Research Article Faber Polynomial Coefficient Bounds for m-Fold Symmetric Analytic and Bi-univalent Functions Involving q-Calculus Zeya Jia, 1 Shahid Khan , 2 Nazar Khan, 3 Bilal Khan , 4 and Muhammad Asif 2 1 School of Mathematics and Statistics, Huanghuai University, Zhumadian, 463000 Henan, China 2 Department of Mathematics and Statistics, Riphah International University, Islamabad 44000, Pakistan 3 Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan 4 School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Road, Shanghai 200241, China Correspondence should be addressed to Shahid Khan; shahidmath761@gmail.com Received 19 August 2021; Accepted 4 October 2021; Published 26 October 2021 Academic Editor: Richard I. Avery Copyright © 2021 Zeya Jia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In our present investigation, by applying q-calculus operator theory, we define some new subclasses of m-fold symmetric analytic and bi-univalent functions in the open unit disk U = fz ∈ ℂ : jzj <1g and use the Faber polynomial expansion to find upper bounds of ja mk+1 j and initial coefficient bounds for ja m+1 j and ja 2m+1 j as well as Fekete-Szego inequalities for the functions belonging to newly defined subclasses. Also, we highlight some new and known corollaries of our main results. 1. 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 expan- sion of the form fz ðÞ = z + 〠 ∞ n=2 a n z n : ð1Þ By S , we mean the subclass of A consisting of univalent functions. The inverse f −1 of univalent function f can be defined as f −1 fz ðÞ ð Þ = z, z ∈ U, f f −1 w ð Þ   = w, w j j < r 0 f ðÞ, r 0 f ðÞ ≥ 1 4 , ð2Þ where g 1 w ð Þ = f −1 w ð Þ = w − a 2 w 2 + 2a 2 2 − a 3   w 3 − 5a 3 2 − 5a 2 a 3 + a 4   w 4 +: ⋯ ð3Þ 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 bi- univalent 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 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 ana- lytic 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 contribu- tions 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 [14–17]. A domain U is said to be m-fold symmetric if f e i 2π/m ð Þ z   = e i 2π/m ð Þ fz ðÞ, z ∈ U, f ∈ A , m ∈ ℕ: ð4Þ Hindawi Journal of Function Spaces Volume 2021, Article ID 5232247, 9 pages https://doi.org/10.1155/2021/5232247