Uniform estimates for polynomial approximation in domains with corners

Let G⊂C be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γj, such that {zj}≔Γj-1∩Γj≠∅, j=1,…,l, where Γ0≔Γl. Denote by αjπ, 0<αj⩽2, the angles at zj's between the curves Γj-1 and Γj, exterior with respect to G. Let Φ be a conformal mapping of the exterior C⧹G¯ of G¯=G∪∂G onto the exterior of the unit disk, normed by Φ′(∞)>0. We assume that there is a neighborhood U of G¯, such that 0