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

Abstract

Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad \Omega := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $\Delta := \{z : |z| > 1\}; \quad w = \Phi(z)$ be the univalent conformal mapping of $\Omega$ ont $\Phi$ normalized by $\Phi(\infty) = \infty,\quad \Phi'(\infty) > 0$. Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition $$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d \sigma_z < \infty,\quad (∗)$$ where $\sigma$ is a two-dimensional Lebesque measure. Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walsh lemma says that $$P_n(z)k ≤ |\Phi(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in \Omega. \quad (∗∗)$$ Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the estimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$ for some regions of complex plane.