Advertisement

# 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.

## References

1. 1.
J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, RI (1960).
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).
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).
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).
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).
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).
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).

## Copyright information

© Springer Science+Business Media, Inc. 2011

## Authors and Affiliations

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