On the behavior of \(m-th\) derivatives of polynomials in bounded and unbounded regions without zero angles in weighted Lebesgue spaces

Abstract

In this paper, we study the growth of the \(m-th\) (\(m\ge 1\)) derivatives of an arbitrary algebraic polynomial in weighted Lebesgue spaces over the whole complex plane. We first study the growth of the \(m-th\) derivatives of an arbitrary algebraic polynomial over unbounded regions of the complex plane, and then we obtain estimates for the growth of the \(m-th\) derivatives of this polynomial over the closure of the given region. Combining both estimates, we find estimates for the growth of the \(m-th\) derivatives of an arbitrary algebraic polynomial over the whole complex plane.