Covering theorem for p-valent functions with Montel's normalization

Let Mp(ω),p≥2 stand for the class of all holomorphic p-valent functions in the unit disk U={z:|z|<1} normalized by f(0)=0, f(ω)=ω, 0<ω<1. Let R(f) denote the Riemann surface obtained as an image of U under the mapping f from the class Mp(ω). We find the maximal value of ρ, for which any surface R(f), f∈Mp(ω), contains an open k-valent disk, k≤p, branching over the disk |w|<ρ.