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  • Proceedings of the Edinburgh Mathematical Society, Volume 56, Issue 2
  • June 2013, pp. 623-635

Analytic functions with decreasing coefficients and Hardy and Bloch spaces

  • Miroslav Pavlović (a1)
  • DOI: http://dx.doi.org/10.1017/S001309151200003X
  • Published online: 26 July 2012
Abstract
Abstract

The following rather surprising result is noted.

(1) A function f(z) = ∑anzn such that an ↓ 0 (n → ∞) belongs to H1 if and only if ∑(an/(n + 1)) < ∞.

A more subtle analysis is needed to prove that assertion (2) remains true if H1 is replaced by the predual, 1(⊂ H1), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following.

(2) f belongs to Hp (1 < p < ∞) if and only if ∑(n + 1)p−2anp < ∞.

A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H1 to H1 is improved by proving that it does not map 1 into H1.

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13.M. Mateljević and M. Pavlović , Lp-behavior of power series with positive coefficients and Hardy spaces, Proc. Am. Math. Soc. 87(2) (1983), 309316.

15.M. Nowak and M. Pavlović , On the Libera operator, J. Math. Analysis Applic. 370(2) (2010), 588599.

17.M. Pavlović , A short proof of an inequality of Littlewood and Paley, Proc. Am. Math. Soc. 134 (2006), 36253627.

19.A. L. Shields and D. L. Williams , Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Am. Math. Soc. 162 (1971), 287302.

22.J. Xiao , Holomorphic Q classes, Lecture Notes in Mathematics, Volume 1767 (Springer, 2001).

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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