On the Bernstein- and Markov-type inequalities

In this talk, we consider asymptotically sharp extensions of the classical Bernstein and Markov inequalities for rational functions on Jordan arcs and curves. Also, we discuss higher-order inequalities for trigonometric polynomials. The asymptotically sharp constants there can be expressed via the normal derivatives of certain Green’s functions with poles at the poles of the rational functions in question. In the proofs key roles are played by Borwein-Erdélyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions, fast decreasing polynomials, and conformal mappings.
This presentation is based on joint work with Béla Nagy and Vilmos Totik.