The modified Fekete-Szegö functional for a subclass of close-to-starlike mappings in complex Banach spaces

In [23], Koepf proved that for a function \(f(\xi )=\xi +\sum \limits _{m=2}^\infty a_m\xi ^m\) $f(\xi )=\xi +\sum _{m=2}^{\infty }{a}_m{\xi }^m$ in the class of normalized close-to-convex functions in the unit disk, \(\begin{aligned} |a_3-\lambda a_2^2|\le \left\{ \begin{array}{ll} 3-4\lambda ,\quad &{} \lambda \in [0, \frac{1}{3}],\\ \frac{1}{3}+\frac{4}{9\lambda },\quad &{} \lambda \in [\frac{1}{3}, \frac{2}{3}],\\ 1,\quad &{} \lambda \in [\frac{2}{3}, 1]. \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill |a_3-\lambda a_2^2|\le \left\{\begin{array}{cc}3-4\lambda ,\hfill & \lambda \in [0,\frac{1}{3}],\hfill \\ \frac{1}{3}+\frac{4}{9\lambda },\hfill & \lambda \in [\frac{1}{3},\frac{2}{3}],\hfill \\ 1,\hfill & \lambda \in [\frac{2}{3},1].\hfill \end{array}\right)\end{array}$$ In this paper, considering the zero of order (i.e., the mapping \(f(x)-x\) $f(x)-x$ has zero of order \(k+1\) $k+1$ at the point \(x=0\) $x=0$ ), we generalize the above classical result and establish the modified Fekete-Szegö functional for s subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space.