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

On a theorem of Hardy and Littlewood


Author: Luis G. Bernal
Journal: Proc. Amer. Math. Soc. 104 (1988), 1078-1080
MSC: Primary 40E05; Secondary 30B10, 30B30
MathSciNet review: 931724
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Abstract: In this paper, we give an extension of a classical theorem of Hardy and Littlewood on power series. Let $ \varphi $ be a strictly positive function defined on some interval $ \left( {\delta ,1} \right)$, satisfying a certain condition of limit. We prove that if $ f\left( x \right)$ is the sum of a convergent power series for $ 0 < x < 1$ with nonnegative coefficients $ {a_n}$ and $ f\left( x \right) \sim \varphi \left( x \right)\;\left( {x \to 1} \right)$, then $ {S_n} \sim \alpha \cdot\varphi \left( {x_0^{1/n}} \right)\left( {n \to \infty } \right)$, where $ {S_n} = {a_0} + {a_1} + \cdots + {a_{n,\;}}{x_0} \in \left( {0,1} \right)$ and $ \alpha $ depends only upon $ \varphi $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0931724-X
PII: S 0002-9939(1988)0931724-X
Keywords: Power series, Hardy-Littlewood theorem, Weierstrass theorem, moment sequence, completely monotonic sequence
Article copyright: © Copyright 1988 American Mathematical Society