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A strengthening of the Carleman-Hardy-Pólya inequality

Author: Finbarr Holland
Journal: Proc. Amer. Math. Soc. 135 (2007), 2915-2920
MSC (2000): Primary 26D15
Published electronically: May 9, 2007
MathSciNet review: 2317969
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Abstract | References | Similar Articles | Additional Information

Abstract: As a consequence of a more general statement proved in the paper, it is deduced that, if $ n\ge1$, and $ a_j>0,\,j=1,2,\dotsc,n$, then

$\displaystyle \left(\frac{\sum_{j=1}^n \root j\of{a_1a_2\dotsm a_j}}{\sum_{j=1}... ...of{a_1a_2\dotsm a_n}}{\sum_{j=1}^n \root j\of{a_1a_2\dotsm a_j}}\le\frac{n+1}n,$

with equality if and only if $ a_1=a_2=\dotsb =a_n$. This is a new refinement of Carleman's classic inequality.

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Additional Information

Finbarr Holland
Affiliation: Mathematics Department, University College, Cork, Ireland

Keywords: Weighted geometric means, Carleman's inequality, convex functions
Received by editor(s): September 23, 2005
Received by editor(s) in revised form: June 8, 2006
Published electronically: May 9, 2007
Communicated by: David Preiss
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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