On Bernstein- and Markov-type inequalities on compact sets of the unit circle

First, we consider covering and distortion theorems for polynomials normalized on a union of circular arcs, and as corollaries, results on polynomials with additional constraints on zeros. Then, we discuss a special form of an asymptotically sharp Markov-type inequality on rather general compact sets on the unit circle. The talk is based on joint works with V. N. Dubinin and K. Konoplev.