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

  • F. G. Abdullayev
  • N.D. Aral
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.


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.


  1. 1.
    J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, RI (1960).zbMATHGoogle Scholar
  2. 2.
    E. Hille, G. Szegö, and J. D. Tamarkin, “On some generalization of a theorem of A. Markoff,” Duke Math. J., 3, 729–739 (1937).MathSciNetCrossRefGoogle Scholar
  3. 3.
    O. Lehto and K. I. Virtanen, Quasiconformal Mapping in the Plane, Springer, Berlin (1973).Google Scholar
  4. 4.
    S. Rickman, “Characterisation of quasiconformal arcs,” Ann. Acad. Sci. Fenn., Ser. A. Math., 395, 1–30 (1966).MathSciNetGoogle Scholar
  5. 5.
    F. G. Abdullayev, “On some properties of orthogonal polynomials over an area in domains of the complex plane. III,” Ukr. Math. J., 53, No. 12, 1934–1948 (2001).CrossRefGoogle Scholar
  6. 6.
    F. G. Abdullayev, “The properties of the orthogonal polynomials with weight having singularity on the boundary contour,” J. Comp. Anal. Appl., 6, No. 1, 43–59 (2004).MathSciNetzbMATHGoogle Scholar
  7. 7.
    V. V. Andrievskii, V. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions on Complex Plane, World Federation, Atlanta (1995).Google Scholar
  8. 8.
    L. V. Ahlfors, Lectures on Quasiconformal Mappings, van Nostrand, Princeton, NJ (1966).zbMATHGoogle Scholar
  9. 9.
    N. Stylianopoulos, “Fine asymptotics for Bergman orthogonal polynomials over domains with corners,” CMFT, 2009, Ankara (2009).Google Scholar
  10. 10.
    V. V. Andrievskii, “Constructive characterization of harmonic functions in domains with quasiconformal boundary,” in: Quasiconformal Continuation and Approximation by Functions in a Set of the Complex Plane [in Russian], Kiev (1985).Google Scholar
  11. 11.
    F. G. Abdullayev, Ph.D. Thesis, Donetsk (1986).Google Scholar
  12. 12.
    F. G. Abdullayev, “Uniform convergence of generalized Bieberbach polynomials in regions with nonzero angles,” Acta Math. Hung., 77, No. 3, 223–246 (1997).MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

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

Personalised recommendations