A NOTE ON THE FEKETE–SZEGÖ PROBLEM FOR CLOSE-TO-CONVEX FUNCTIONS WITH RESPECT TO CONVEX FUNCTIONS

Bogumiła Kowalczyk, Adam Lecko, H. M. Srivastava

Abstract: We discuss the sharpness of the bound of the Fekete–Szegö functional for close-to-convex functions with respect to convex functions. We also briefly consider other related developments involving the Fekete–Szegö functional $|a_3-\lambda a_2^2|$ ($0\le \lambda \le 1$) as well as the corresponding Hankel determinant for the Taylor–Maclaurin coefficients ${\left\{a_n\right\}}_{n\in \mathbb{N}\setminus \left\{1\right\}}$ of normalized univalent functions in the open unit disk $𝔻$, $\mathbb{N}$ being the set of positive integers.

Keywords: analytic functions; convex functions; Fekete–Szegö problem; Hankel determinant; Taylor-Maclaurin coefficients; close-to-convex functions with respect to a convex function; Carathéodory class; Schwarz functions

Classification (MSC2000): 30C45

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