A quadratic approximation to the Sendov radius near the unit circle

Define $S(n,\beta)$ to be the set of complex polynomials of degree $n\ge2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $\vert P\vert _\beta$ to be the distance between $\beta$ and the closest root of the derivative $P'$. Finally, define $r_n(\beta)=\sup \{ \vert P\vert _\beta : P \in S(n,\beta) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta)\le1$.

In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\beta)\sim1+C_1(1-\vert\beta\vert)+C_2(1-\vert\beta\vert)^2$ for $\vert\beta\vert$near $1$. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.