On a theorem of Hardy and Littlewood
Author:
Luis G. Bernal
Journal:
Proc. Amer. Math. Soc. 104 (1988), 10781080
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 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 (19,735d)
 [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. 411478.
 [3]
J.
Karamata, Über die HardyLittlewoodschen Umkehrungen des
Abelschen Stetigkeitssatzes, Math. Z. 32 (1930),
no. 1, 319–320 (German). MR
1545168, http://dx.doi.org/10.1007/BF01194636
 [4]
E.
C. Titchmarsh, Hanshu lun, Translated from the English by Wu
Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
(33 #5850)
 [5]
David
Vernon Widder, The Laplace Transform, Princeton Mathematical
Series, v. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
(3,232d)
 [1]
 P. Dienes, The Taylor series: An introduction to the theory of a complex variable, Dover, New York, 1957. MR 0089895 (19:735d)
 [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. 411478.
 [3]
 J. Karamata, Über die HardyLittlewoodschen Umkerhrungen des Abelschen Steligkeitssatzes, Math. Z. 32 (1930), pp. 319320. MR 1545168
 [4]
 E. C. Titchmarsh, The theory of functions, 2nd ed. (corrected), Oxford Univ. Press, New York, 1968. MR 0197687 (33:5850)
 [5]
 D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, N. J., 1946. MR 0005923 (3:232d)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919880931724X
PII:
S 00029939(1988)0931724X
Keywords:
Power series,
HardyLittlewood theorem,
Weierstrass theorem,
moment sequence,
completely monotonic sequence
Article copyright:
© Copyright 1988 American Mathematical Society
