The best possible quadratic refinement of Sendov's conjecture

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial’s derivative. If we define $r(\beta )$ to be the greatest possible distance between $\beta$ and the closest root of the derivative, then Sendov’s conjecture claims that $r(\beta )\le 1$. In this paper, we assume (without loss of generality) that $0 \le \beta \le 1$andmakethestrongerconjecturethat$r(\beta) \le 1-(3/10)\beta(1-\beta)$. We prove this new conjecture for all polynomials of degree 2 or 3, for all real polynomials of degree 4, and for all polynomials of any degree as long as all their roots lie on a line or $\beta$ is sufficiently close to 1.