The Gauss-Lucas theorem relates the location of zeros of a polynomial to the location of zeros of its derivative:

Suppose $f(z)\in \mathbb{C}[z]$ is a non-constant polynomial with roots $\alpha_1,\ldots,\alpha_n$, and let $$K = K(\alpha_1,\ldots,\alpha_n) \subset \mathbb{C}$$ denote the convex hull of these points. Then all roots of $f'(z)$ lie inside $K$.

If we instead consider a polynomial with real coefficients, Rolle's theorem gives a stronger condition on the location of zeros of the derivative:

Suppose $f(x)\in\mathbb{R}[x]$ is a non-constant polynomial with real roots $\alpha_1\leq \cdots \leq \alpha_n$, counted with multiplicity. Then for any $i < j$, the closed interval $$ I = [\alpha_i, \alpha_j] \subset \mathbb{R} $$ contains some root of $f'(x)$.

The second statement in "local" in the sense that knowing only two roots of $f$ gives us some information about where the roots of $f'$ lie. In the first statement, knowing a subset of the roots will not in general determine the convex hull.

Question: For $f(z)\in\mathbb{C}[z]$, is there any information we get about the location of roots of $f'(z)$ if we know only the locations of (say) 3 non-collinear* roots of $f(z)$?

Some guesses which are **false**:
Given three roots $\alpha_1, \alpha_2, \alpha_3$ of $f(z)$,

$f'(z)$ must contain a root inside the triangle spanned by $\alpha_1, \alpha_2, \alpha_3$.

$f'(z)$ must contain a root inside the circle passing through $\alpha_1, \alpha_2, \alpha_3$.

*as suggested by Gerry Myerson, the case when $\alpha_i$ are collinear may require separate analysis. But even this case is unresolved up to my understanding.

slightlyrelated question which you may find interesting which I asked on this site some time ago: mathoverflow.net/questions/189245/… $\endgroup$ – Trevor J Richards Mar 22 '17 at 21:407more comments