2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68714Define $S(n,β)$ 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 $β$. For a polynomial $P$, define $|P|_β$ to be the distance between $β$ and the closest root of the derivative $P'$. Finally, define $r_n(β)=\sup \{|P|_β: P \in S(n,β) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(β) \le 1$. 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 $r_n(β) \sim 1 + C_1 (1-|β|) + C_2 (1-|β|)^2$ for $|β|$ near 1. We also consider some consequences of this approximation.22 pages, AMS-LaTeX, no figuresComplex Variables30C15text