On the subadditivity of the transfinite diameter
Published online by Cambridge University Press: 24 October 2008
Extract
Let M be a bounded and closed set of points in the complex z plane; d(M), a set-function which is of great importance in potential and function theory, may then be defined as follows. n points z1, z2, …, zn in M are so chosen that the product of the mutual distances
has the greatest possible value Then it can be proved that
exists. Thus the set-function d(M), named by Fekete the transfinite diameter of M, is defined.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 4 , October 1941 , pp. 373 - 383
- Copyright
- Copyright © Cambridge Philosophical Society 1941
References
† Fekete, M., ‘Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten’, Math. Z. 17 (1923), 228–49.CrossRefGoogle Scholar
‡ Loc. cit. p. 239; for a certain generalization of this relation see Szegö, G., ‘Bemerkungen zu einer Arbeit von Herrn Fekete…’, Math. Z. 21 (1924), 203–8.CrossRefGoogle Scholar
† For an elementary proof see Fekete, M., ‘Über die Verallgemeinerung der Picard-Landauschen und Picard-Schottkyschen Sätze…’, Math. Ann. 106 (1932), 598Google Scholar, note 12.
‡ Fekete, M., Relations between the transfinite diameter of an arc and its length, Sefer Magnes, Jerusalem (1938), pp. 401–13Google Scholar (Hebrew with an abstract in English).
§ Pólya, and Szegö, , ‘Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen’, J. reine angew. Math. 165 (1931), 4–49.Google Scholar
∥ See for example Kellog, , Foundations of potential theory (Berlin, 1929), p. 331.CrossRefGoogle Scholar
¶ Pólya and Szegö, loc. cit. This property of c(M) follows elementarily from definition (2) of d(M) and the identity (3). This was pointed out by Prof. Fekete in a seminar on the transfinite diameter held at the Hebrew University in 1934. Compare also Nevanlinna, R., Eindeutige analytische Funktionen (Berlin, 1936), p. 120CrossRefGoogle Scholar, Hilfssatz 1.
† Löwner, K., ‘Über Extremumssätze bei der konformen Abbildung des Äusseren des Einheitskreises’, Math. Z. 3 (1919), 65–77.CrossRefGoogle Scholar
† Carathéodory, C., ‘Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten’, Math. Ann. 72 (1912), 107–14.CrossRefGoogle Scholar
† Carathéodory, C., ‘Über die Begrenzung einfach-zusammenhängender Gebiete’, Math. Ann. 73 (1913), 323–70.CrossRefGoogle Scholar
‡ Bieberbach, L., ‘Über die Koeffizienten derjenigen Potenzreihen, …’, Berl. Sitzungsber. (1916), pp. 940–55.Google Scholar
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