Let denote the unit ball in , and let be the class of all holomorphic mappings . The authors define the Bergman metric for the unit ball as
where denotes the Hermitian scalar product in and .
For , , and , the Fréchet derivative of at of order is defined by
The main result of the paper is the following.
Let , , , . Then
It is a generalization of the classical Schwarz-Pick lemma (take ) and the result by H. H. Chen [Sci. China, Ser. A 46, No. 6, 838–846 (2003; Zbl 1097.47509)] (take ).
As a consequence of the main result, the authors obtain a Schwarz-Pick estimate for partial derivatives of a mapping , which, in case , is much better than the one obtained by Z. H. Chen and Y. Liu [Acta Math. Sin., Engl. Ser. 26, No. 5, 901–908 (2010; Zbl 05795520)].