An estimate for the Green’s function

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 $g_{\Omega (K)}(z_0,\infty )$.