Seeking a 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 conjecture that there is a constant $c>0$ so that $r(\beta )\le 1-c\beta (1-\beta )$ for all $\beta \in [0,1]$. We find such constants for complex polynomials of degree $2$ and $3$, for real polynomials of degree $4$, for all polynomials whose roots lie on a line, for all polynomials with exactly one distinct critical point, and when $\beta$ is sufficiently close to $1$. In addition, we show that experimental data suggests that $c\approx 0.233$.