Abstract: Let $\mathcal{C}$ be the familiar class of normalized close-to-convex functions in the unit disk. In [17], Koepf demonstrated that, as to a function $f(\xi)=\xi+\sum\limits_{m=2}^\infty a_m\xi^m$ in the class $\mathcal{C}$, $$ \max\limits_{f\in \mathcal{C}}|a_3-\lambda a_2^2|\leq \left\{\begin{array}{ll} 3-4\lambda, \quad & \lambda\in[0, \frac{1}{3}], \\[3mm] \frac{1}{3}+\frac{4}{9\lambda}, \quad & \lambda\in[\frac{1}{3}, \frac{2}{3}], \\[3mm] 1, \quad & \lambda\in[\frac{2}{3}, 1]. \end{array}\right.$$ By applying this inequality, it can be proven that $||a_3|-|a_2||\leq 1$ for close-to-convex functions. Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.

Key words: Fekete and Szegö inequality, successive coefficients difference bound, close-to-starlike mappings, complex Banach space