TY - JOUR

T1 - An estimate for the Green’s function

AU - Solynin, Alexander Yu

N1 - Publisher Copyright:
© 2014 American Mathematical Society.

PY - 2014/9/1

Y1 - 2014/9/1

N2 - Let K be a continuum on ℂ and let gΩ(K)(z,∞) be the Green’s function of Ω(K) = ℂ¯\K. In a recent paper, V. Totik proved that gΩ(K)(z0,∞) ≤ C dist(z0,∞)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 C. In this note we prove a sharp version of Totik’s inequality and discuss a conjectural sharp lower bound for gΩ(K)(z0,∞).

AB - Let K be a continuum on ℂ and let gΩ(K)(z,∞) be the Green’s function of Ω(K) = ℂ¯\K. In a recent paper, V. Totik proved that gΩ(K)(z0,∞) ≤ C dist(z0,∞)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 C. In this note we prove a sharp version of Totik’s inequality and discuss a conjectural sharp lower bound for gΩ(K)(z0,∞).

KW - Extremal problem

KW - Green’s function

UR - http://www.scopus.com/inward/record.url?scp=84924777564&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2014-12018-1

DO - 10.1090/S0002-9939-2014-12018-1

M3 - Article

AN - SCOPUS:84924777564

VL - 142

SP - 3067

EP - 3074

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 9

ER -