A theorem of Beurling and Tsuji is best possible
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- by Shinji Yamashita PDF
- Proc. Amer. Math. Soc. 72 (1978), 286-288 Request permission
Abstract:
We shall show that Beurling-Tsuji’s theorem (see Theorem A) is, in a sense, best possible. For each pair $a, b \in (0, + \infty )$ there exists a function f holomorphic in $\{ |z| < 1\}$ such that the Euclidean area of the Riemannian image of each non-Euclidean disk of non-Euclidean radius a, is bounded by b, and such that f has finite angular limit nowhere on the unit circle.References
- Arne Beurling, Ensembles exceptionnels, Acta Math. 72 (1940), 1–13 (French). MR 1370, DOI 10.1007/BF02546325
- Peter A. Lappan, Fatou points of harmonic normal functions and uniformly normal functions, Math. Z. 102 (1967), 110–114. MR 222265, DOI 10.1007/BF01112077
- Ch. Pommerenke, On Bloch functions, J. London Math. Soc. (2) 2 (1970), 689–695. MR 284574, DOI 10.1112/jlms/2.Part_{4}.689
- Masatsugu Tsuji, Beurling’s theorem on exceptional sets, Tohoku Math. J. (2) 2 (1950), 113–125. MR 40435, DOI 10.2748/tmj/1178245640
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 72 (1978), 286-288
- MSC: Primary 30D40
- DOI: https://doi.org/10.1090/S0002-9939-1978-0507324-6
- MathSciNet review: 507324