A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL

In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f is an element of P(D-n, B-N) is C1+alpha at z(0) subset of E-r subset of partial derivative D-n with f(0) = 0 and f(z(0)) = w(0) is an element of partial derivative B-N for any n, N >= 1, then there exist a nonnegative vector lambda(f) = (lambda(1), 0, ... , lambda(r), 0, ... , 0)(T) is an element of R-2n satisfying lambda(i) >= for -1/2(2n-1) for 1 <= i <= r such that (Df(z(0)'))(T) w(0)' = diag(lambda(f))z(0)', where z(0)' and w(0)' are real versions of z(0) and w(0), respectively.