A quadratic approximation to the Sendov radius near the unit circle

Abstract

Define $S(n,\beta)$ to be the set of complex polynomials of degree $n \ge 2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $|P|_\beta$ to be the distance between $\beta$ and the closest root of the derivative $P'$. Finally, define $r_n(\beta)=\sup \{|P|_\beta : P \in S(n,\beta) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta) \le 1$. 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 $r_n(\beta) \sim 1 + C_1 (1-|\beta|) + C_2 (1-|\beta|)^2$ for $|\beta|$ near 1. We also consider some consequences of this approximation.