Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space

Abstract

Let C be a complex plane, C̄ := C∪ {∞}; G ⊂ C be a bounded Jordan region, with 0 ∈ G and the boundary L := ∂G be a simple closed rectifiable Jordan curve, Ω := C̄ r Ḡ = extL; ∆ := {w : |w| > 1}. Let w = Φ(z) be the univalent conformal mapping of Ω onto the ∆ normalized by Φ(∞) = ∞, limz→∞ Φ(z) z > 0. For t > 1, let us set Lt := {z : |Φ(z)| = t}, L1 ≡ L, Gt := intLt, Ωt := extLt. For z ∈ C and S ⊂ C let d(z, S) := dist(z, S) = inf{|ζ− z| : ζ ∈ S}. Let h(z) be a weight function defined in GR0 for some fixed R0 > 1 and let ℘n denote the class of arbitrary algebraic polynomials Pn(z) of degree at most n ∈ N := {1, 2, . . .}. For any p > 0 we denote