On the Bernstein - Walsh-type lemmas in regions of the complex plane - Abdullayev, F. G.; Aral, N. D. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
Let $G \subset C$ be a finite region bounded by a Jordan curve $L := \partial G,\quad Ω := \text{ext} \; \overline{G}$ (respect to $\overline{C}$), $Δ := \{z : |z| > 1\}; \quad w = Φ(z)$ be the univalent conformal mapping of $Ω$ ont $Φ$ normalized by $Φ(∞) = ∞,\quad Φ'(∞) > 0$.Let $A_p(G),\; p > 0$, denote the class of functions $f$ which are analytic in $G$ and satisfy the condition$$||f||^p_{A_p(G)} := \int\int_G |f(z)|^p d σ_z < ∞,\quad (∗)$$where $σ$ is a two-dimensional Lebesque measure.Let $P_n(z)$ be arbitrary algebraic polynomial of degree at most $n$. The well-known Bernstein – Walshlemma says that$$P_n(z)k ≤ |Φ(z)|^{n+1} ||P_n||_{C(\overline{G})}, \; z \in Ω. \quad (∗∗)$$Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying theestimation (∗∗) when we replace the norm $||P_n||_{C(\overline{G})}$ by $||P_n||_{A_2(G)}$for some regions of complex plane.