Certain new applications of Faber polynomial expansion for some new subclasses of υ -fold symmetric bi-univalent functions associated with q -calculus

: In this article, we deﬁne the q -difference operator and Salagean q -differential operator for υ -fold symmetric functions in open unit disk U by ﬁrst applying the concepts of q -calculus operator theory. Then, we considered these operators in order to construct new subclasses for υ -fold symmetric bi-univalent functions. We establish the general coefﬁcient bounds | a υ k + 1 | for the functions in each of these newly speciﬁed subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefﬁcient bounds for the function h that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main ﬁndings are also highlighted

The power series for the inverse function g(w) is given by g(w) = w − a 2 w 2 + (2a 2 2 − a 3 )w 3 − Q(a)w 4 +, · · · , (1.2) Where Q(a) = (5a 3 2 − 5a 2 a 3 + a 4 ). An analytic function h is called bi-univalent in U if h and h −1 are univalent in U and class of all biunivalent functions are denoted by Σ. In 1967, for h ∈ Σ, Levin [32] showed that |a 2 | < 1.51 and after twelve years Branan and Clunie [8] gave the improvement of |a 2 | and proved that |a 2 | ≤ √ 2. Furthermore, for h ∈ Σ, Netanyahu [34] proved that max |a 2 | = 4 3 and an intriguing subclass of analytic and bi-univalent functions was proposed and studied by Branan and Taha [9], who also discovered estimates for the coefficients of the functions in this subclass. Recently, the investigation of numerous subclasses of the analytic and bi-univalent function class Σ was basically revitalized by the pioneering work of Srivastava et al. [41]. In 2012, Xu et al. [44] defined a general subclass of class Σ and investigated coefficient estimates for the functions belonging to the new subclass of class Σ. Recently, several different subclasses of class Σ were introduced and investigated by a number of authors (see for details ([23, 29, 38]). In these recent papers only non-sharp estimates on the initial coefficients were obtained.
Faber polynomials was introduced by Faber [15] and first time he used it to determine the general coefficient bounds |a k | for k ≥ 4. Gong [16] interpreted significance of Faber polynomials in mathematical sciences, particularly in Geometric Function Theory. In 1913, Hamidi et al. [18] first time used the Faber polynomials expansion technique on meromorphic bi-starlike functions and determined the coefficient estimates. The Faber polynomials expansion method for analytic bi-closeto-convex functions was examined by Hamidi and Jahangiri [21,22], who also discovered some new coefficient bounds for new subclasses of close-to-convex functions. Furthermore, many authors [3,4,7,11,12,14,20] used the same technique and determined some interesting and useful properties for analytic bi-univalent functions. For h ∈ Σ, by using the Faber polynomial expansions methods, only a few works have been done so far and we recognized very little over the bounds of Maclaurin's series coefficient |a k | for k ≥ 4 in the literature. Recently only a few authors, used the Faber polynomials expansion technique and determined the general coefficient bounds |a k | for k ≥ 4, (see for detail [6,11,24,39,40,42]).
A domain U is said to be the υ-fold symmetric if and every h υ has the series of the form The class S υ represents the set of all υ-fold symmetric univalent functions. For υ = 1, then S υ reduce to the class S of univalent functions. If the inverse g υ of univalent h is univalent then h is called υ-fold symmetric bi-univalent functions in U and denoted by Σ υ . The series expansion of inverse function g υ investigated by Srivastava et al. in [43]: For υ = 1, the series in (1.4) reduces to the (1.2) of the class Σ. In [43] Srivastava et al. defined a subclass of υ-fold symmetric bi-univalent functions and investigated coeffiients problem for υfold symmetric bi-univalent functions. Hamidi and Jahangiri [19] defined υ-fold symmetric bistarlike functions and discussed the unpredictability of the coefficients of υ-fold symmetric bi-starlike functions. Many researchers have used the q-calculus and fractional q-calculus in the field of Geometric Function Theory (GFT) and they defined and studied several new subclasses of analytic, univalent and bi-univalent functions. In 1909, Jackson ( [26,27]), gave the idea of q-calculus operator and defined the q-difference operator (D q ) while in [25], Ismail et al. was the first who used D q in order to define a class of q-starlike functions in open unit disk U. The most signifcant usages of q-calculus in the perspective of GFT was basically furnished and the basic (or q−) hypergeometric functions were first used in GFT in a book chapter by Srivastava (see, for details, [37]). For more study about q-calculus operator theory in GFT, see the following articles [5,28,33]. Now we recall, some basic definitions and concepts of the q-calculus which will be used to define some new subclasses of the this paper.
For a non-negative integer t, the q-number [t, q], (0 < q < 1), is defined by and the q-number shift factorial is given by For q → 1−, then [t, q]! reduces to t!.
The q-generalized Pochhammer symbol is defined by Remark 1.1. For q → 1−, then [t, q] k reduces to (t) k = Γ(t+k) Γ(t) . Definition 1.2. Jackson [27] defined the q-integral of function h(z) as follows: Jackson [26] introduced the q-difference operator for analytic functions as follows: Definition 1.3. [26]. For h ∈ A, the q-difference operator is defined as: Note that, for k ∈ N and z ∈ U and Here, we introduce the q-difference operator for υ-fold symmetric functions related to the q-calculus as follows: . Then q-difference operator will be defined as Now we define Salagean q-differential operator for υ-fold symmetric functions as follows: Definition 1.5. For m ∈ N, the Salagean q-differential operator for h υ ∈ Σ υ is defined by Remark 1.6. For υ = 1, we have Salagean q-differential operator for analytic functions proved in [17].
Motivated by the following articles [1,10,25] and using the q-analysis in order to define new subclasses of class Σ υ , we apply Faber polynomial expansions technique in order to determine the estimates for the general coefficient bounds |a υk+1 |. We also derive initial coefficients |a υ+1 | and |a 2υ+1 | and obtain Feketo-Sezego coefficient bounds for the functions belonging to the new subclasses of Σ υ .
Or equivalently by using subordination, we can write the above conditions as:

