Bloch constants in one and several variables

We give covering theorems in one variable for holomorphic functions on the unit disc with k-fold symmetry. In the case of convex maps we give a generalization, shown to us by D. Minda, to the case where a₂ = ... = a_k = 0. In several variables we determine the Bloch constant (equivalently the Koebe constant) for convex maps of B_n with k-fold symmetry, k ≥ 2. We also estimate and in some cases compute the Bloch constant for starlike maps of B_n with k-fold symmetry. We compare the Bloch constant with the Koebe constant for such maps and determine values of n and k for which equality holds.