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An estimate for the Green's function


Author: Alexander Yu. Solynin
Journal: Proc. Amer. Math. Soc. 142 (2014), 3067-3074
MSC (2010): Primary 30C75, 31A15
Published electronically: May 14, 2014
MathSciNet review: 3223363
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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 )$.


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Additional Information

Alexander Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409
Email: alex.solynin@ttu.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12018-1
Keywords: Green's function, extremal problem
Received by editor(s): August 1, 2012
Received by editor(s) in revised form: September 4, 2012, and September 11, 2012
Published electronically: May 14, 2014
Additional Notes: This research was supported by NSF grant DMS-1001882
Dedicated: In memory of Promarz M. Tamrazov, an excellent mathematician, a friend, and a wonderful person
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.