New Subclasses of Biholomorphic Mappings and the Modified Roper-Suffridge Operator

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The Roper-Suffridge operator on the unit ball $B^{n}$ in $\mathbb{C}^{n}$ is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator $\Phi_{G, \gamma}(f)](z)$ preserves the properties of $S^*_{\Omega}(A,B)$, as well as strong and almost spirallikeness of type $\beta$ and order $\alpha$ on $B^{n}$. Thus, the mappings in $S^*_{\Omega}(A,B)$, as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.