Summary: Let \(\mathcal{C}\) be the familiar class of normalized close-to-convex functions in the unit disk. W. Koepf [Proc. Am. Math. Soc. 101, 89–95 (1987; Zbl 0635.30019)] proved that for a function \(f(z)=z+\sum_{k=2}^\infty a_k\,z^k\) in the class \(\mathcal{C} \), \[ |a_3-\lambda a_2^2|\leq \begin{cases} 3-4\lambda, & \lambda\in\bigl[0, \frac{1}{3}\bigr],\\ \frac{1}{3}+\frac{4}{9\lambda}, & \lambda\in\bigl[\frac{1}{3}, \frac{2}{3}\bigr],\\ 1, & \lambda\in\bigl[\frac{2}{3}, 1\bigr] \end{cases} \quad \text{and}\quad \bigl| |a_3|-|a_2| \bigr|\leq 1.\] Recently, Q. Xu et al. [Acta Math. Sci., Ser. B, Engl. Ed. 43, No. 5, 2075–2088 (2023; Zbl 1524.32019)] generalized the above results to a subclass of close-to-quasiconvex mappings of type \(B\) defined on the open unit polydisc in \(\mathbb{C}^n\), and to a subclass of close-to-starlike mappings defined on the open unit ball of a complex Banach space, respectively. In the first part of this paper, by using different methods, we obtain the corresponding results of norm type and functional type on the open unit ball in a complex Banach space. We next give the coefficient inequalities for a subclass of \(g\)-starlike mappings of complex order \(\lambda\) on the open unit ball of a complex Banach space, which generalize many known results. Moreover, the proofs presented here are simpler than those given in the related papers.