A boundary Schwarz lemma for holomorphic mappings from the polydisc to   the unit ball
  In this paper, we prove a general Schwarz lemma at the boundary for holomorphic mappings from the polydisc to the unit ball in any dimensions. For the special case of one complex variable, the obtained results give the classic boundary Schwarz lemma. 
Yang Liu, Zhihua Chen and Yifei Pan
math.CV