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

Article

First Online:

- 65 Downloads
- 1 Citations

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 where σ is a two-dimensional Lebesgue measure. Let 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.

*A*_{ p }(*G*);*p*> 0; we denote a class of functions*f*analytic in*G*and satisfying the condition$$ \left\| f \right\|_{Ap}^p(G): = \int\limits_G {{{\left| {f(z)} \right|}^p}d{\sigma_z} < \infty, } $$

(*)

*P*_{ n }(*z*) be arbitrary algebraic polynomial of degree at most*n*: The well-known Bernstein–Walsh lemma says that$$ \left\| {{P_n}(z)} \right\| \leq {\left| {\Phi (z)} \right|^{n + 1}}{\left\| {{P_n}} \right\|_{C\left( {\bar{G}} \right)}},\quad z \in \Omega . $$

(**)

## 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.

## References

- 1.J. L. Walsh,
*Interpolation and Approximation by Rational Functions in the Complex Domain*, American Mathematical Society, Providence, RI (1960).zbMATHGoogle Scholar - 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.O. Lehto and K. I. Virtanen,
*Quasiconformal Mapping in the Plane*, Springer, Berlin (1973).Google Scholar - 4.S. Rickman, “Characterisation of quasiconformal arcs,”
*Ann. Acad. Sci. Fenn., Ser. A. Math.*,**395**, 1–30 (1966).MathSciNetGoogle Scholar - 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.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.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.L. V. Ahlfors,
*Lectures on Quasiconformal Mappings*, van Nostrand, Princeton, NJ (1966).zbMATHGoogle Scholar - 9.N. Stylianopoulos, “Fine asymptotics for Bergman orthogonal polynomials over domains with corners,”
*CMFT, 2009*, Ankara (2009).Google Scholar - 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.F. G. Abdullayev,
*Ph.D. Thesis*, Donetsk (1986).Google Scholar - 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