Estimates for Coefﬁcients of Bi-Univalent Functions Associated with a Fractional q -Difference Operator

: In the present paper, we discuss a class of bi-univalent analytic functions by applying a principle of differential subordinations and convolutions. We also formulate a class of bi-univalent functions inﬂuenced by a deﬁnition of a fractional q -derivative operator in an open symmetric unit disc. Further, we provide an estimate for the function coefﬁcients | a 2 | and | a 3 | of the new classes. Over and above, we study an interesting Fekete–Szego inequality for each function in the newly deﬁned classes. S.A.-O.; funding K.N.


Preliminaries
In mathematics, functions and symmetric functions are very common in theory and applications. They are applied to various fields including group theory, Lie algebras and the algebraic geometry and may others, to mention but a few. In the concept of geometric functions, the set A has has been introduced as the set of all analytic class of normalized functions defined on the open symmetric disc D = {z ∈ C, |z| < 1} such that they possess the following formula (see [1]) f (z) = z + ∞ ∑ n=2 a n z n . (1) The univalent class of functions in A is denoted by S. The best known subclasses of S containing starlike, close to convex and convex functions, respectively, are denoted by ST, C and CV. If f is a function satisfying (1) and g is another function defined by g(z) = z + ∞ ∑ n=2 b n z n , then the Hadamard product or the convolution of the functions f and g is offered by Ruscheweyh [2] as ( f * g)(z) = z + ∞ ∑ n=2 a n b n z n .
Then, for two arbitrary analytic functions f and g in A, we say f is the subordinate function to the function g, expressed as f ≺ g or f (z) ≺ g(z), if there can be found a function w ∈ B such that f (z) = g(w(z)) (see [1]). A unified treatment of the familiar subclass of univalent functions has been considered by Ma and Minda [3], following a principle of differential sub-ordinations. Here, we present P as the class of functions p which are analytic in D provided p(0) = 1 and Re p(z) > 0 and, for each z ∈ D, the functions p(z) can be written in the form Such a class of functions is the so-called Caratheodory class of functions (see [1]).
Due to the one-quarter theorem of Koebe (see [4]), it is shown that every function f in S has an inverse function f −1 such that We say a function f in the set A is bi-univalent in D if D ⊆ f (D) and both of the functions f and its inverse f −1 are univalent functions in D. The class of such bi-univalent functions defined in D is denoted by Σ. For a useful summary and applicable examples in the class of bi-univalent functions we refer to [5]. The class Σ of bi-univalent functions was discussed by Lewin [6] to derive the second coefficient bound. Srivastava et al. [7] have introduced and investigated a new class of analytic and bi-univalent functions. In recent years, some researchers have studied a number of different subclasses of Σ in the context of theory of geometric functions [8][9][10][11].
In recent decades, the fractional q-calculus was applied in the approximation theory which has a new generalization of the classical operator. The concept of q-calculus operator has been broadly been applied in various fields including optimal control, q-difference, hypergeometric series, quantum physics, fractional subdiffusion equations and q-integral equations [12][13][14][15][16]. The concept of p-calculus was first applied in this context by Lupas [17]. Jackson in [18] defined the q-analogues of the ordinary derivative. He also studied the q-integral operator and investigated some applications of this theory.
Nowadays, a number of researchers have studied q-analogues of an analytic class of bi-univalent functions. We may refer readers to Darus [19] who investigated a generalized q-differential operator by utilizing the q-hypergeometric functions. Furthermore, he worked on an operator to derive some useful applications. Zhang et al. [20] discussed a q-Starlike functions associated with generalized conic domain Ω k,α . Kanas et al. [21] defined the operator and later, Arif et al. [22] extended this operator for multivalent functions. Srivastava et al. [23] defined the q-starlike class of functions over conic regions. Furthermore, they presented some of their applications. Srivastava et al. [24] studied q-Noor integral operators and some of their applications. The q-calculus theory in a fractional sense and its real applications in the geometric class of functions of complex analysis and related fields are investigated in [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].
Let q be a fixed number with 0 < q < 1 and n ∈ N. Then, we recall the following useful definitions and notations (see [18]) The q-analogue of the Γ-function is defined by The q-analogue of the difference operator of non-integer order α is defined by where α = 0 and q ∈ (0, 1).
For n ∈ N, the following equation is satisfied Therefore for f ∈ A, we obtain that Stimulated by the previous result in the paper, we estimate the initial coefficients for a new subclass of bi-univalent functions defined by q-analogue difference operator with non-integer order α . Additionally, we establish corresponding Fekete-Szego type fractional inequality for this functions. Furthermore, we estimate bounds for |a 2 | and |a 3 |.

