TY - JOUR
T1 - An estimate for the Green’s function
AU - Solynin, Alexander Yu
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 -