The maximal range problem for a bounded domain

Let Ω⊂CΩ⊂C be a bounded domain such that 0∈Ω0∈Ω. Denote by Pn,n∈N≔{1,2,…}, the set of all complex polynomials of degree at most nn. Let Pn(Ω)≔{p∈Pn:p(0)=0,p(D)⊂Ω}, where D≔{z:|z|<1}. We relate the maximal polynomial rangeΩn≔⋃p∈Pn(Ω)p(D) to the geometry of ΩΩ.