On the growth of $m$-th $(m\geq 1)$ derivatives of algebraic polynomials in the whole complex plane for the domains with cusps

Abstract

UDC 517.51; 517.53

We study the growth of  $m$-th $(m\geq 1)$ derivatives of an arbitrary algebraic polynomial in bounded and unbounded domains with piecewise Dini-smooth boundary that have interior and exterior zero angles in weighted Lebesgue spaces. First, we study the growth of the $m$-th derivatives of an arbitrary algebraic polynomial in unbounded domains of the complex plane and then obtain estimates for the growth of the $m$-th derivatives of this polynomial in the closure of this domain. Combining both estimates, we find estimates for the growth of the $m$-th derivatives of an arbitrary algebraic polynomial in the whole complex plane, depending on the geometry of the given domain and the weight function.