LAUSR

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