An estimate for the Green's function

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)3067-3074
JournalProceedings of the AMS
StatePublished - Sep 1 2014

Fingerprint Dive into the research topics of 'An estimate for the Green's function'. Together they form a unique fingerprint.

Cite this