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Analytic functions with decreasing coefficients and Hardy and Bloch spaces

Published online by Cambridge University Press:  26 July 2012

Miroslav Pavlović
Miroslav Pavlović*
Affiliation:
Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11001 Beograd, PO Box 550, Serbia ([email protected])
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Abstract

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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.

Keywords

analytic functionsHardy spacesBloch type spacesLibera operator
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 623 - 635
Copyright
Copyright © Edinburgh Mathematical Society 2013

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