Faber Polynomial Coefficient Bounds for m-Fold Symmetric Analytic and Bi-univalent Functions Involving q-Calculus

School of Mathematics and Statistics, Huanghuai University, Zhumadian, 463000 Henan, China Department of Mathematics and Statistics, Riphah International University, Islamabad 44000, Pakistan Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China


Introduction, Definitions, and Motivation
Let A denote the class of all analytic functions f ðzÞ in the open unit disk U = fz : jzj < 1g and have the series expansion of the form f z ð Þ = z + 〠 ∞ n=2 a n z n : By S, we mean the subclass of A consisting of univalent functions. The inverse f −1 of univalent function f can be defined as where According to the Koebe one-quarter theorem [1], an analytic function f is called bi-univalent in U if both f and f −1 are univalent in U. Let Σ denote the class all biunivalent functions in U. For f ∈ Σ, Lewin [2] showed that ja 2 j < 1:51 and Brannan and Cluni [3] proved that ja 2 j ≤ ffiffi ffi 2 p . Netanyahu [4] showed that max ja 2 j = 4/3: Brannan and Taha [5] introduced a certain subclass of bi-univalent functions for class Σ. In recent years, Srivastava et al. [6], Frasin and Aouf [7], Altinkaya and Yalcin [8,9], and Hayami and Owa [10] studied the various subclasses of analytic and bi-univalent function. For a brief history, see [11].
In [12], Faber introduced Faber polynomials, and after that, Gong [13] studied Faber polynomials in geometric function theory. In their published works, some contributions have been made to finding the general coefficient bounds |a n | by applying Faber polynomial expansions. By using Faber polynomial expansions, very little work has been done for the coefficient bounds ja n j for n ≥ 4 of Maclaurin's series. For more studies, see [14][15][16][17].
A domain U is said to be m-fold symmetric if The univalent function hðzÞ maps the unit disk U into a region with m-fold symmetry and can be defined as A function f is said to be m-fold symmetric [18] if it has the series expansion of the form The class of all m-fold symmetric univalent functions is denoted by S m , and for m = 1, then S m = S.
In [19], Srivastava et al. proved the inverse f −1 m series expansion for f ∈ Σ m , which is given as follows: Here, we will denote m-fold symmetric bi-univalent functions by Σ m . For m = 1, equation (7) coincides with equation (3) of the class Σ: The coefficient problem for f ∈ Σ m is one of the favorite subjects of geometric function theory in these days (see [20][21][22][23]).
The quantum (or q-) calculus has great importance because of its applications in several fields of mathematics, physics, and some related areas. The importance of q -derivative operator ðD q Þ is pretty recognizable by its applications in the study of numerous subclasses of analytic functions. Initially, in 1908, Jackson [24] introduced a q -derivative operator and studied its applications. Further, in [25], Ismail et al. defined a class of q-starlike functions; after that, Srivastava [26] studied q-calculus in the context of univalent function theory; also, numerous mathematicians studied q-calculus in the context of univalent function theory: Further, the q-analogue of the Ruscheweyh differential operator was defined by Kanas and Raducanu [27] and Arif et al. [28] discussed some of its applications for multivalent functions while Zhang et al. in [29] studied q-starlike functions related with the generalized conic domain. Srivastava et al. published the articles (see [30,31]) in which they studied the class of q-starlike functions. For some more recent investigations about q-calculus, we may refer to [32][33][34].
For a better understanding of the article, we recall some concept details and definitions of the q-difference calculus. Throughout the article, we presume that Definition 1. The q-factorial ½n q ! is defined as and the q-generalized Pochhammer symbol ½t n,q , t ∈ ℂ, is defined as Remark 2. For n = 0, then ½n q ! = 1, and ½t n,q = 1.
Definition 3. The q-number ½t q for q ∈ ð0, 1Þ is defined as Definition 4 (see [24]). The q-derivative (or q-difference) operator D q of a function f is defined, in a given subset of ℂ, by provided that f ′ ð0Þ exists.
From Definition 4, we can observe that for a differentiable function f in a given subset of ℂ. It is also known from (1) and (12) that Here, in this paper, we use the q-difference operator to define new subclasses of m-fold symmetric analytic and biunivalent functions and then apply the Faber polynomial expansion technique to determine the general coefficient bounds ja mk+1 j and initial coefficient bounds ja m+1 j and ja 2m+1 j as well as Fekete-Szego inequalities.

