$L_p$-convergence of Bieberbach polynomials

The author proves the estimate
$$ \|p_n-\omega\|_{L_p(G)}\leqslant\frac{c_{p,\varepsilon}}{(\ln\ln n)^{\frac18(1-\theta)-\varepsilon}} $$
where $G\subset\mathbf C$; $p_n(z)\equiv p_n$ are Bieberbach polynomials for the pair $(G,0)$; $\omega(0)=0$, $\omega'(0)=1$, $\omega(z)=\omega\colon G\to\{z;|z|<R\}$ is a conformal mapping, $\varepsilon>0$, $p\in[1,\infty)$, $0<\theta\equiv\theta(G)<q$. The boundary $\partial G$ is more general than Lipschitz.
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