya operator (see [22,23]). [18]). In present paper, we defined a general subclass ΣH s q,b (τ, λ, µ) of bi-univalent functions related to the Bell numbers by using q−Srivastava Attiya operator. Using the principles of subordination, the estimates for the coefficients |a 2 |, |a 3 | and a 3 − δa 2 2 of the functions of the form 1.1 in the class ΣH s q,b (τ, λ, µ) have been obtained. For some particular choices of τ, λ, µ and s the bounds determined.

Coefficient estimates
Let Ω be the class of analytic functions of the form in the unit disk D satisfying the condition |w(z)| < 1. There is an important relation between the classes Ω and P as follows: Define the functions p and s in P given by It follows that Definition 2.1. A function f ∈ Σ is said to be in the class ΣH s q,b (τ, λ, µ) if the following conditions hold true for all z, w ∈ D: Remark 2.1. We note that, for suitable choices parameters, the class ΣH s q,b (τ, λ, µ) reduces to the following classes.
. Then a function f ∈ Σ is said to be in the class ΣH s q,b (τ, µ) if the following subordinations hold for all z, w ∈ D: . Then a function f ∈ Σ is said to be in the class ΣH s q,b (µ) if the following subordinations hold for all z, w ∈ D: and . Then a function f ∈ Σ is said to be in the class ΣH s q,b (τ, λ) if the following subordinations hold for all z, w ∈ D: . Then a function f ∈ Σ is said to be in the class ΣH s q,b (λ) if the following subordinations hold for all z, w ∈ D: and . Then a function f ∈ Σ is said to be in the class ΣH s q,b if the following subordinations hold for all z, w ∈ D: . Then a function f ∈ Σ is said to be in the class ΣH (τ, λ, µ) if the following subordinations hold for all z, w ∈ D: The following theorem derives the estimates for the coefficients |a 2 | and |a 3 | for the functions given by 1.1 that belong to the class ΣH s q,b (τ, λ, µ). Theorem 2.1. Let f given by (1.1) be in the class ΣH s q,b (τ, λ, µ). Then and Proof. Let f ∈ΣH s q,b (τ, λ, µ) and g = f −1 . Then, there are analytic functions u, v ∈ Ω satisfying In other words, by using 2.1 in 2.6 and 2.7 we write (2.9) From 2.8 and 2.9, we have 1 + (µ + λq) τ Comparing the coefficients on the both sides of above last equalities, we have the relations (2.13) Therefore, from the Eqs 2.10 and 2.12 , we find that and which upon applying Lemma 1.1, yields On the other hand, by using 2.11 and 2.13, we obtain which yields We now, investigate the upper bound of |a 3 | . For this, by using 2.11 and 2.13, we have Therefore for substituting 2.15 in 2.17, we have On the other hand, according to the Lemma 1.2 and 2.14, we write and so, from 2.19 and 2.20, we have If we apply triangle inequality to equation 2.21, we obtain Since the function p(e iθ z) (θ ∈ R) is in the class P for any p ∈ P, there is no loss of generality in assuming p 1 > 0. Write p 1 = p, p ∈ [0, 2]. Thus, for |x| ≤ 1 and |y| ≤ 1 we obtain which upon applying Lemma 1.1, yields upper bound of |a 3 | .
2. If f (z) given by (1.1) be in the class ΣH s q,b (τ, λ, µ) and δ ∈ C, then Proof. From the Eqs 2.16 and 2.18 we obtain and (2.24) Therefore, by using the equalities 2.23 and 2.24 for δ ∈ C, we have After the necessary arrangements, we rewrite the above last equality as where K and L are given by 2.22. Taking the absolute value of 2.25, from Lemma 1.1 we obtain the desired inequality.

Conclusions
In this paper, we defined a general subclass of bi-univalent functions related with q−Srivastava Attiya operator by using the Bell numbers and subordination. For the functions belonging to this class, we obtained non-sharp bounds for the initial coefficients and the Fekete-Szegö functional. Some interesting corollaries and applications of the results are also discussed.