Main Results
Using the Faber polynomial expansion of functions f ∈ A of the form (1), the coefficients of its inverse map g = f −1 may be expressed as [15] given by for an expansion of K −n n−1 (see [37]). In particular, the first three terms of K −n n−1 are In general, for any p ∈ ℕ and n ≥ 2, an expansion of K p n−1 is as (see [15]) where E p n−1 = E p n−1 ða 2 , a 3 , ⋯Þ, and by [37], while a 1 = 1, and the sum is taken over all nonnegative integers μ 1 , ⋯, μ n satisfying μ 1 + μ 2 +⋯+μ n = m, Evidently, E n−1 n−1 ða 2 , ⋯, a n Þ = a n−1 2 (see [14]), or equivalently, while a 1 = 1, and the sum is taken over all nonnegative integers μ 1 , ⋯, μ n satisfying μ 1 + μ 2 +⋯+μ n = m, It is clear that E n n ða 1 , ⋯, a n Þ = E n 1 , and the first and last polynomials are E n n = a n 1 and E 1 n = a n : Similarly, using the Faber polynomial expansion of functions f ∈ A of the form (6), that is, The coefficients of its inverse map g = f −1 m may be expressed as Theorem 9. For b ∈ ℂ \ f0g, let f ∈ R b ðφ, m, qÞbe given by (6), and ifa mj+1 = 0, 1 ≤ j ≤ k − 1, then Proof. By definition, for the function f ∈ R b ðφ, m, qÞ of the form (6), we have and for its inverse map g = f −1 m , we have where

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On the other hand, since f ∈ R b ðφ, m, qÞ and g = f −1 m ∈ R b ðφ, m, qÞ by definition, we have where Comparing the coefficients of (27) and (31), we have Similarly, comparing coefficients of (28) and (32), we have Note that for a mj+1 = 0, 1 ≤ j ≤ k − 1, we have and so Now taking the absolute of (36) and (37) and using the fact that jφ 1 j ≤ 2, jc k j ≤ 1, and jd k j ≤ 1, we have which completes the proof of Theorem 9.
For m = 1 and k = n − 1, in Theorem 9, we obtain the following corollary.
Corollary 11 (see [35]). For b ∈ ℂ \ f0g, let f ∈ R b ðφÞ, and ifa j+1 = 0, 1 ≤ j ≤ n, then a n j j ≤ 2 b j j n , for n ≥ 3: ð40Þ ðφ, m, qÞ be given by (6), and then a m+1 j j≤ Proof. Replacing k by 1 and 2 in (33) and (34), respectively, we have From (42) and (44), we have Journal of Function Spaces Adding (43) and (45), we have Taking the absolute value (47), we have a m+1 j j≤ Now, the bounds given for ja m+1 j can be justified since From (43), we have Next, we subtract (45) from (43), and we have or After some simple calculation and by taking the absolute, we have Using the assertion (46), we have From (50) and (54), we note that Now, we rewrite (45) as Taking the absolute value, we have Finally, from (51), we have Taking the absolute value, we have For m = 1 and k = n − 1, in Theorem 12, we obtain the following corollary.

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Corollary 14 (see [35]). For b ∈ ℂ \ f0g, let f ∈ B b ðφÞ be given by (1), and then Theorem 15. Let f ∈ S * Σ m ðφ, qÞbe given by (6), and Proof. By definition, for the function f ∈ S * Σ m ðφ, qÞ of the form (6), we have where the first few coefficients of F k ða m+1 , a 2m+1 , ⋯, a mk+1 Þ are In general, where For the inverse map g = f −1 m ∈ S * Σ m ðφ, qÞ, we obtain where On the other hand, since f ∈ S * Σ m ðφ, qÞ and g = f −1 m ∈ S * Σ m ðφ, qÞ by definition, we have where Comparing the coefficients of (63) and (70), we have Similarly, comparing the coefficients of (67) and (71), we have Note that for a mj+1 = 0, 1 ≤ j ≤ k − 1, we have and so mk Taking the absolute values of (75) and (76) and using the fact that jφ 1 j ≤ 2, jc k j ≤ 1, and jd k j ≤ 1, we have 6 Journal of Function Spaces Hence, Theorem 15 is complete.