A Note on the Converse Sendov Problem

Abstract

For a polynomial of degree $n$ whose zeros lie in the closed unit disk, we determine the largest possible distance from a prescribed critical point of modulus $r$ to the nearest zero. If $n$ is even, the sharp radius is $\sqrt{1-r^2}$; if $n$ is odd, the sharp radius is strictly smaller for $r\in(0,1)$ and depends on $n$. Equality cases are also determined. The proof is based on the logarithmic-derivative identity and elementary geometric considerations.