symmetry S S Article Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial Adel A. Attiya 1,2 , Abdel Moneim Lashin 2,3 , Ekram E. Ali 1,4 and Praveen Agarwal 5,6,7,8, *   Citation: Attiya, A.A.; Lashin, A.M.; Ali, E.E.; Agarwal, P. Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial. Symmetry 2021, 13, 302. https://doi.org/10.3390/ sym13020302 Academic Editor: Dmitriy V. Dolgy Received: 31 October 2020 Accepted: 30 January 2021 Published: 10 February 2021 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional clai- ms in published maps and institutio- nal affiliations. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and con- ditions of the Creative Commons At- tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, College of Science, University of Ha’il, Ha’il 81451, Saudi Arabia; aattiy@mans.edu.eg (A.A.A.); ekram_008eg@yahoo.com (E.E.A.) 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt; aylashin@mans.edu.eg 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4 Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt 5 Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India 6 International Center for Basic and Applied Sciences, Jaipur 302029, India 7 Department of Mathematics, Harish-Chandra Research Institute, Allahabad 211 019, India 8 Department of Mathematics, Netaji Subhas University of Technology, New Delhi 110078, India * Correspondence: praveen.agarwal@anandice.ac.in or goyal.praveen2011@gmail.com Abstract: In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of n − th (n ≥ 3) Taylor–Maclaurin coefficients | a n | is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established. Keywords: faber polynomial; coefficient bounds; uniformly convex; uniformly starlike; univalent functions; bi-univalent functions 1. Introduction Faber polynomials, which were introduced by Faber in 1903 [1], play an important role in the theory of functions of a complex variable and different areas of mathematics and there is a rich literature [2–7] describing their properties and their applications. Given a function h(z) of the form h(z)= z + b 0 + b 1 z −1 + b 2 z −2 + ..., consider the expansion ςh ′ (ζ ) h(ζ ) − w = ∞ ∑ n=0 Ψ n (w)ζ −n , valid for all ζ in some neighborhood of ∞. The function Ψ n (w)= w n + n ∑ k=1 a nk w n−k is a polynomial of degree n, called the n-th Faber polynomial with respect to the function h(z). In particular, Ψ 0 (w)= 1, Ψ 1 (w)= w − b 0 , Ψ 2 (w)= w 2 − 2b 0 w +(b 2 0 − 2b 1 ), Ψ 3 (w)= w 3 − 3b 0 w 2 +(3b 2 0 − 3b 1 )w +(b 3 0 + 3b 1 b 0 − 3b 2 ). Let Ψ n (0)= F n (b 0 , b 1 , ... , b n ), n ≥ 0, see ([8], p. 118). Let A denote the class of all functions of the form: f (z)= z + ∞ ∑ n=2 a n z n , (1) Symmetry 2021, 13, 302. https://doi.org/10.3390/sym13020302 https://www.mdpi.com/journal/symmetry