Abstract
Let R I (m, n) be the classical domain of type I in ℂm×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Fréchet derivative Df(\(\mathop Z\limits^ \circ \)) at a smooth boundary fixed point \(\mathop Z\limits^ \circ \)of R I (m, n) for a holomorphic self-mapping f of R I (m, n). We provide a necessary and sufficient condition such that the boundary points of R I (m, n) are smooth, and give some properties of the smooth boundary points of R I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R I (m, n), which may be applied to get some optimal estimates in several complex variables.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11571105 and 11471111) and Natural Science Foundation of Zhejiang Province (Grant No. LY14A010017). The authors thank the referees for their comments and suggestions.
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Liu, T., Tang, X. A boundary Schwarz lemma on the classical domain of type I . Sci. China Math. 60, 1239–1258 (2017). https://doi.org/10.1007/s11425-015-0225-7
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DOI: https://doi.org/10.1007/s11425-015-0225-7