The faber polynomial expansion method and application
Using the Faber polynomial technique for the analytic function h, then the coefficient of its inverse map g can be written as follows (see [2,4]): and Q i is a homogeneous polynomial in the variables a 2 , a 3 , ...a k , for 7 ≤ i ≤ k. Particularly, the first three term of −k k−1 are 1 2 In general, for r ∈ N and k ≥ 2, an expansion of r k of the form: where, and by [2], we have The sum is taken over all non negative integer µ 1 , ..., µ k which is satisfying Clearly, The coefficient of inverse map g υ can be expressed of the form: , a 2υ+1 , ...a υk+1 )w υk+1 .
Now the bounds given for |a υ+1 | can be justified since .
Taking the square-root of (2.35), we have .
Proof. We can prove Theorem 2.9 by using the similar method of Theorem 2.1. Proof. We can prove Theorem 2.10 by using the similar method of Theorem 2.3.

Conclusions
In this article, first of all, we used the q-difference operator for υ-fold symmetric functions in order to define some new subclasses of the υ-fold symmetric bi-univalent functions in the open symmetric unit disk U. We also used the basic concepts of q-calculus and defined the Salagean q-differential operator for υ-fold symmetric functions. We considered this operator and investigated a new subclass of υ-fold symmetric bi-univalent functions. Faber Polynomial expansion method and q-analysis are used in order to determined general coefficient bounds |a υ+1 | for functions in each of these newly defined υ-fold symmetric bi-univalent functions classes. Feketo-Sezego problems and initial coefficient bounds |a υ+1 | and |a 2υ+1 | for the function belonging to the subclasses of υ-fold symmetric bi-univalent functions in open symmetric unit disk U are also investigated.