Some properties for certain class of bi-univalent functions deﬁned by q -C˘atas¸ operator with bounded boundary rotation

: Throughout the paper, we introduce a new subclass H n , q ,λ, l α,µ,ρ, m ,β f ( z ) by using the Bazileviˇc functions with the idea of bounded boundary rotation and q -analogue C˘atas¸ operator. Also we ﬁnd the estimate of the coe ﬃ cients for functions in this class. Finally, in the concluding section, we have chosen to reiterate the well-demonstrated fact that any attempt to produce the rather straightforward ( p , q )-variations of the results, which we have presented in this article, will be a rather trivial and inconsequential exercise, simply because the additional parameter p is obviously redundant.


Introduction
Let A denote the class of analytic functions of the form: a k z k (z ∈ U : U = {z ∈ C : |z| < 1}). (1.1) Let S be the subclass of A consisting of univalent functions in U and let K, S λ , S * and C be the usual subclasses of S consisting of functions which are, respectively, close-to-convex, λ -spiral-like, starlike (w.r.t. the origin) and convex in U.
By the Koebe one-quarter theorem [15], we know that the image of U under every univalent function f ∈ A contains the disc with the center in the origin and radius 1/4. Therefore, every univalent function f has an inverse f −1 satisfies It is easy to see that the inverse function has the form A function f ∈ A is said to be bi-univalent in U if both f and its inverse map g = f −1 are univalent in U. Let denote the class of bi-univalent functions in U in the form (1.1).
For interesting examples about the class (see [8,11,16,18,25,46]). The pioneering work of Srivastava et al. [45] actually revived the study of bi-univalent functions in recent years. In a substantially large number of work subsequent to the work of Srivastava et al. [45], several distinct subclasses of the bi-univalent function class were presented and examined similarly by many authors. For example, the function classes H (τ, µ, λ, δ; α) and H (τ, µ, λ, γ; β) were defined and the estimates on the Taylor-Maclaurin coefficients |a 2 | and |a 3 | were obtained by Srivastava et al. [43]. The upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions were obtained by Caglar et al. [13]. Several new subclasses of the class of m-fold symmetric bi-univalent functions were introduced and the initial estimates of the Taylor-Maclaurin series as well as some Fekete-Szegö functional problems for each of their defined function classes were obtained by Tang et al. [50] and Srivastava et al. [42]. Several other well-known mathematicians gave their findings on this subject (e.g. [40,41,44]).
As we know that the fractional q-calculus and the fractional of q-derivative operators in Geometric Function Theory were investigated by sturdy of researchers (see [2,3,7,9,10,17,[19][20][21][22][23][24][25][37][38][39][47][48][49]). The q-calculus is an important tool which is used to study various applications in mathematics, physics and chemistry and some basic sciences subjects. In the study of Geometric Function Theory, the versatile applications of the q-derivative operator D q make it remarkably significant. Inspired by the above-mentioned works, in recent years, important researches have played a significant part in the development of Geometric Function Theory of complex analysis. Several convolutional and fractional calculus q-operators were defined by many researchers, which were surveyed in the above-cited work by Srivastava [37]. For a function f (z) ∈ A given by (1.1) and 0 < q < 1. Jackson's q-derivative (or q-difference) D q of a function defined on a subset of the complex space C is defined as follows: For a function f (z) ∈ A, Aouf and Madian [10] defined the q-analogue Cȃtaş operator as follows (see also [7,9] with p = 1]): And introduced recurrent relation as follows: We note that I n q (λ, l) f (z) generalized many operators such as Cȃtaş operator, Multiplier operator and Sȃlȃgean operator etc., for more details see [1, 5-7, 14, 28, 35].
c n z n ∈ P µ m (ρ), then The result is sharp. Equality is attained for the odd coefficients and even coefficients, respectively, for the functions We note that for µ = ρ = 0 in Lemma 1, we obtain the result obtained by Goswami et al. [16,Lemma 1] for the class P m . The object of this paper is to introduce a new subclass of the class by using the definition of Bazilevič functions, bi-univalent functions with bounded boundary rotation and q-analogue Cȃtaş operator. As well as I calculate the coefficient estimates for functions in the subclass H n,q,λ,l α,µ,ρ,m,β f (z). Also I get coefficients bounds for the subclasses of our main class.
Putting µ = 0 and m = 2 in Theorem 1, we obtain the following corollary.
Putting α = 1 in Theorem 1, we obtain the following corollary.

Conclusions
Throughout the paper, I used the definition of Bazilevič function, bi-univalent functions with bounded boundary rotation and the definition of q-analogue Cȃtaş operator to introduce the new subclass H n,q,λ,l α,µ,ρ,m,β f (z). I estimated the coefficients bounds for the functions belong to the subclass H n,q,λ,l α,µ,ρ,m,β f (z). In addition, through this paper we presented coefficients bounds for the functions belong to the subclasses of our main class.
Srivastava [37, p. 340] discussed the connection between the classical q-analysis, which we used in this article, and its so-called trivial and inconsequential (p, q)-variation involving an obviously superfluous parameter p. Specifically, the results in this article for the q-analogues (0 < q < 1), can easily translated into the corresponding (p, q)-variants (0 < q < p ≤ 1) by following the observation by Srivastava [37, p. 340] who applied some obvious parametric and argument variations, the additional parameter p being redundant. As clearly and significantly pointed out by Srivastava et al. [46,48], some group of authors have made use of the so-called trivial and inconsequential (p, q)-variation by introducing a seemingly redundant parameter p in the already known results dealing with the classical q-analysis. For further details, see the survey-cum-expository review article by Srivastava [37, p. 340].

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