On Bernstein-Walsh-type lemmas in regions of the complex plane

ABDULLAYEV F., Aral N.

Ukrainian Mathematical Journal, vol.63, no.3, pp.337-350, 2011 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 63 Issue: 3
  • Publication Date: 2011
  • Doi Number: 10.1007/s11253-011-0507-y
  • Journal Name: Ukrainian Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.337-350
  • Uşak University Affiliated: No

Abstract

Let G ⊂ ℂ be a finite region bounded by a Jordan curve L:= ∂G, let Ω:= ext Ḡ (with respect to ℂ̄), Δ:= {z:{pipe}z(pipe} > 1}, and let w = Φ(z) be a univalent conformal mapping of Ω onto Δ normalized by Φ(∞) = infin, Φ′(∞) > 0. By Ap(G); p > 0, we denote a class of functions f analytic in G and satisfying the condition where σ is a two-dimensional Lebesgue measure. Let Pn(z) be arbitrary algebraic polynomial of degree at most n. The well-known Bernstein-Walsh lemma says that, First, we study the problem of estimation (**) for the norm (*). Second, we continue studying estimation (**) by replacing the norm {double pipe}Pn{double pipe}C(Ḡ) with {double pipe}Pn{double pipe}A2(Ḡ) for some regions of the complex plane. © 2011 Springer Science+Business Media, Inc.