On the Coefficient Inequality for a Subclass of Strongly Starlike Mappings of Order $\alpha$ in Several Complex Variables

Abstract

Let $\mathcal {SS}_\alpha ^*$ be the familiar class of strongly starlike functions of order $\alpha$ in the unit disk. Xu et al. (Results Math 72:343–357, 2017) proved that for a function $f(z)=z+\sum \nolimits _{k=2}^\infty a_kz^k$ in the class $\mathcal {SS}_\alpha ^*$, then

$$\begin{aligned} |a_3-\lambda a_2^2|\le \alpha \max \{1,\ \alpha |3-4\lambda |\}, \quad \lambda \in \mathbb {C}. \end{aligned}$$

In this paper, we investigate the corresponding problem for the subclass of strongly starlike mappings of order $\alpha$ defined on the unit ball in a complex Banach space, on the unit polydisk in $\mathbb {C}^n$ and the bounded starlike circular domain in $\mathbb {C}^n$, respectively.