Bloch Constants of Bounded Symmetric Domains

Let D1 and D2 be two irreducible bounded symmetric domains in the complex spaces V1 and V2 respectively. Let E be the Euclidean metric on V2 and h the Bergman metric on V1. The Bloch constant b(D1,D2) is defined to be the supremum of E(f ′(z)x, f ′(z)x) 1 2 /hz(x, x)1/2, taken over all the holomorphic functions f : D1 → D2 and z ∈ D1, and nonzero vectors x ∈ V1. We find the constants for all the irreducible bounded symmetric domains D1 and D2. As a special case we answer an open question of Cohen and Colonna.