A bounded analytic function in the unit disk with a level set component of infinite length

Authors:
K. F. Barth and J. G. Clunie

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

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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 where is a Blaschke Product.

**[1]**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****[2]**J. B. Garnett, F. W. Gehring and P. W. Jones,*Level sets for univalent functions and a theorem of Fejér and Riesz*(to appear).**[3]**W. K. Hayman and J. M. G. Wu,*Level sets of univalent functions*, Comment. Math. Helv.**56**(1981), no. 3, 366–403. MR**639358**, https://doi.org/10.1007/BF02566219**[4]**Peter W. Jones,*Bounded holomorphic functions with all level sets of infinite length*, Michigan Math. J.**27**(1980), no. 1, 75–79. MR**555839****[5]**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****[6]**George Piranian and Allen Weitsman,*Level sets of infinite length*, Comment. Math. Helv.**53**(1978), no. 2, 161–164. MR**490872**, https://doi.org/10.1007/BF02566072

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0660605-9

Keywords:
Bounded functions,
level set,
level curve

Article copyright:
© Copyright 1982
American Mathematical Society