In this paper, we prove a Schwarz lemma at the boundary for holomorphic mappings f between Hilbert balls, and obtain related consequences. Especially, we obtain estimations of ∥ Df ( z 0 )∥ on the holomorphic tangent space for holomorphic mappings f or for homogeneous polynomial mappings f between Hilbert balls. Next, we prove the boundary rigidity theorem for holomorphic self-mappings of a Hilbert ball which have an interior fixed point. We obtain two distortion theorems for various subclasses of starlike mappings on the Euclidean unit ball $${\mathbb{B}^n}$$ B n in ℂ n , as applications of the boundary Schwarz lemma for holomorphic mappings between the Euclidean unit balls. Finally, certain coefficient bounds for subclasses of starlike mappings on the unit ball of a complex Hilbert space are also obtained as an application of the estimation of ∥ Df ( z 0 )∥ on the holomorphic tangent space for homogeneous polynomial mappings f between Hilbert balls.

中文翻译：

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复杂希尔伯特球边界处的 Schwarz 引理及其对星状映射的应用

在本文中，我们在希尔伯特球之间的全纯映射 f 的边界处证明了 Schwarz 引理，并获得相关结果。特别是，我们在全纯切线空间上获得了对 ∥ Df ( z 0 )∥ 的估计，用于全纯映射 f 或希尔伯特球之间的齐次多项式映射 f。接下来，我们证明了具有内部不动点的希尔伯特球的全纯自映射的边界刚性定理。我们获得了欧几里德单位球 $${\mathbb{B}^n}$$ B n in ℂ n 上星状映射的各种子类的两个失真定理，作为边界施瓦茨引理在欧几里德单位球之间的全纯映射的应用. 最后，