The Fekete-Szego Theorem for Close-to-convex Functions Associated with The Koebe Type Function

This paper deals with the class S containing functions which are analytic and univalent in the open unit disc U = {z ∈ ℂ : |z| < 1}. Functions f in S are normalized by f(0) = 0 and f′(0) = 1 and has the Taylor series expansion of the form f(z)=z+∑n=2∞anzn f\left( z \right) = z + \sum\limits_{n = 2}^\infty {{a_n}{z^n}} . In this paper we investigate on the subclass of S of close-to-convex functions denoted as C_gα(λ, δ) where function f ∈ C_gα(λ, δ) satisfies Re{ eiλzf′(z)gα(z) } {\mathop{\rm Re}\nolimits} \left\{ {{e^{i\lambda }}{{zf'\left( z \right)} \over {g\alpha \left( z \right)}}} \right\} for | λ |<π2 \left| \lambda \right| < {\pi \over 2} , cos(λ) > δ, 0 ≤ δ < 1, 0 ≤ α ≤ 1 and gα=z(1−αz)2 {g_\alpha } = {z \over {{{\left( {1 - \alpha z} \right)}^2}}} . The aim of the present paper is to find the upper bound of the Fekete-Szego functional |a₃ − µa₂²| for the class C_gα(λ, δ). The results obtained in this paper is significant in the sense that it can be used in future research in this field, particularly in solving coefficient inequalities such as the Hankel determinant problems and also the Fekete-Szego problems for other subclasses of univalent functions.