On the precise form of the inverse Markov inequality for convex sets

Let $\Pi_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in a convex compact set $K\subset \mathbb{C}$. The Turán type inverse Markov factor $M_n(K)$ is defined by $M_n(K)=\inf_{P\in \Pi_n(K)} \left(\|P'\|_{C(K)}/\|P\|_{C(K)}\right)$. Extending two well-known results due to Levenberg and Poletsky (2002) and Révész (2006), we obtain (up to a constant factor) the precise form of $M_n(K)$ in terms of $n$, $d$ and $w$, where $d>0$ is the diameter and $w\ge 0$ is the minimal width of $K$.