The estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation in several complex variables

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation on the unit ball of complex Banach spaces. Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C ⁿ . Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results.