The Generalized Schwarz–Pick Estimates of Arbitrary Order on the Unit Polydisk

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Let Ω be a homogeneous circular convex domain (a bounded symmetric domain) in \({\mathbb{C}^N}\) containing the origin. The generalization of the Schwarz–Pick estimates of partial derivatives of arbitrary order are established for holomorphic mappings from the unit polydisk D n to Ω associated with the Caratheodory metric. The metric enables us to obtain some known results from a unified perspective and also leads some new results. In particular, our results are valid for the Cartan domains. When Ω = B N , the Schwarz–Pick estimates of high order reduces to that of Liu and Chen.