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
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
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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 3, pp. 291–302, March, 2011.
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Abdullayev, F.G., Aral, N. On Bernstein–Walsh-type lemmas in regions of the complex plane. Ukr Math J 63, 337–350 (2011). https://doi.org/10.1007/s11253-011-0507-y
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DOI: https://doi.org/10.1007/s11253-011-0507-y