On a polynomial lemma of Andrievskii (Q1086697)

The lemma in question is as follows. Let G be a region bounded by a quasiconformal Jordan curve, and let \(0\in G\). Then for every polynomial P of degree \(n\geq 2\) with \(P(0)=0\), we have \[ \max \{| P(z)|: z\in \bar G\}\leq c(G)\sqrt{\log n}[\iint _{G}| P'(z)| ^ 2db_ z]^{1/2}, \] where c(G) depends on G only. A new and simpler proof of this lemma is given, the assumption about \(\partial G\) is relaxed, and the lemma is extended to rational functions with poles off Ḡ.