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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The derivative of the atomic function is not in $ B\sp{2/3}$


Authors: Charles L. Belna and Benjamin Muckenhoupt
Journal: Proc. Amer. Math. Soc. 63 (1977), 129-130
MSC: Primary 30A78
MathSciNet review: 0586555
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Abstract | References | Similar Articles | Additional Information

Abstract: H. A. Allen and C. L. Belna have shown that the derivative of the atomic function $ A(z) = \exp [(z + 1)/(z - 1)]$ is in $ {B^p}$ for $ 0 < p < 2/3$, where $ {B^p}$ is the containing Banach space for the Hardy class $ {H^p}(0 < p < 1)$. Here we show that $ A'(z)$ does not belong to any of the other $ {B^p}$ spaces.


References [Enhancements On Off] (What's this?)

  • [1] P. R. Ahern and D. N. Clark, On inner functions with $ {B^p}$ derivative, Michigan Math. J. 23 (1976), 107-118. MR 0414884 (54:2976)
  • [2] H. A. Allen and C. L. Belna, Singular inner functions with derivative in $ {B^p}$, Michigan Math. J. 19 (1972), 185-188. MR 45 #8844. MR 0299796 (45:8844)
  • [3] P. L. Duren, B. W. Romberg and A. L. Shields, Linear functionals on $ {H^p}$ spaces with $ 0 < p < 1$, J. Reine Angew. Math. 238 (1969), 32-60. MR 41 #4217. MR 0259579 (41:4217)
  • [4] D. J. Newman and H. S. Shapiro, The Taylor coefficients of inner functions, Michigan Math. J. 9 (1962), 249-255. MR 26 #6371. MR 0148874 (26:6371)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0586555-2
PII: S 0002-9939(1977)0586555-2
Article copyright: © Copyright 1977 American Mathematical Society



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