Let G ⊂ ℂ be a finite region bounded by a Jordan curve L := ∂G, let \( \Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G} \) (with respect to \( \overline{\mathbb{C}} \)), let Δ := {w : |w| > 1}, and let w = Φ(z) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let h(z) be a weight function and let A p (h,G), p > 0 denote a class of functions f analytic in G and satisfying the condition
where σ is a two-dimensional Lebesgue measure.
Moreover, let P n (z) be an arbitrary algebraic polynomial of degree at most n ∈ ℕ. The well-known Bernstein–Walsh lemma states that
In this present work we continue the investigation of estimation (*) in which the norm \( {\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)} \) is replaced by \( {\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0 \), for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 5, pp. 579–597, May, 2014.
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Abdullayev, F.G., Özkartepe, P. On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary. Ukr Math J 66, 645–665 (2014). https://doi.org/10.1007/s11253-014-0962-3
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DOI: https://doi.org/10.1007/s11253-014-0962-3