Certain class of bi-univalent functions defined by quantum calculus operator associated with Faber polynomial

1 Department of Mathematics, College of Science and Arts in Al-Badaya, Qassim University, Buraidah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt 3 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, TN, India 4 Department of Mathematics, College of Science and Arts in Al-Badaya, Qassim University, Buraidah, Saudi Arabia


Introduction and preliminary definitions
In his survey-cum-expository review article, Srivastava [1] presented and motivated about brief expository overview of the classical q-analysis versus the so-called (p, q)-analysis with an obviously redundant additional parameter p. We also briefly consider several other families of such extensively and widely-investigated linear convolution operators as (for example) the Dziok-Srivastava, Srivastava-Wright and Srivastava-Attiya linear convolution operators, together with their extended and generalized versions. The theory of (p, q)-analysis has important role in many areas of mathematics and physics. Our usages here of the q-calculus and the fractional q-calculus in geometric function theory of complex analysis are believed to encourage and motivate significant further developments on these and other related topics (see Srivastava and Karlsson [2, pp. 350-351], Srivastava [3,4]). Our main objective in this survey-cum-expository article is based chiefly upon the fact that the recent and future usages of the classical q-calculus and the fractional q-calculus in geometric function theory of complex analysis have the potential to encourage and motivate significant further researches on many of these and other related subjects. Jackson [5,6] was the first that gave some application of q-calculus and introduced the q-analogue of derivative and integral operator (see also [7,8]), we apply the concept of q-convolution in order to introduce and study the general Taylor The Hadamard product (or convolution) of f and h, given by (1.1) and (1.2), respectively, is defined by (1. 3) If f and F are analytic functions in ∆, we say that f is subordinate to F, written as f (z) ≺ F(z), if there exists a Schwarz function s, which is analytic in ∆, with s(0) = 0, and |s(z)| < 1 for all z ∈ ∆, such that f (z) = F(s(z)), z ∈ ∆. Furthermore, if the function F is univalent in ∆, then we have the following equivalence ( [9,10] The Koebe one-quarter theorem (see [11]) prove that the image of ∆ under every univalent function f ∈ S contains the disk of radius 1 4 . Therefore, every function f ∈ S has an inverse f −1 that satisfies A function f ∈ A is said to be bi-univalent in ∆ if both f and f −1 are univalent in ∆. Let Σ represent the class of bi-univalent functions in ∆ given by (1.1). The class of analytic bi-univalent functions was first familiarised by Lewin [12], where it was shown that |a 2 | < 1.51. Brannan and Clunie [13] enhanced Lewin's result to |a 2 | < √ 2 and later Netanyahu [14] proved that |a 2 | < 4 3 . Note that the functions with their corresponding inverses are elements of Σ (see [15,16]). For a brief history and exciting examples in the class Σ (see [17]). Brannan and Taha [18] (see also [16]) presented certain subclasses of the bi-univalent functions class Σ similar to the familiar subclasses S * (α) and K (α) of starlike and convex functions of order α (0 ≤ α < 1), respectively (see [17,19,20]). Ensuing Brannan and Taha [18], a function f ∈ A is said to be in the class S * Σ (α) of bi-starlike functions of order α (0 < α ≤ 1), if each of the following conditions are satisfied: where the function g is the analytic extension of f −1 to ∆, given by A function f ∈ A is said to be in the class K Σ (α) of bi-convex functions of order α (0 < α ≤ 1), if each of the following conditions are satisfied: The classes S * Σ (α) and K Σ (α) of bi-starlike functions of order α and bi-convex functions of order α (0 < α ≤ 1), corresponding to the function classes S * (α) and K (α), were also introduced analogously. For each of the function classes S * Σ (α) and K Σ (α), they found non-sharp estimates on the first two Taylor-Maclaurin coefficients |a 2 | and |a 3 | ( [16,18] ).

