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 $|a_3-\lambda a_2^2|$ ($0\leq\lambda\leq 1$) as well as the corresponding Hankel determinant for the Taylor--Maclaurin coefficients $\{a_n\}_{n\in \mathbb{N}\smallsetminus\{1\}}$ of normalized univalent functions in the open unit disk $\mathbb{D}$, $\mathbb{N}$ being the set of positive integers.