The maximal range problem for a quasidisk Original Research Article

Let G⊂C and D⊂C be simply connected domains such that 0∈G∩D. Denote by Pn, n∈N≔{1,2,…}, the set of all complex polynomials of degree at most n. LetPn(G,D)≔{p∈Pn: p(0)=0,p(G)⊂D}.Our main purpose is to find how large, i.e., how close to D, the “maximal polynomial range”Dn(G)≔⋃p∈Pn(G,D) p(G)can be. We consider G to be a quasidisk and D to be an arbitrary domain whose boundary consists of more than two points.