Turán type reverse Markov inequalities for compact convex sets

For a compact convex set K⊂C the well-known general Markov inequality holds asserting that a polynomial p of degree n must have ∥p′∥⩽c(K)n2∥p∥. On the other hand for polynomials in general, ∥p′∥ can be arbitrarily small as compared to ∥p∥.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds ∥p′∥⩾(n/2)∥p∥ for the unit disk D and for the unit interval I≔[-1,1]. Although partial results provided general lower estimates of 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 bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K ∥p′∥⩾c(K)n∥p∥ holds true, while ∥p′∥⩽C(K)n∥p∥ occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.