Bi-Bazilevič functions of complex order involving Ruscheweyh type q-difference operator

Abstract In this paper, we define a new subclass of bi-univalent functions involving q-difference operator in the open unit disk. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|.


Introduction and Preliminaries
Let A denote the class of functions of the form f (z) = z + ∞ n=2 a n z n , (1) which are analytic in the open unit disk = {z : z ∈ C and |z| < 1}.
The convolution or Hadamard product of two functions f, h ∈ A is denoted by f * h and is defined as (f * h)(z) = z + ∞ n=2 a n b n z n , where f (z) is given by (1) and h(z) = z + ∞ n=2 b n z n . An analytic function f is subordinate to an analytic function h, written f (z) ≺ h(z) (z ∈ ), provided there is an analytic function w defined on with w(0) = 0 and |w(z)| < 1 satisfying f (z) = h(w(z)).
By S we shall denote the class of all functions in A which are univalent in . Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S * (α) of starlike functions of order α in and the class K(α) of convex functions of order α in .
Ma and Minda [13] unified various subclasses of starlike functions and convex functions which consist of functions f ∈ A satisfying the subordinations respectively, here (and throughout this paper) φ with positive real part in the unit disk , φ(0) = 1, φ (0) > 0 and φ maps onto a region starlike with respect to 1 and symmetric with respect to the real axis. Such a function has the form It is well known that every function f ∈ S has an inverse f −1 , defined by A function f ∈ A is said to be bi-univalent in if both f and f −1 are univalent in , in the sense that f −1 has a univalent analytic continuation to . Let Σ denote the class of bi-univalent functions in given by (1). A function f is bi-starlike of Ma-Minda type or bi-convex of Ma-Minda type if both f and f −1 are respectively Ma-Minda starlike or convex. These classes are denoted respectively by S * Σ (φ) and K Σ (φ). Now we recall here the notion of q-operator i.e. q-difference operator that play vital role in the theory of hypergeometric series, quantum physics and in the operator theory. The application of q-calculus was initiated by Jackson [8], recently Kanas and Răducanu [11] have used the fractional q-calculus operators in investigations of certain classes of functions which are analytic in U.
Let 0 < q < 1. For any non-negative integer n, the q-integer number n is defined by In general, we will denote for a non-integer number x. Also the q-number shifted factorial is defined by Clearly, lim q→1 − [n] q = n and lim q→1 − [n] q ! = n!.
Bi-Bazilevic functions of complex order involving Ruscheweyh type operator

[7]
For 0 < q < 1, the Jackson's q-derivative operator (or q-difference operator) of a function f ∈ A given by (1) defined as follows [8] From (5), we have where [n] q is given by (4). For a function h(z) = z n we obtain where h is the ordinary derivative. Let t ∈ R and n ∈ N. The q-generalized Pochhammer symbol is defined by and for t > 0 the q-gamma function is defined by Using the q-difference operator, Kannas and Raducanu [11] defined the Ruscheweyh q-differential operator as below. For f ∈ A, where We note that Making use of (6) and (7), we have [8]

Gangadharan Murugusundaramoorthy and Serap Bulut
From (8), we note that Also we have where For our study, we will use the short presentation Recently there has been triggering interest to study bi-univalent function class Σ and obtained non-sharp coefficient estimates on the first two coefficients |a 2 | and |a 3 | of (1). But the coefficient problem for each of the following Taylor-Maclaurin coefficients is still an open problem (see [2,3,4,12,14,18]). Many researchers (see [1,7,9,17]) have recently introduced and investigated several interesting subclasses of the biunivalent function class Σ. Motivated by the earlier work of Bulut [5], Deniz [6], Inayat Noor [10] and Srivastava et al. [16], in the present paper we introduce new families of Bazilevic functions of complex order of the function class Σ, involving the operator D q (R δ q f (z)), and find estimates on the coefficients |a 2 | and |a 3 | for functions in the new subclass of function class Σ. Several related classes are also considered, and connection to earlier known results are made. (1) is said to be in the class S q Σ (γ, λ, δ; φ) if the following conditions are satisfied: where z, w ∈ , γ ∈ C \ {0}, δ > −1, λ ≥ 0 and the function g = f −1 is given by (3).

Gangadharan Murugusundaramoorthy and Serap Bulut
On specializing the parameters λ and δ, one can state the various new subclasses of Σ.

Coefficient Bounds for the class S q Σ (γ, λ, δ; φ)
We begin by finding the estimates on the coefficients |a 2 | and |a 3 | for functions in the class S q Σ (γ, λ, δ; φ). In order to derive our main results, we shall need the following lemma.

[12]
Gangadharan Murugusundaramoorthy and Serap Bulut Adding (20) and (22), we obtain Using (24) in (25), we get Applying Lemma 2.1 for the coefficients p 2 and q 2 , we immediately have This gives the bound on |a 2 | as asserted in (11). Next, in order to find the bound on |a 3 |, by subtracting (22) from (20), we get Using (23) and (24) in (26), we get Applying Lemma 2.1 once again for the coefficients p 1 , q 1 , p 2 and q 2 , we readily get (12). This completes the proof of Theorem 2.1.

Corollaries and Consequences
By setting λ = 0 in Theorem 2.1, we have the following Theorem.
Let the function f (z) given by (1) be in the class S q Σ (γ, δ; φ). Then By setting λ = 1 in Theorem 2.1, we have the following result.