Abstract
We study properties of a two-point version of the transfinite diameter of a set. By using relations obtained for its calculation, we prove a two-point version of the well-known Pólya theorem on an estimate from above for the Hankel determinants of a holomorphic function.
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REFERENCES
M. Fekete, “Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzah igen Koeffizienten, ” Math. Z., 17, 228–249 (1923).
F. Leja, “Une généralisation de l'écart et du diametre transfini d'un ensemble, ” Ann. Soc. Pol. Math., 22, 35–42 (1949).
F. Leja, “Proprietés des points extremaux des ensembles plans et leur application à la représentation conforme, ” Ann. Pol. Math., 3, 319–342 (1957).
A. A. Gonchar and E. A. Rakhmanov, “Equilibrium measure and distribution of zeros of extremal polynomials, ” Mat. Sb., 125 (167), 117–127 (1984).
H. N. Mhaskar and E. B. Saff, “Weighted analogues of capacity, transfinite diameter, and Chebyshev constant, ” Constr. Approxim., 8, 105–124 (1992).
E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren der mathematischen Wissenschaften, Vol. 316, Springer, Berlin (1997).
V. I. Buslaev, “On the convergence of continuous T-fractions, ” Tr. Mat. Inst. Ros. Akad. Nauk, 235 (2001).
D. Pólya, “Beitrag zur Veralldemeinerung des Verzerrungssatzes auf mehrfach zusammenhngende Gebiete. III, ” Sitzungsber. Preuss. Acad. Wiss. Phys.-Math. K1, 55–62 (1929).
G. M. Goluzin, Geometrical Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1966).
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Buslaeva, S.F. On a Two-Point Version of Transfinite Diameter. Ukrainian Mathematical Journal 53, 1383–1390 (2001). https://doi.org/10.1023/A:1014327604550
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DOI: https://doi.org/10.1023/A:1014327604550