ON THE SENDOV CONJECTURE FOR 6TH DEGREE POLYNOMIALS

The Sendov conjecture asserts that if p(z) = PI(k = 1)n(z - z(k)) is a polynomial with zeros \z(k)\ less-than-or-equal-to 1, then each disk \z - z(k)\ less-than-or-equal-to 1, (1 less-than-or-equal-to k less-than-or-equal-to n) contains a zero of p'(z). This conjecture has been verified in general only for polynomials of degree n = 2, 3, 4, 5. If p(z) is an extremal polynomial for this conjecture when n = 6, it is known that if a zero \z(j)\ less-than-or-equal-to lambda-6 = 0.626997... then \z - z(j)\ less-than-or-equal-to 1 contains a zero of p'(z). (The conjecture for n = 6 would be proved if lambda-6 = 1.) It is shown that lambda-6 can be improved to lambda-6 = 63/64 = 0.984375.