Communications in Mathematics and Applications Vol. 11, No. 1, pp. 57–64, 2020 ISSN 0975-8607 (online); 0976-5905 (print) Published by RGN Publications http://www.rgnpublications.com DOI: 10.26713/cma.v11i1.1331 Research Article On Initial Chebyshev Polynomial Coefficient Problem for Certain Subclass of Bi-Univalent Functions F. Müge Sakar 1,*, and Ertu˘ grul Do ˘ gan 2, 1 Department of Business Administration, Faculty of Management and Economics, Dicle University, Diyarbakır, Turkey 2 Institute of Science, Batman, Batman University, Turkey *Corresponding author: mugesakar@hotmail.com Abstract. In this paper, we firstly, introduced the subclass R Σ (τ, α, γ; t) satisfying subordinate conditions. Subsequently, considering this defined subclass, initial coefficient estimations are established using by Chebyshev polynomials expansions, and Fekete-Szegö inequalities are also derived for functions belonging to the said subclass. Furthermore, Some relevant consequences of these results are also discussed. Keywords. Initial coefficients problem; Bi-univalent function; Chebyshev polinomials; Fekete-Szegö problem MSC. 30C45; 30C50 Received: December 2, 2019 Accepted: December 18, 2019 Copyright © 2020 F. Müge Sakar and Ertu˘ grul Do ˘ gan. 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. 1. Introduction Let A denote the class of functions of the form f ( z) = z + ∞  n=2 a n z n (1) which are analytic in the open unit disc Δ = { z : z ∈ C and | z|< 1}. In addition, we indicate by S the class of all univalent functions in Δ.