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

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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
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