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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hardy spaces of close-to-convex functions and their derivatives


Authors: Finbarr Holland and John B. Twomey
Journal: Trans. Amer. Math. Soc. 246 (1978), 359-372
MSC: Primary 30D55
MathSciNet review: 515543
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Abstract: Let $ f(z) = \sum\nolimits_1^\infty {{a_n}} {z^n}$ be close-to-convex on the unit disc. It is shown that (a) if $ \lambda > 0$, then f belongs to the Hardy space $ {H^\lambda }$ if and only if $ {\sum {{n^{\lambda - 2}}\left\vert {{a_n}} \right\vert} ^\lambda }$ is finite and that (b) if $ 0 < \lambda < 1$, then $ f' \in {H^\lambda }$ if and only if either $ \sum {{n^{2\lambda - 2}}} {\left\vert {{a_n}} \right\vert^\lambda }$ or, equivalently, $ \int_0^1 {{M^\lambda }(r,f')} dr$ is convergent. It is noted that the first of these results does not extend to the full class of univalent functions and that the second is best possible in a number of different senses.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515543-2
PII: S 0002-9947(1978)0515543-2
Keywords: Membership of the Hardy space $ {H^\lambda }$, close-to-convex functions, functions with positive real part, Riesz products, Taylor coefficients
Article copyright: © Copyright 1978 American Mathematical Society