On Carleman and Knopp's Inequalities

A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is p...

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Bibliographic Details
Published in:Journal of approximation theory 2002, Vol.117 (1), p.140-151
Main Authors: Kaijser, Sten, Persson, Lars-Erik, Öberg, Anders
Format: Article
Language:English
Subjects:
Algebra, geometri och analys
Algebra, geometry and mathematical analysis
Analys
Carleman's inequality
geometric means
historical remarks
inequalities
Knopp's inequality
MATEMATIK
Mathematical analysis
MATHEMATICS
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Summary:A sharpened version of Carleman's inequality is proved. This result unifies and generalizes some recent results of this type. Also the “ordinary” sum that serves as the upper bound is replaced by the corresponding Cesaro sum. Moreover, a Carleman-type inequality with a more general measure is proved and this result may also be seen as a generalization of a continuous variant of Carleman's inequality, which is usually referred to as Knopp's inequality. A new elementary proof of (Carleman–)Knopp's inequality and a new inequality of Hardy–Knopp type is pointed out.
ISSN:0021-9045
1096-0430
1096-0430
DOI:10.1006/jath.2002.3684