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 C be a finite region bounded by a Jordan curve L := G,Ω := ext G (respect to C), Δ := {z : |z| > 1}; w = Φ(z) be the univalent conformal mapping of Ω ont Φ normalized by Φ(∞) = ∞,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) := _G |f(z)|^p d σ_z < ∞,(∗) 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 – Walsh lemma says that P_n(z)k ≤ |Φ(z)|^n+1 ||P_n||_C(G), z Ω. (∗∗) Firstly, we study the estimation problem (∗∗) for the norm (∗). Secondly, we continue studying the estimation (∗∗) when we replace the norm ||P_n||_C(G) by ||P_n||_A_2(G) for some regions of complex plane.