LetPn= {pp{z) = z + a2z2+…+ anzn} and define μ(p) = min{|z|:p′(z)=0}pεPn. The paper deals with the following problem: Given ⊆Pn, how to obtain the best bound r(f) such thatpεPn, μ(p) ⩾ r(f) impliespεf. For classesfwith a certain test-property the extremal polynomials are of type [(z + r)n− rn]/(nrn-1). The main result states thatf⊆Pnhas the test-property, if the polynomials offcan be described by a criterion of Dieudonné-type. This result contains classical theorems of Alexander and Kakeya.