An estimate for the Green's function

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Abstract

Let $K$ be a continuum on ${\mathbb{C}}$ and let $g_{\Omega(K)}(z,\infty)$ be the Green's function of $\Omega(K)=\overline{{\mathbb{C}}}\setminus K$. In a recent paper, V.~Totik proved that $g_{\Omega(K)}(z_0,\infty)\le C\,dist(z_0,\infty)^{1/2}$ with some non-sharp constant $C$ depending only on the diameter of $K$. He also used this inequality to prove new results on polynomial approximation in $\mathbb{C}$. In this note we prove a sharp version of Totik's inequality and discuss a conjectural sharp lower bound for $g_{\Omega(K)}(z_0,\infty)$.
Original language English 3067-3074 Proceedings of the AMS Published - Sep 1 2014

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