Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Analytic functions with decreasing coefficients and Hardy and Bloch spaces

Miroslav Pavlović

Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11001 Beograd, PO Box 550, Serbia (pavlovic@matf.bg.ac.rs)

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(⊂ H 1), 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−2 an p < ∞.

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 H 1 to H 1 is improved by proving that it does not map 1 into H 1.

(Received May 27 2011)

Keywords

  • analytic functions;
  • Hardy spaces;
  • Bloch type spaces;
  • Libera operator

2010 Mathematics subject classification

  • Primary 30D55