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The Schwarz-Pick lemma of high order in several variables. (English) Zbl 1208.32001

Let 𝔹 n denote the unit ball in n , and let Ω n,m be the class of all holomorphic mappings f:𝔹 n 𝔹 m . The authors define the Bergman metric for the unit ball 𝔹 n as

H n (z;β):=(1-z 2 )β 2 +|β,z| 2 (1-z 2 ) 2 ,z𝔹 n ,β n ,

where , denotes the Hermitian scalar product in n and z:=(z,z) 1/2 .

For fΩ n,m , k, and z𝔹 n , the Fréchet derivative of f at z of order k is defined by

D k (f,z,β):= |α|=k k! α! k f(z) z 1 α 1 z n α n β α ,β n ·

The main result of the paper is the following.

Let fΩ n,m , k, z𝔹 n , β n {0}. Then

H m (f(z);D k (f,z,β))(k!) 2 1+|β,z| ((1-z 2 )β 2 +|β,z| 2 ) 1/2 2(k-1) (H n (z;β)) k ·

It is a generalization of the classical Schwarz-Pick lemma (take n=m=k=1) and the result by H. H. Chen [Sci. China, Ser. A 46, No. 6, 838–846 (2003; Zbl 1097.47509)] (take k=1).

As a consequence of the main result, the authors obtain a Schwarz-Pick estimate for partial derivatives of a mapping fΩ n,m , which, in case m=1, 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 1243.32002)].

32A10Holomorphic functions (several variables)
32F45Invariant metrics and pseudodistances