Bernstein-Nikol’skii-Markov-type inequalities for algebraic polynomials in aweighted Lebesgue space in regions with cusps

Abstract

: In this paper, we study Bernstein-Nikol’skii-Markov type inequalities for arbitrary algebraic polynomials with respect to a weighted Lebesgue space, where the weight functions have some singularities on a given contour. We consider curves which can contain a finite number of exterior and interior corners with power law tangency of the boundary arcs at those points where the weight functions have both zeros and poles of finite order. The estimates are given for the growth of the module of derivatives for algebraic polynomials on the closure of a region bounded by a given curve, depending on the behavior of weight functions, on the property of curve, and on the degree of contact of the boundary arcs, which form zero angles on the boundary.