On the uniform convergence of the generalized Bieberbach polynomials in regions with K-quasiconformal boundary (Q2744436)

Let \(G\) be a bounded domain on the complex plane such that the boundary \(L:= \partial G\) is a \(K\)-quasiconformal Jordan curve. Given a point \(z_o \in G\), let \(\varphi\) be the conformal mapping of \(G\) onto the disk \(\{|w|<r\}\) such that \(\varphi(z_o) =0\) and \(\varphi'(z_o) =1\). Given \(p>0\), let \(\varphi_p\) be the single-valued branch of \([\varphi'(\cdot)]^{\frac 2p}\) such that \(\varphi_p(z_o) = 1\). Let \(\mathcal P_n\) be the set of all polynomials \(p\) of degree at most \(n\) such that \(p(z_o)=0\) and \(p'(z_o) =1\). Then there exists a unique (generalized Bieberbach) polynomial \(\pi_{n,p} \in \mathcal P_n\) such that \(\|\varphi_p -\pi_{n,p}\|_{L_p(G)}\leq \|\varphi_p -p\|_{L_p(G)}\) for all \(p\in \mathcal P_n\). NEWLINENEWLINENEWLINEMain theorem. If \(p>2-\frac {K^2+1}{2K^4}\) then there exists a positive constant \(C\) such that NEWLINE\[NEWLINE\|\varphi_p-\pi_{n,p}\|_{\mathcal C(G)} \leq Cn^{-\gamma +\varepsilon}NEWLINE\]NEWLINE for all \(n\geq 2\) and \(\varepsilon > 0\), where \(\gamma\) is expressed explicitly by \(p\) and \(K\).