Turán-Erőd type inequalities in L^q norm

In 1939 Turán, as a certain converse to the well-known Bernstein and Markov inequalities, initiated the study of estimates of norms of derivatives of polynomials from below by the norm of polynomials. As adding a constant can increase the norm of the polynomial while leaving the derivative unchanged, any sound setup requires a normalization. Turán normalized the class of polynomials assuming that all the zeroes should belong to the fundamental set K, where the norm is taken. Turán obtained that under such normalization, we have for degree n polynomials the estimate that ||p'|| is at least n/2 ||p|| on the disk D, and at least $\sqrt{n}/6||p|| $ on the interval I=[-1,1]. Already in 1939 Erőd started to study the same type of inequalities on other domains of the complex plane, but the progress for general domains was exteremly slow - actually nothing in the whole XXth Century. The first general result for arbitrary compact convex sets is due to Levenberg and Poletsky in 2002.
We present our results on the maximum norm, which settled both the order of magnitude and the right dependence on geometry of convex domains. Then we pass on to the more intricate question of analogous estimates in the L^q norm on the boundary of K. Our proofs are all elementary, using much of the geometry of convex sets.