tailieunhanh - Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion
In the present study, we give and look into a new subclass of analytic and bi-close-to-convex functions in the open unit disk. Making use of the Faber series, we have an upper bound for the general coefficient of functions in this class. We also demonstrate the invisible behavior of the beginning coefficients of a special subclass of bi-close-to-convex functions. | Turk J Math (2017) 41: 888 – 895 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Coefficient bounds for a new subclass of analytic bi-close-to-convex functions by making use of Faber polynomial expansion 1 2,∗ ¨ ¨ Fethiye M¨ uge SAKAR1 , Hatun Ozlem GUNEY Department of Business Administration, Faculty of Management and Economics, Batman University, Batman, Turkey 2 Department of Mathematics, Faculty of Science, Dicle University, Diyarbakır, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: Recently, in the literature, we can see quite a few papers about general coefficient bounds for subclasses of bi-univalent functions. However, we can find just a few papers about general coefficient estimates for subclasses of bi-close-to-convex functions. In the present study, we give and look into a new subclass of analytic and bi-close-to-convex functions in the open unit disk. Making use of the Faber series, we have an upper bound for the general coefficient of functions in this class. We also demonstrate the invisible behavior of the beginning coefficients of a special subclass of bi-close-to-convex functions. Key words: Analytic functions, bi-close-to convex functions, Faber polynomials, bi-univalent functions, coefficient estimates 1. Introduction We know that a function is univalent if it never takes the same value twice. We also know that a function is bi-univalent if both it and its inverse are univalent. Let A denote the class of functions f that are analytic in the open unit disk U = {z : z ∈ C and |z| α in U and C(α) indicate the class of functions f ∈ S that are close-to-convex of U, namely, Re zgg(z) { ′ } (z) order α in U, namely, if a function g is in S ∗ (0) = S ∗ so that Re zfg(z) > α in U [6, 10]. We note that S ∗ (α) ⊂ C(α) ⊂ S and that |an | ≤ n for f ∈ S by the Bieberbach conjecture [3, 6]. The Koebe 1/4 theorem [6] asserts
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