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

  • F. G. Abdullayev
  • N.D. Aral
Article
Let \( G \subset {\mathbb C} \) be a finite region bounded by a Jordan curve \( L: = \partial G \), let \( \Omega : = {\text{ext}}\bar{G} \) (with respect to \( {\overline {\mathbb C}} \)), \( \Delta : = \left\{ {z:\left| z \right| > 1} \right\} \), and let \( w = \Phi (z) \) be a univalent conformal mapping of Ω onto Δ normalized by \( \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 \). By A p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition
$$ \left\| f \right\|_{Ap}^p(G): = \int\limits_G {{{\left| {f(z)} \right|}^p}d{\sigma_z} < \infty, } $$
(*)
where σ is a two-dimensional Lebesgue measure. Let P n (z) be arbitrary algebraic polynomial of degree at most n: The well-known Bernstein–Walsh lemma says that
$$ \left\| {{P_n}(z)} \right\| \leq {\left| {\Phi (z)} \right|^{n + 1}}{\left\| {{P_n}} \right\|_{C\left( {\bar{G}} \right)}},\quad z \in \Omega . $$
(**)
First, we study the problem of estimation (**) for the norm (*). Second, we continue studying estimation (**) by replacing the norm \( {\left\| {{P_n}} \right\|_{C\left( {\bar{G}} \right)}} \) with \( {\left\| {{P_n}} \right\|_{{A_2}(G)}} \) for some regions of the complex plane.

Keywords

Complex Plane Orthogonal Polynomial Quasiconformal Mapping Jordan Curve Univalent Conformal Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • F. G. Abdullayev
    • 1
  • N.D. Aral
    • 1
  1. 1.MersinTurkey

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