A quantitative Gauss-Lucas theorem (Q2143066)

Let \(K\) be a convex subset of the complex plane and let \(K_{\varepsilon}\) be the \(\varepsilon\)-neighborhood of \(K\). For a degree-\(n\) polynomial \(P_n\), the Gauss-Lucas theorem states that if \(P_n\) has all zeros in \(K\) then the same is true for the zeros of the derivative \(P_n'\). In this paper, the following quantitative version of this classical result is proven: For any \(\varepsilon>0\), there is an \(\alpha_{\varepsilon}<1\) such that if \(P_n\) has \(k\geq\alpha_{\varepsilon}n\) zeros in \(K\) then \(P_n'\) has at least \(k-1\) zeros in \(K_{\varepsilon}\). The second main result consist in quantitative bounds for \(\alpha_{\varepsilon}\) in terms of \(\varepsilon\).