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.

中文翻译：

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ℂn中某个双全纯映射子类的失真结果

让${\mathbb{C}}^n$是n维复变量的空间和${\mathbb{D}}^n$成为单位 polydisc${\mathbb{C}}^n$. 对于定义在${\mathbb{D}}^n$. 此外，在单位球上定义的相应子类的雅可比行列式失真定理${\mathbb{C}}^n$以任意规范成立。我们的结果允许复杂向量的每个分量具有不同的维度，这扩展了之前与星状映射的某些子类密切相关的几个工作。