关键词: Bieberbach conjecture, homogeneous polynomial expansions, $g$-parametric starlike mappings of complex order $\gamma$, $k$-fold symmetric mapping, a zero of order $k+1$

Abstract: In this paper, the class of starlike functions of complex order $\gamma\, (\gamma\in \mathbb{C}-\{0\})$ is extended from the case on unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$ to the case on the unit ball $B$ in a complex Banach space or the unit polydisk $\mathbb{U}^n$ in $\mathbb{C}^n$. Let $g$ be a convex function in $\mathbb{U}$. We mainly establish the sharp bounds of all terms of homogeneous polynomial expansions for a subclass of $g$-parametric starlike mappings of complex order $\gamma$ on $B$ (resp. $\mathbb{U}^n$) when the mappings $f$ are $k$-fold symmetric, $k\in \mathbb{N}.$ Our results partly solve the Bieberbach conjecture in several complex variables and generalize some prior works.

Key words: Bieberbach conjecture, homogeneous polynomial expansions, $g$-parametric starlike mappings of complex order $\gamma$, $k$-fold symmetric mapping, a zero of order $k+1$