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

DOI:
https://doi.org/10.1090/S0002-9939-1988-0931724-X

MathSciNet review:
931724

Full-text PDF

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 be a strictly positive function defined on some interval , satisfying a certain condition of limit. We prove that if is the sum of a convergent power series for with nonnegative coefficients and , then , where and depends only upon .

**[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**, https://doi.org/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**

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

DOI:
https://doi.org/10.1090/S0002-9939-1988-0931724-X

Keywords:
Power series,
Hardy-Littlewood theorem,
Weierstrass theorem,
moment sequence,
completely monotonic sequence

Article copyright:
© Copyright 1988
American Mathematical Society