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

Abdullayev F., Imashkyzy M.

Analysis and Mathematical Physics, vol.15, no.3, 2025 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 15 Issue: 3
  • Publication Date: 2025
  • Doi Number: 10.1007/s13324-025-01055-9
  • Journal Name: Analysis and Mathematical Physics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Keywords: Algebraic polynomial, Asymptotically conformal curves, Bernstein-Markoff inequality, Quasicircle, Quasiconformal mapping, Walsh inequality
  • Uşak University Affiliated: Yes

Abstract

In this paper, we study the growth of the m-th (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 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.