Faber polynomial expansion of functions f ∈ A
The Faber polynomials introduced by Faber [21] play an important role in various areas of mathematical sciences, especially in Geometric Function Theory of Complex Analysis (see, for details, [22]). In 2013, Hamidi and Jahangiri [23][24][25] took a new approach to show that the initial coefficients of classes of bi-starlike functions e as well as provide an estimate for the general coefficients of such functions subject to a given gap series condition.Recently,their idea of application of Faber polynomials triggered a number of related publications by several authors (see, for example, [26][27][28] and also references cited threin) investigated some interesting and useful properties for analytic functions. Using the Faber polynomial expansion of functions f ∈ A has the form (1.1), the coefficients of its inverse map may be expressed as such that U i with 7 ≤ i ≤ m is a homogeneous polynomial in the variables a 2 , a 3 , ..., a m , In particular, the first three terms of K −m m−1 are In general, an expansion of K −n m (n ∈ N) is (see [29][30][31][32][33]) where D n m = D n m (a 2 , a 3 , ...) and while a 1 = 1 and the sum is taken over all non-negative integers i 1 ...i m satisfying Evidently D m m (a 1 , a 2 , ...a m ) = a m 1 .

Quantum calculus operator
Srivastava [1] made use of several operators of q-calculus and fractional q-calculus and recollecting the definition and representations. The q-shifted factorial is defined for κ, q ∈ C and n ∈ N 0 = N ∪ {0} as follows By using the q-Gamma function Γ q (z), we get where (see [34]) Also, we note that and, the q-Gamma function Γ q (z) is known where [m] q symbolizes the basic q-number defined as follows Using the definition formula (1.7) we have the next two products: (i) For any non-negative integer m, the q-shifted factorial is given by (ii) For any positive number r, the q-generalized Pochhammer symbolis defined by It is known in terms of the classical (Euler's) Gamma function Γ (z), that where (κ) m is the familiar Pochhammer symbol defined by For 0 < q < 1, the q-derivative operator (or, equivalently, the q-difference operator) El-Deeb et al. [35] defined D q for f * h given by (1.3) is defined by (see [5,6]) where, as in the definition (1.7) [m] q := For κ > −1 and 0 < q < 1, El-Deeb et al. [35] (see also) defined the linear operator H κ,q h : where the function M q,κ+1 is given by A simple computation shows that From the definition relation (1.9), we can easily verify that the next relations hold for all f ∈ A: (1.10) Remark 1. Taking precise cases for the coefficients b m we attain the next special cases for the operator H κ,q h : (i) For b m = 1, we obtain the operator I κ q defined by Srivastava [32] and Arif et al. [36] as follows , υ > 0, we obtain the operator N κ υ,q defined by El-Deeb and Bulboacȃ [37] and El-Deeb [38] as follows where , α > 0, n ≥ 0, we obtain the operator M κ,α n,q defined by El-Deeb and Bulboacȃ [39] and Srivastava and El-Deeb [40] as follows (1.14) (iv) For b m = ρ m−1 (m − 1)! e −ρ , ρ > 0, we obtain the q-analogue of Poisson operator defined by El-Deeb et al. [35] (see [41]) as follows , n ∈ Z, ℓ ≥ 0, µ ≥ 0, we obtain the q-analogue of Prajapat operator defined by El-Deeb et al. [35] (see also [42]) as follows , we obtain the q-analogue of the Pascal distribution operator defined by Srivastava and El-Deeb [28] (see also [35,43,44]) as follows The purpose of the paper is to present a new subclass of functions L q,κ Σ (η; h; Φ) of the class Σ, that generalize the previous defined classes. This subclass is defined with the aid of a general H κ,q h linear operator defined by convolution products composed with the aid of q-derivative operator. This new class extend and generalize many preceding operators as it was presented in Remark 1, and the main goal of the paper is find estimates on the coefficients |a 2 |, |a 3 |, and for the Fekete-Szegö functional for functions in these new subclasses. These classes will be introduced by using the subordination and the results are obtained by employing the techniques used earlier by Srivastava et al. [16]. This last work represents one of the most important study of the bi-univalent functions, and inspired many investigations in this area including the present paper, while many other recent papers deals with problems initiated in this work, like [33,[44][45][46][47][48], and many others. Inspired by the work of Silverman and Silvia [49] (also see [50]) and recent study by Srivastava et al [51], in this article, we define the following new subclass of bi-univalent functions M q,κ Σ (ϖ, ϑ, h) as follows: and let the function f ∈ Σ be of the form (1.1) and h is given by (1.2), the function f is said to be in the class M q,κ Σ (ϖ, ϑ, h) if the following conditions are satisfied: and with κ > −1, 0 < q < 1, 0 ≤ ϑ < 1 and z, w ∈ ∆, where the function g is the analytic extension of f −1 to ∆, and is given by (1.4).
Definition 2. Let ϖ = 0 and let the function f ∈ Σ be of the form (1.1) and h is given by (1.2), the function f is said to be in the class M q,κ Σ (ϑ, h) if the following conditions are satisfied: and with κ > −1, 0 < q < 1, 0 ≤ ϑ < 1 and z, w ∈ ∆, where the function g is the analytic extension of f −1 to ∆, and is given by (1.4).
Definition 3. Let ϖ = π and let the function f ∈ Σ be of the form (1.1) and h is given by (1.2), the function f is said to be in the class HM q,κ Σ (ϑ, h) if the following conditions are satisfied: with κ > −1, 0 < q < 1, 0 ≤ ϑ < 1 and z, w ∈ ∆, where the function g is the analytic extension of f −1 to ∆, and is given by (1.4).
Recall the following Lemma which will be needed to prove our results.
Lemma 1. (Caratheodory Lemma [11]) If ϕ ∈ P and ϕ(z) = 1 + ∞ n=1 c n z n then |c n | ≤ 2 for each n, this inequality is sharp for all n where P is the family of all functions ϕ analytic and having positive real part in ∆ with ϕ(0) = 1.
We firstly introduce a bound for the general coefficients of functions belong to the class M q,κ Σ (ϖ, ϑ; h) . Theorem 2. Let the function f given by (1.1) belongs to the class M q,κ Σ (ϖ, ϑ; h). If a k = 0 for 2 ≤ k ≤ m − 1, then Proof.
and H κ,q h g(w) and g = f −1 ∈ M q,κ Σ (γ, η, ϑ; h) , we know that there are two positive real part functions: and H κ,q h g(w) Using (2.1) and comparing the corresponding coefficients in (2.3), we obtain and similarly, by using (2.2) in the equality (2.4), we have under the assumption a k = 0 for 0 ≤ k ≤ m − 1, we obtain A m = −a m and so and Taking the absolute values of (2.7) and (2.8), we conclude that Applying the Caratheodory Lemma 1, we obtain which completes the proof of Theorem. □ Theorem 3. Let the function f given by (1.1) belongs to the class M q,κ Σ (ϖ, ϑ; h), then

