A theorem of Beurling and Tsuji is best possible

Author:
Shinji Yamashita

Journal:
Proc. Amer. Math. Soc. **72** (1978), 286-288

MSC:
Primary 30D40

MathSciNet review:
507324

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Abstract: We shall show that Beurling-Tsuji's theorem (see Theorem A) is, in a sense, best possible. For each pair there exists a function *f* holomorphic in 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.

**[1]**Arne Beurling,*Ensembles exceptionnels*, Acta Math.**72**(1940), 1–13 (French). MR**0001370****[2]**Peter A. Lappan,*Fatou points of harmonic normal functions and uniformly normal functions*, Math. Z.**102**(1967), 110–114. MR**0222265****[3]**Ch. Pommerenke,*On Bloch functions*, J. London Math. Soc. (2)**2**(1970), 689–695. MR**0284574****[4]**Masatsugu Tsuji,*Beurling’s theorem on exceptional sets*, Tôhoku Math. J. (2)**2**(1950), 113–125. MR**0040435****[5]**M. Tsuji,*Potential theory in modern function theory*, Maruzen Co., Ltd., Tokyo, 1959. MR**0114894**

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DOI:
https://doi.org/10.1090/S0002-9939-1978-0507324-6

Article copyright:
© Copyright 1978
American Mathematical Society