let $f(z)\in\mathbf{C}[z]$ be a nonzero polynomial. by the Gauss-Lucas theorem we have that $$\text{conv}(\{\text{roots of }f\})\subseteq\bigcap_{F'=f}\text{conv(\{roots of F\})}$$
Question. can the above inclusion be proper?
Example. suppose $f(z)=(z-\alpha)(z-\beta)$ has degree two and two distinct roots. pick an antiderivative $F'(z)=f(z)$ with $F(\alpha)=0$. then $\alpha$ is a double root of $F$ and so $F(z)=c(z-\alpha)^2(z-\gamma)$. by Gauss-Lucas, we know that $[\alpha,\beta]\subseteq [\alpha,\gamma]$. therefore $\gamma$ is on the ray from $\alpha$ to $\beta$. doing the same thing but replacing $\beta$ with $\alpha$, we get $G'=f$ with $\text{conv(\{root of G\})}=[\delta,\beta]$ for some $\delta$ on the ray from $\beta$ to $\alpha$. in total, the inclusion is an equality.