Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $.

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

  • [1] P. Dienes, The Taylor series: an introduction to the theory of functions of a complex variable, Dover Publications, Inc., New York, 1957. MR 0089895
  • [2] G. H. Hardy and J. E. Littlewood, Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive, Proc. London Math. Soc. (2) 11 (1911), pp. 411-478.
  • [3] J. Karamata, Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), no. 1, 319–320 (German). MR 1545168, 10.1007/BF01194636
  • [4] E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
  • [5] David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 40E05, 30B10, 30B30

Retrieve articles in all journals with MSC: 40E05, 30B10, 30B30

Additional Information

Keywords: Power series, Hardy-Littlewood theorem, Weierstrass theorem, moment sequence, completely monotonic sequence
Article copyright: © Copyright 1988 American Mathematical Society