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 polynomials derivative. If we define $r(beta)$ to be the greatest possible distance between $beta$ and the closest root of the derivative, then Sendovs 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.
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