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On the Chebyshev coefficients for a general subclass of univalent function

In this work, considering a general subclass of univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class. | Turk J Math (2018) 42: 2885 – 2890 © TÜBİTAK doi:10.3906/mat-1510-53 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article On the Chebyshev coefficients for a general subclass of univalent functions Şahsene ALTINKAYA∗,, Sibel YALÇIN, Department of Mathematics, Faculty of Arts and Science, Bursa Uludağ University, Bursa, Turkey Received: 15.10.2015 • Accepted/Published Online: 03.03.2017 • Final Version: 27.11.2018 Abstract: In this work, considering a general subclass of univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class. Key words: Chebyshev polynomials, analytic and univalent functions, coefficient bounds, subordination 1. Introduction and definitions Let D be the open unit disk {z ∈ C : |z| 0. f (z) The Fekete–Szegö functional a3 − µa22 for normalized univalent functions of the form given by (1) is well known for its rich history in geometric function theory. Its origin was in the disproof by Fekete and Szegö ∗Correspondence: sahsene@uludag.edu.tr 2010 AMS Mathematics Subject Classification: 30C45, 30C50 2885 This work is licensed under a Creative Commons Attribution 4.0 International License. ALTINKAYA and YALÇIN/Turk J Math of the 1933 conjecture of Littlewood and Paley that the coefficients of odd univalent functions are bounded by unity (see [5]). The functional has since received great attention, particularly in many subclasses of the family of univalent functions. Nowadays, it seems that this topic had become of interest among researchers (see, for example, [1, 2, 7, 8]). Chebyshev polynomials have become increasingly important in numerical analysis, from both theoretical and practical points of view. There are four kinds of Chebyshev polynomials. The majority of books and research papers dealing with specific orthogonal polynomials of the Chebyshev family contain mainly results of Chebyshev polynomials of the first and second kinds Tn (x)