Right order Turan-type converse Markov inequalities for convex domains on the plane

For a convex domain $K$ in the complex plane, the well-known general Bernstein-Markov inequality holds asserting that a polynomial $p$ of degree $n$ must have $\mathrm{\mid }\mathrm{\mid }{p}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }<c(K)n^2\mathrm{\mid }\mathrm{\mid }p\mathrm{\mid }\mathrm{\mid }$. On the other hand for polynomials in general, $\mathrm{\mid }\mathrm{\mid }{p}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }$ can be arbitrarily small as compared to $\mathrm{\mid }\mathrm{\mid }p\mathrm{\mid }\mathrm{\mid }$. The situation changes when we assume that the polynomials in question have all their zeroes in the convex body $K$. This was first investigated by Tur'an, who showed the lower bounds $\mathrm{\mid }\mathrm{\mid }{p}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }\ge (n\mathrm{/}2)\mathrm{\mid }\mathrm{\mid }p\mathrm{\mid }\mathrm{\mid }$ for the unit disk $D$ and $\mathrm{\mid }\mathrm{\mid }{p}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }>c\sqrt{n}\mathrm{\mid }\mathrm{\mid }p\mathrm{\mid }\mathrm{\mid }$ for the unit interval $I:=[-1,1]$. Although partial results provided general lower estimates of lower order, as well as certain classes of domains with lower bounds of order $n$, it was not clear what order of magnitude the general convex domains may admit here. Here we show that for all compact and convex domains $K$ with nonempty interior and polynomials $p$ with all their zeroes in $K$ $||p’|| > c(K) n ||p||$holdstrue,while$||p’|| < C(K) n ||p||$occursforany$K$. Actually, we determine $c(K)$ and $C(K)$ within a factor of absolute numerical constant.