Reverse Markov Inequality

Let K be a compact convex set in C . For each point z0 ∈ ∂K and each holomorphic polynomial p = p(z) having all of its zeros in K , we prove that there exists a point z ∈ K with |z − z0| ≤ 20 diamK/ √ deg p such that |p′(z)| ≥ (deg p) 1/2 20(diamK) |p(z0)|; i.e., we have a pointwise reverse Markov inequality. In particular, ‖p‖K ≥ (deg p) 20(diamK) ‖p‖K .