2024/11/11 by Zhang, Teng
#30C10 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.07105
Let \(F(z) = ∏k=1n(z - zk)\) be a monic complex polynomial of degree \(n\) whose zeros satisfy \(max1 ≤ k ≤ n |zk| ≤ 1\). Pawłowski [Trans. Amer. Math. Soc. 350(11) (1998)] considered the radius \(γn\) of the smallest disk, centered at the centroid \((1)/(n)∑k=1n zk\), containing at least one critical point of \(F\), establishing the bound γn ≤ \frac2 n(1)/(n-1)n(2)/(n-1) + 1. In this paper, inspired by the spirit of Borcea's variance conjectures and leveraging the classical Schoenberg inequality, we significantly refine Pawłowski's estimate by proving succinctly and elegantly that γn ≤ √((n - 2)/(n - 1)). This result also represents a rare and noteworthy application of Schoenberg's inequality to the geometry of polynomial critical points.