Sergei Kalmykov, Far Eastern Federal University, Russia Asymptotically sharp
Bernstein type inequalities for polynomials on analytic arcs Abstract Bernstein (or Riesz) type polynomial inequalities are well known. On
the complex plane Bernstein inequality was extended to compact sets bounded by
smooth Jordan curves and the asymptotically sharp constant can be expressed
via the normal derivative of Green’s function [1]. There is a general
conjecture by Totik for Jordan arcs that the
asymptotically sharp Bernstein factor can be expressed as the maximum of the
two normal derivatives of Green’s function. It was proved for the subarcs [2], and later, for general subsets of the unit
circle [3]. In this talk we consider the case of an arbitrary analytic Jordan
arc. The proofs of the main results (see [4]) are based on facts from
potential and interpolation theories, Borwein-Erdélyi
inequality for derivative of rational functions on the unit circle, Gonchar-Grigorjan estimate of the norm of the holomorphic
component of meromorphic function, Totik’s construction of fast decreasing polynomials, and
conformal mappings. This is based on a joint
work with Béla Nagy. 1. B.
Nagy, V. Totik, Sharpening of Hilbert’s lemniscate theorem, Journal d’Analyse
Mathematique, 2005, 96, 191-223. 2. B.
Nagy, V. Totik, Bernstein’s inequality for
algebraic polynomials on circular arcs, Constructive Approximation, 2013,
37(2), 2013, 223-232. 3. B.
Nagy, V. Totik, Riesz-type
inequalities on general sets, JMAA, 2014, 416(1), 344-351. 4. S.I.
Kalmykov, B. Nagy, Polynomial and rational
inequalities on Jordan arcs and domains, JMAA, 2015, 430(2), 874-894. |