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    Rights statement: This is an Accepted Manuscript of an article published by Taylor & Francis in American Mathematical Monthly on 05/04/2022, available online: http://www.tandfonline.com/10.1080/00029890.2022.2051407

    Accepted author manuscript, 223 KB, PDF document

    Available under license: CC BY-NC: Creative Commons Attribution-NonCommercial 4.0 International License

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An equality underlying Hardy's inequality

Research output: Contribution to Journal/Magazine › Comment/debate › peer-review

Published
Article number6
<mark>Journal publication date</mark>31/05/2022
<mark>Journal</mark>American Mathematical Monthly
Issue number6
Volume129
Number of pages5
Pages (from-to)582-586
Publication StatusPublished
Early online date5/04/22
<mark>Original language</mark>English

Abstract

A classical inequality of G. H. Hardy states that Cx≤2x for x in l2, where C is the Cesàro (alias averaging) operator. This inequality has been strengthened to (C−I)x≤x. It has also been shown that CTx≤Cx for x in l2. We present equalities that imply these inequalities, together with the reverse inequalities (C−I)x≥(1/√2)x and Cx≤√2CTx. We also present companion results involving the shift operator.