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

OpenAlex lv

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