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.
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Acknowledgments
The author cordially thanks to the referees for their thorough reviewing with useful suggestions and comments made to the paper. This work was supported by the National Natural Science Foundation of China (No. 11471111, 11101139) and NSF of Zhejiang province (No. LY14A010013).
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Communicated by V. Ravichandran.
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Wang, J. Distortion Theorem for Locally Biholomorphic Bloch Mappings on the Unit Ball \(\mathcal {B}^{n*}\) . Bull. Malays. Math. Sci. Soc. 38, 1657–1667 (2015). https://doi.org/10.1007/s40840-014-0109-6
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DOI: https://doi.org/10.1007/s40840-014-0109-6