A boundary Schwarz Lemma for holomorphic mappings between unit balls of different dimensions

In this paper, we give a general boundary Schwarz lemma for holomorphic mappings between unit balls in any dimensions. It is proved that if the mapping $f\in C^{1+\alpha }$ at $z_0\in \mathrm{\partial }{\mathbb{B}}^n$ with $f(z_0)=w_0\in \partial \mathbb B^N$forany$n,N\geq 1$,thentheJacobianmatrix$J_f(z_0)$mapsthetangentspace$T_{z_0}(\partial \mathbb B^n)$to$T_{w_0}(\partial \mathbb B^N)$,andtheholomorphictangentspace$T^{(1,0)}{z_0}(\partial \mathbb B^n)$to$T^{(1,0)}{w_0}(\partial \mathbb B^N)$ as well.