  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 $0 \le \beta \le
1$ and make the stronger conjecture that $r(\beta) \le 1-(3/10)\beta(1-\beta)$.
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.
