Can the roots of the derivative of the polynomial in complex variable be as close as we want them to be from the roots of the polynomial itself?

Question

The (probably) famous Gauss-Lucas theorem states that the roots of the derivative $P'(z)$ are contained in the convex hull of the roots of $P(z)$, where $P(z)$ is complex variable polynomial.

I am interested here in could it be the case that we always have some polynomial of any degree (except $1$) $P(z)$ such that some root of its derivative is "at a small as we want distance" from some root of $P(z)$.

To be more precise, here is the statement of the question:

Is it true that for every $\varepsilon>0$ and for every $n\in \mathbb N \setminus \{1\}$ there exists polynomial $P(z)$ in complex variable of degree $n$ with $n$ different roots such that there is root $z_a$ of $P'(z)$ and root $z_b$ of $P(z)$ which are such that we have $|P'(z_a)-P(z_b)|<\varepsilon$

Answer 1

Let $P(z) = (z-\epsilon)(z-2\epsilon)\cdots (z-n\epsilon).$ Using the mean value theorem we can see that every root of $P'$ is less than $\epsilon$ away from some root of $P.$

Answer 2

Let $P(z)=(z-\epsilon)(z+\epsilon)\prod_{i=1}^{n-2}(z-i)$. This is a perturbation of the polynomial $z^2\prod_{i=1}^{n-2}(z-i)$, removing the double root.