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Abstract

Let $E$ be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of $\mathbb{C}_\infty\setminus E$ with pole at infinity and $M_n(E)=\sup \frac{\|P'\|_E}{\|P\|_E}$ where the {supremum} is taken over all polynomials $P|_E\not\equiv 0$ of degree at most $n$, and $\|f\|_E=\sup \{|f(z)| : z\in E\}$. The paper deals with recent results concerning a connection between the smoothness of $g_E$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence $\{M_n(E)\}_{n=1,2,\dots}$. Some additional conditions are given for special classes of sets.