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On Boundary Values of an Analytic Transformation of a Circle into a Riemann Surface
Published online by Cambridge University Press: 22 January 2016
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Let f(z) be a nonconstant analytic transformation of the unit circle U : ∣z∣ < 1 into a Riemann surface ℜ. As an extension of a classical theorem of F. and M. Riesz, the author proved in Theorem 3.4 of [2] that if the image of U is relatively compact in ℜ and has universal covering surface of hyperbolic type, and if, at every point of a set on ∣z∣ = 1 of positive inner linear measure, there terminates a curve along which f(z) has limit, then the set of such limits has positive inner logarithmic capacity. This theorem was followed by the first proposition in Kuramochi [1], which asserts that, if ℜ has a null boundary and the image of U excludes a set of positive logarithmic capacity on ℜ and if, at every point of a set E on ∣z∣ = 1, there terminates a curve along which f(z) has limit in the union of a set of inner logarithmic capacity zero on ℜ and the boundary components of ℜ then the inner linear measure of E is zero.
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- Research Article
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- Copyright © Editorial Board of Nagoya Mathematical Journal 1956
References
[ 2 ]
Ohtsuka, M.: Dirichlet problems on Riemann surfaces and conformal mappings, Nagoya Math. J., 3 (1951), pp. 91–137.Google Scholar
[ 3 ]
Ohtsuka, M.: Generalisations of Montel-Lindelöf’s theorem on asymptotic values, ibid., 10 (1956), pp. 129–163.Google Scholar
[ 4 ]
, F. and Riesz, M.: Über die Randwerte einer analytischen Funktion, 4 Congrès Scand. Stockholm, (1916), pp. 27–44.Google Scholar
[ 5 ]
Tsuji, M.: Some metrical theorems on Fuchsian groups, Ködai Math. Sem. Report, nos. 4-5 (1950), pp. 89–93.Google Scholar
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