On the Coefficient Inequality for a Subclass of Strongly Starlike Mappings of Order $$\alpha $$ α 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.