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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
DOI: https://doi.org/10.1090/S0002-9939-2014-12018-1
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 )$.


References [Enhancements On Off] (What's this?)

  • [1] N. A. Lebedev, Printsip ploshchadei v teorii odnolistnykh funktsii, Izdat. ``Nauka'', Moscow, 1975 (Russian). MR 0450540 (56 #8834)
  • [2] Zeev Nehari, Conformal mapping, Dover Publications Inc., New York, 1975. Reprinting of the 1952 edition. MR 0377031 (51 #13206)
  • [3] George Pólya and Gabor Szegő, Problems and theorems in analysis. II, Theory of functions, zeros, polynomials, determinants, number theory, geometry; translated from the German by C. E. Billigheimer; reprint of the 1976 English translation. Classics in Mathematics, Springer-Verlag, Berlin, 1998. MR 1492448
  • [4] A. Yu. Solynin, Polarization and functional inequalities, Algebra i Analiz 8 (1996), no. 6, 148-185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 6, 1015-1038. MR 1458141 (98e:30001a)
  • [5] P. M. Tamrazov, Extremal conformal mappings and poles of quadratic differentials, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1033-1043 (Russian). MR 0235105 (38 #3417)
  • [6] Vilmos Totik, Christoffel functions on curves and domains, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2053-2087. MR 2574887 (2011b:30006), https://doi.org/10.1090/S0002-9947-09-05059-4

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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.

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