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

Let $K\subset \mathbb{C}$ be a convex compact set, and let ${\mathrm{\Pi }}_n(K)$ be the class of polynomials of exact degree $n$, all of whose zeros lie in $K$. The Turán type inverse Markov factor is defined by $M_n(K)={\inf }_{P\in {\mathrm{\Pi }}_n(K)}\left(\mathrm{\parallel }{P}^{\mathrm{\prime }}{\mathrm{\parallel }}_{C(K)}\mathrm{/}\mathrm{\parallel }P{\mathrm{\parallel }}_{C(K)}\right)$. A combination of two well-known results due to Levenberg and Poletsky (2002) and Révész (2006) provides the lower bound $M_n(K)\ge c(wn\mathrm{/}{d}^2+\sqrt{n}\mathrm{/}d)$, $c:=0.00015$, where $d>0$ is the diameter of $K$ and $w\ge 0$ is the minimal width (the smallest distance between two parallel lines between which $K$ lies). We prove that this bound is essentially sharp, namely, $M_n(K)\le 28(wn\mathrm{/}{d}^2+\sqrt{n}\mathrm{/}d)$ for all $n,w,d$.