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Examples of Weak Boundary Components

Published online by Cambridge University Press:  22 January 2016

Tohru Akaza  and
Kôtaro Oikawa
Tohru Akaza
Affiliation:
Kanazawa University, Nagoya University and Hirosima University
Kôtaro Oikawa
Affiliation:
Kanazawa University, Nagoya University and Hirosima University
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Let D be a plane domain and Γ be a component of the boundary of D consisting of a single point. According to Sario [5] we shall call Γ a weak boundary component of D if its image under any conformal mapping of D consists of a single point. A weak boundary component has been introduced by Grötzsch [2], who called it “vollkommen punktförmig”. If Γ is not weak we shall say that it is unstable (Sario [5]). We know that the weakness depends merely on the configuration of D in a neighborhood of Γ (see [4], p. 274).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 18 , April 1961 , pp. 165 - 170
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

References

[1] Akaza, T., On the weakness of some boundary components. Nagoya Math. J., 17 (1960), 219–223.Google Scholar
[2] Grötzsch, H., Eine Bemerkung zum Koebeschen Kreisnormierungsprinzip. Ber. Verh. Sachs. Akad. Wiss. Leipzig. Math.-Nat. Kl., 87 (1935), 319–324.Google Scholar
[3] Jurchescu, M., Modulus of a boundary component. Pacific J. Math., 8 (1958), 791–809.Google Scholar
[4] Oikawa, K., On the stability of boundary components. Ibid., 10 (1960), 263–294.Google Scholar
[5] Sario, L., Strong and weak boundary components. J. Anal. Math. 5 (1956/57), 389–398.CrossRefGoogle Scholar