Description
In this talk we consider asymptotically sharp extensions of
the classical Bernstein and Markov inequalities for rational functions
on Jordan arcs and curves. 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 the
question. In the proofs key roles are played by Borwein-Erdelyi
inequality on the unit circle, Gonchar-Grigorjan type estimate of the
norm of the holomorphic part of a meromorphic functions, Totik's
construction of fast decreasing polynomials, and conformal mappings.
Also we discuss Bernstein-type inequalities for polynomials and
rational functions with certain restrictions on the interval [-1,1]
and a circular arc.
This presentation is based on a joint work with V. Dubinin, B. Nagy,
and V. Totik.