A boundary Schwarz lemma on the classical domain of type I

Abstract

Let R I (m, n) be the classical domain of type I in ℂ m×n with 1 ≤ m ≤ n. We obtain the optimal estimates of the eigenvalues of the Frechet derivative Df( $$\mathop Z\limits^ \circ $$ ) at a smooth boundary fixed point $$\mathop Z\limits^ \circ $$ of R I (m, n) for a holomorphic self-mapping f of R I (m, n). We provide a necessary and sufficient condition such that the boundary points of R I (m, n) are smooth, and give some properties of the smooth boundary points of R I (m, n). Our results extend the classical Schwarz lemma at the boundary of the unit disk Δ to R I (m, n), which may be applied to get some optimal estimates in several complex variables.