Conclusions and observations
In the current paper, we mainly get upper bounds of the initial Taylors coefficients of bi-univalent functions related with q− calculus operator. By fixing b m as demonstrated in Remark 1, one can effortlessly deduce results correspondents to Theorems 2 and 3 associated with various operators listed in Remark 1. Further allowing q → 1 − as itemized in Remark 2 we can outspread the results for new subclasses stated in Remark 2. Moreover by fixing ϖ = 0 and ϖ = π in Theorems 2 and 3, we can easily state the results for f ∈ M q,κ Σ (ϑ; h) and f ∈ HM q,κ Σ (ϑ; h). Further by suitably fixing the parameters in Theorem 4, we can deduce Fekete-Szegö functional for these function classes.
By using the subordination technique, we can extend the study by defining a new class where Ψ(z) the function Ψ is an analytic univalent function such that ℜ (Ψ) > 0 in ∆ with Ψ(0) = 1, Ψ ′ (0) > 0 and Ψ maps ∆ onto a region starlike with respect to 1 and symmetric with respect to the real axis and is given by Ψ(z) = z + B 1 z + B 2 z 2 + B 3 z 3 + · · · , (B 1 > 0). Also, motivating further researches on the subject-matter of this,we have chosen to draw the attention of the interested readers toward a considerably large number of related recent publications (see, for example, [1,2,4]). and developments in the area of mathematical analysis. In conclusion,we choose to reiterate an important observation, which was presented in the recently-published review-cum-expository review article by Srivastava ( [1], p. 340), who pointed out the fact that the results for the above-mentioned or new q− analogues can easily (and possibly trivially) be translated into the corresponding results for the so-called (p; q)−analogues(with 0 < |q| < p ≤ 1)by applying some obvious parametric and argument variations with the additional parameter p being redundant.