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)



$ L\sp{p}$-behavior of power series with positive coefficients and Hardy spaces

Authors: Miodrag Mateljević and Miroslav Pavlović
Journal: Proc. Amer. Math. Soc. 87 (1983), 309-316
MSC: Primary 30D55
MathSciNet review: 681840
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For the power series $ f(x) = \sum\nolimits_1^\infty {{a_n}{x^n}} $ with $ {a_n} \geqslant 0$, certain weighted $ {L^p}$-norms of $ f$ on $ [0,1]$ are estimated from above and below in terms of the coefficients $ {a_n}$. Some consequences of this are obtained. For example, some known results concerning Hardy spaces may be extended to a wider class of spaces.

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

  • [1] R. Askey, $ {L^p}$ behaviour of power series with positive coefficients, Proc. Amer. Math. Soc. 19 (1968), 303-305. MR 0223786 (36:6834)
  • [2] R. Askey and R. P. Boas, Jr., Some integrability theorems for power series with positive coefficients, Mathematical Essays dedicated to A. J. Macintyre, Ohio Univ. Press, Athens, 1970. MR 0277956 (43:3689)
  • [3] P. L. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [4] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1931), 403-439. MR 1545260
  • [5] -, Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, J. Reine Angew. Math. 157 (1927), 141-158.
  • [6] -, Some new properties of Fourier constants, Math. Ann. 97 (1926), 159-209.
  • [7] W. K. Hayman, Multivalent functions, Cambridge Univ. Press, London and New York, 1958. MR 0108586 (21:7302)
  • [8] F. Holland and J. B. Twomey, On Hardy classes and the area function, J. London Math. Soc. 17 (1978), 275-283. MR 0486532 (58:6255)
  • [9] -, Conditions for membership of Hardy spaces, Aspects of Contemporary Complex Analysis (edited by D. A. Brannan and J. G. Clunie), Academic Press, New York, 1980, pp. 425-433. MR 623486 (82h:30035)
  • [10] J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. II, Proc. London Math. Soc. 42 (1936), 52-89.
  • [11] M. Mateljević and M. Pavlović, On the integral means of derivatives of the atomic function (to appear).
  • [12] J. W. Noonan and D. K. Thomas, The integral means of regular functions, J. London Math. Soc. 9 (1975), 557-560. MR 0364625 (51:879)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30D55

Retrieve articles in all journals with MSC: 30D55

Additional Information

Keywords: Hardy spaces, Hardy-Stein identity
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society