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.