Distortion results for a certain subclass of biholomorphic mappings in ℂⁿ

Abstract

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.

Keywords:

• Biholomorphic mappings
• distortion theorem of Jacobi-determinant type
• distortion theorems of the Fréchet-derivative type
• starlike mappings

AMS Subject Classifications:

• 32H02
• 30C45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project is supported by the National Natural Science Foundation of China (Grant No. 12061035) and Doctoral Research Foundation of Jiangxi Science and Technology Normal University of China (Grant No. 2019BSQD017).