Hostname: page-component-5cf477f64f-54txb Total loading time: 0 Render date: 2025-04-03T20:15:01.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic capacity for two segments

Published online by Cambridge University Press:  22 January 2016

Takafumi Murai
Takafumi Murai*
Affiliation:
Department of Mathematics, School of Science, Nagoya University, Nagoya, 464-01, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The analytic capacity γ(E) of a compact set E in the complex plane C is defined by γ(E) = sup , where — f′(∞) is the 1/z-coeffieient of f(ζ) at infinity and the supremum is taken over all bounded analytic functions f(ζ) outside E with supremum norm less than or equal to 1. Analytic capacity γ(·) plays various important roles in the theory of bounded analytic functions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 122 , June 1991 , pp. 19 - 42
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[1] Ferrari, C., Sulla transformazione conforme di due cerchi in due profili alari, Memorie della R. Accad. delle Scienze di Torino, Serie II, 67 (1930), 1–15.Google Scholar
[2] Garnett, J., Analytic capacity and measure, Lecture Notes in Math., 297, Springer-Verlag, Berlin, 1972.Google Scholar
[3] Garrick, E., Potential flow about arbitrary biplane wing sections, Technical Report No. 542, N.A.C.A. (1936), 47–75.Google Scholar
[4] Hancock, H., Lectures on the theory of elliptic functions, Dover, New York, 1958.Google Scholar
[5] Komatu, Y., (Japanese), Theory of conformal mappings II, Kyoritsu, Tokyo, 1947.Google Scholar
[6] Mandelbrot, B. B., The fractal geometry and nature, Freeman, San Francisco, 1982.Google Scholar
[7] Milne-Thomson, L. M., Theoretical hydrodynamics, Fifth edition, Macmillan, London, 1968.Google Scholar
[8] Murai, T., A real variable method for the Cauchy transform, and analytic capacity, Lecture Notes in Math., 1307, Springer-Verlag, Berlin, 1988.Google Scholar
[9] Murai, T., The power 3/2 appearing in the estimate of analytic capacity, Pacific J. Math., 143 (1990), 313–340.Google Scholar
[10] Nehari, Z., Conformal mapping, McGraw-Hill, New York, 1952.Google Scholar
[11] Pommerenke, Ch., Über die analytische Kapazität, Ark. der Math., 11 (1960), 270–277.Google Scholar
[12] Sario, L. and Oikawa, K., Capacity functions, Springer-Verlag, Berlin, 1969.Google Scholar
[13] Sasaki, T., (Japanese), Applications of conformal mappings, Fuzanbo, Tokyo, 1939.Google Scholar
[14] Vitushkin, , (Russian), Example of a set of positive length but of zero analytic capacity, Dokl. Akad. Nauk. SSSR, 127 (1959), 246–249.Google Scholar