Distortion Theorem for Locally Biholomorphic Bloch Mappings on the Unit Ball $$\mathcal {B}^{n*}$$ B n ∗

Abstract

In this note, we establish a distortion theorem for locally biholomorphic Bloch mappings \(f\) satisfying \(||f||_{0}=1\) and \(\det f'(0)=\alpha \in (0,1],\) where \(\Vert f\Vert _{0}=\mathrm {sup}\{(1-|z|^{2})^\frac{n+1}{2n}|\det f'(z)| ^\frac{1}{n}:z\in \mathcal {B}^{n}\}.\) This result extends the result of Bonk, Minda, and Yanagihara of one complex variable to higher dimensions. Moreover, a lower estimate for the radius of the largest univalent ball in the image of \(f\) centered at \(f(0)\) is given.