On the Cluster Sets of Analytic Functions in a Jordan Domain

Makoto OHTSUKA

doi:10.2969/jmsj/00210001

Makoto OHTSUKA

著者情報

ジャーナルフリー

1950 年 2 巻 1-2 号 p. 1-15

DOI https://doi.org/10.2969/jmsj/00210001

詳細

Published: 1950 Received: 1948/10/11 Available on J-STAGE: 2006/08/29 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/08/29 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Right : (1) We use + for sums of disjoint sets.
(2) D_γ denotes the closure of D_γ: ditto concerning M_γ (E) Y_γ (E) etc.
(3) We will call this a function of class α.
(4) Cf. W. Gross: Zum Verhalten der konformen Abbildung am Rande. Math. Zeit. 3 (1919).
(5) An example will be shown at the end of n°3.
(6) F. Iversen: Sur quelques propriétés des fonctions monogènes au voisinage d'un point singulier. Öfv. af Finska Vet-Soc. Förh. 58 (1916). W. Seidel: On the cluster values of analytic functions Trans. Amer. Math. Soc. 54 (1932).
(7) J. L. Doob: On a theorem of Gross and Iversen. Ann. of Math. 33 (1932). K. Noshiro: On the singularities of analytic functions. Jap. Jour. Math. 17 (1940).
(8) Cf. S. Ishikawa: On the cluster sets of analytic functions. Nippon Sugaku-Butsurigaku Kaishi. 13 (1939) (in Japanese).
(9) W. Gross: Zum Verhalten analytischer Funktionen in der Umgebung singulärer Stellen. Math. Zeit. 2 (1918).
(10) Q₁⊃Q₂ represents that Q₂ is nearer to z₀ than Q₁.
(11) S. Kametani and T. Ugaheri: A remark on Kawakami's extension of Löwner's lemma. Proc. Imp. Acad. Tokyo. 18 (1942).
(12) M. Tsuji: On an extension of Löwner's theorem. Proc. Imp. Acad. Tokyo, 18 (1942).
(13) F. Iversen: Recherches sur les fonctions inverses des fonctions méromorphes. Thèse de Helsingfors. 1914. K. Noshiro: loc. cit. (7).
(14) Capacity means logarithmic capacity.
(15) O. Frostman: Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions. Meddel. Lunds Univ. Mat. Sem. 3 (1935).
(16) We denote Koebe's theorem for a function of class α by generalized Koebe's theorem. Cf. W. Gross: Über die Singularitäten analytischer Funktionen. Mh. Math. u. Physik. 29 (1918).
(17) M. L. Cartwright: On the behaviour of analytic functions in the neighbourhood of its essential singularities. Math. Ann. 112 (1936).
(18) Here, theorems for a function of class α are considered.
(19) Cf. R. Nevanlinna: Eindeutige analytische Funktionen. Berlin. 1936.
(20) W. Seidel: On the distribution of values of bounded analytic function. Trans. Amer. Math. 36 (1934).
(21) G. Hössjer: Bemerkung über einen Satz von E. Lindelöf. Fysiogr. Sällsk. Lunds. Förh. 6 (1937). G. Hössjer assumed the continuity of f(z) on the closed Jordan domain except for z₀, but here it is unnecessary.
(22) φ represents an empty set.
(23) K. Noshiro: loc. cit. (7).
(24) W. Gross has obtained already some similar results. But our results are different from his in several points. W. Gross: loc. cit. (9).
(25) We will call it briefly the branch point (in the w-plane).
(26) Giving two sequences of points {z_n} and {z_n'} which converge to z₀ on C₁ and C₂ respectively and proving that any curve in D connecting two points z_n and z_n' meets at least one of given domains in D, he concluded the existence of a domain having z₀ on its boundary among these domains. But it seems hasty to conclude so.
(27) That is, there runs only a finite number of curves near any point in D.
(28) Area means the inner extent in Jordan's sense.
(29) Since Q_ij (j=1, 2,……, p) don't pass z₀, some neighbourhood of z₀ in D is included in ∩^p_j=1D_ij. Connect z₀ with a point z₁ in G_n by a curve in D. If this curve does not meet Q_ij (j=1, 2,……, p,), z₁ will belong to ∩^p_j=1D_ij, otherwise there will exist a cross-cut Q_i0 which the curve intersects at the first time counting from z₀. Since one side of Q_i0 belongs to ∩^p_j=1D_ij and some part of Q_i0 lies in G_n, it is possible to enter into G_n staying inside ∩^p_j=1D_ij. Accordingly (∩^p_f=1D_ij)∩Gn is a non empty open set.

訂正日: 2006/08/29 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

責任著者(Corresponding author)

訂正情報