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
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Final published version
Licence: CC BY: Creative Commons Attribution 4.0 International License
Research output: Contribution to Journal/Magazine › Comment/debate › peer-review
Research output: Contribution to Journal/Magazine › Comment/debate › peer-review
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TY - JOUR
T1 - An equality underlying Hardy's inequality
AU - Jameson, Graham
PY - 2022/5/31
Y1 - 2022/5/31
N2 - 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.
AB - 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.
U2 - 10.1080/00029890.2022.2051407
DO - 10.1080/00029890.2022.2051407
M3 - Comment/debate
VL - 129
SP - 582
EP - 586
JO - American Mathematical Monthly
JF - American Mathematical Monthly
SN - 0002-9890
IS - 6
M1 - 6
ER -