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On the decay of singular inner functions

Part of: Classical measure theory Function theory on the disc

Published online by Cambridge University Press:  02 December 2020

Thomas Ransford
Thomas Ransford*
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, QCG1V 0A6, Canada
*
e-mail:ransford@mat.ulaval.ca
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Abstract

It is known that if $S(z)$ is a non-constant singular inner function defined on the unit disk, then $\min _{|z|\le r}|S(z)|\to 0$ as $r\to 1^-$ . We show that the convergence can be arbitrarily slow.

Keywords

Singular inner functionHausdorff measure

MSC classification

Primary: 30J15: Singular inner functions
Secondary: 28A78: Hausdorff and packing measures
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 902 - 905
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Research supported by grants from NSERC and the Canada research chairs program.

References

Carleson, L., Selected problems on exceptional sets . Van Nostrand Mathematical Studies, 13, Van Nostrand, Princeton, NJ, 1967.Google Scholar
Mashreghi, J., Derivatives of inner functions . Fields Institute Monographs, 31, Springer, New York NY, 2013. http://dx.doi.org/10.1007/978-1-4614-5611-7 Google Scholar