An apparently easy
problem due to Ilyeff states: If all zeros z1,z2,⋯,zn of a complex polynomial
P(z) lie in |z|≦ 1 then there is always a zero of P′(z) in each of the disks
|z −zj|≦ 1,j = 1,⋯,n. If true, the conjecture is best possible as one can see from the
example P(z) = zn− 1. In full generality the conjectured result was proved only for
polynomials of degree ≦ 4. In this paper the conjecture is proved for quintics and
extensions of earlier results are obtained for zeros of higher derivatives of polynomials
having multiple roots.
