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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A bounded analytic function in the unit disk with a level set component of infinite length
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by K. F. Barth and J. G. Clunie PDF
Proc. Amer. Math. Soc. 85 (1982), 562-566 Request permission

Abstract:

The authors construct a bounded analytic function in the unit disk with a level set component of infinite length. The example is of the form $\exp iB(z)$ where $B(z)$ is a Blaschke Product.
References
  • C. Belna and G. Piranian, A Blaschke product with a level-set of infinite length, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 79–81. MR 820211
  • J. B. Garnett, F. W. Gehring and P. W. Jones, Level sets for univalent functions and a theorem of Fejér and Riesz (to appear).
  • W. K. Hayman and J. M. G. Wu, Level sets of univalent functions, Comment. Math. Helv. 56 (1981), no. 3, 366–403. MR 639358, DOI 10.1007/BF02566219
  • Peter W. Jones, Bounded holomorphic functions with all level sets of infinite length, Michigan Math. J. 27 (1980), no. 1, 75–79. MR 555839
  • George Piranian, Inner functions with a level-set of infinite length, Complex analysis Joensuu 1978 (Proc. Colloq., Univ. Joensuu, Joensuu, 1978) Lecture Notes in Math., vol. 747, Springer, Berlin, 1979, pp. 309–313. MR 553057
  • George Piranian and Allen Weitsman, Level sets of infinite length, Comment. Math. Helv. 53 (1978), no. 2, 161–164. MR 490872, DOI 10.1007/BF02566072
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 85 (1982), 562-566
  • MSC: Primary 30D50
  • DOI: https://doi.org/10.1090/S0002-9939-1982-0660605-9
  • MathSciNet review: 660605