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

Abstract

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.