Summary: Let \(\mathbb{C}^n\) be the space of \(n\)-dimensional complex variables and \(\mathbb{D}^n\) be the unit polydisc in \(\mathbb{C}^n\). We obtain the distortion theorems of the Fréchet-derivative type and the Jacobi-determinant type for a certain subclass of normalized biholomorphic mappings defined on \(\mathbb{D}^n\). Also, the distortion theorem of Jacobi-determinant type for the corresponding subclass defined on the unit ball in \(\mathbb{C}^n\) with arbitrary norm is established. Our results allow each component of complex vectors to have different dimensions, which extends severl previous works being closely related to some subclasses of starlike mappings.