ON ILYEFFS CONJECTURE

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 Pr(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. © 1969 by Pacific Journal of Mathematics.