Bernstein type inequality can be regarded as an extremal problem, i.e.
it can be rewritten as a problem where finding best or asymptotically
sharp constants is the question. Asymptotically minimal polynomials
minimize the supremum norm in an asymptotic sense. We discuss these
problems and conjectures connecting these two, different questions.
This is a joint work with Sergei Kalmykov and is in progress.