Lucas' theorem refined

We prove a refined version of the classical Lucas' theorem: if p is a polynomial with zeros a₁,...,a_n+1 all having modulus one and φ is the Blaschke product whose zeros are those of the derivative p′, then the compression of the shift S(φ) has its numerical range circumscribed about by the (n + 1)-gon a₁...a_n+1 with tangent points the midpoints of the n + 1 sides of the polygon. This is proved via a special matrix representation of S(φ) and is a generalization of the known case for n = 2.