Coefficient Estimates for the Function Class Σ α q
In this section, the coefficients |a 2 | and |a 3 | for a function are estimated in the class Σ α q in some details.

Definition 1.
Let a function f be estimated by (1) and g(w) = f −1 (w). Then f ∈ Σ α q , 0 < α < 1, provided the following subsequent subordination conditions hold: and where the functionsψ(z) : D → C are analytic of positive real parts. Consequently, we have the appropriate form:ψ Let a function f be estimated by (1) and and

Proof.
Since g = f −1 , the right-hand side of (4) and (5) have the expansions and By using the definition of subordination, we derive where φ(z),φ(z) ∈ B. We define two functions u 1 , u 2 as the subsequent form It can be easily shown that u 1 , u 2 ∈ P and write the relations of φ(z) andφ(z) of the restated subsequent form Therefore, from Equations (11), (13) and (14), we obtain and Equating the corresponding coefficients (9) and (15) reveals Once again, equating the corresponding coefficients (10) and (16) implies Comparing the coefficients (17) and (19) yields and Moreover, by using (20)-(22), the above reveals , which, by applying Lemma 1, we derive the desired estimate on |a 2 | as presented in (7).
To complete the proof of this theorem, we find the estimate on |a 3 |. Subtracting Equation (20) from Equation (18) and using (21), we get Hence, following Equations (22) and (23) gives Finally, by applying Lemma 1 on the coefficients of φ 2 andφ 2 in the last equation, we derive the desired estimate on |a 3 | as presented in (8).
In the next theorem, we calculate the Fekete-Szego result for the class Σ α q .
Proof. By using Equations (22) and (23), we get The preceding equation is indeed equivalent to . Therefore, we have where ψ(λ) is defined in (48). Sinceψ 1 > 0 and the coefficientψ 2 is a real number, then we have This completes the proof of Theorem 4.

Remark 1. Let
This remark is a straightforward consequence of Theorem 4.

Applications of Coefficient Estimates for
Let a function f be estimated by (1), g = f −1 and 0 ≤ ζ < 1. Then f ∈ Σ α q (ζ), 0 < α < 1 if the following subsequent subordination conditions hold: Hence, it is evident that

Corollary 2. Let f be a function estimated by
where

Coefficient Estimates for the Function Class Σ α,β q
In this section, we define the class Σ α,β q and obtain coefficient estimates for |a 2 | and |a 3 | for the function in this class.
Proof. First, for the argument inequalities in (32) and (33), there exist φ(z),φ(z) ∈ B such that and We expand the right-hand side of (36) and (37) as and Now, by equating the coefficients (15) and (38), we establish In the same manner, by equating the coefficients (16) and (39), we obtain By omparing the coefficients (40) and (42), we get By using (42) and (45), it follows that So, following (45) and (46), we find Hence, the desired bounds on |a 2 | are obtained. The proof will be completed by finding the bound for the coefficient. Therefore, subtracting Equation (43) from Equation (41), we obtain Put the value of a 2 2 , given in (45) into (47) to get Therefore, we conclude that Similarly, by putting the value of a 2 2 of (46) into (47), we conclude the following bound: This finishes the proof of Theorem 3.
Similarly, from Theorem 4, we get the Fekete-Szego result for the class Σ α,β q .