Volume 4,  Issue 3 (GI8), 2003

Article 53

CARLEMAN'S INEQUALITY - HISTORY, PROOFS AND SOME NEW GENERALIZATIONS

MARIA JOHANSSON, LARS-ERIK PERSSON AND ANNA WEDESTIG

DEPARTMENT OF MATHEMATICS,
LULEA UNIVERSITY OF TECHNOLOGY,
SE-971 87 LULEA , SWEDEN.
E-Mail: marjoh@sm.luth.se
E-Mail: larserik@sm.luth.se
E-Mail: annaw@sm.luth.se

Received 4 December, 2002; Accepted 25 March, 2003.
Communicated by: S. Saitoh


ABSTRACT.    Carleman's inequality reads
$\displaystyle a_{1}+\sqrt{a_{1}a_{2}}+...+\sqrt[k]{a_{1}...a_{k}}<e\left( a_{1}+a_{2}+....\right) ,$    

where $ a_{k}$ , $ k=1,2,....,$ are positive numbers. In this paper we present some simple proofs of and several remarks (e.g. historical) about the inequality and its corresponding continuous version. Moreover, we discuss and comment on some very new results. We also include some new proofs and results e.g. a weight characterization of a general weighted Carleman type inequality for the case 0 $ <$ p $ \leq $ q $ <$ $ \infty .$ We also include some facts about T. Carleman and his work.


Key words:
Inequalities, Carleman's inequality, Pólya-Knopp's inequality, Sharp constants, Proofs, Weights, Historical remarks.

2000 Mathematics Subject Classification:
26D15.


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Report of the General Inequalities 8 Conference; September 15-21, 2002, Noszvaj, Hungary
Compiled by Zsolt Páles

Convolution Inequalities and Applications
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The Hardy-Landau-Littlewood Inequalities with Less Smoothness
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Continuity Properties of Convex-type Set-Valued Maps
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Carleman's Inequality - History, Proofs and Some New Generalizations
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Andersson's Inequality and Best Possible Inequalities
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On Some Results Involving the Cebysev Functional and its Generalisations
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Separation and Disconjugacy
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New Norm Type Inequalities for Linear Mappings
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An Integral Approximation in Three Variables
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On Some Spectral Results Relating to the Relative Values of Means
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On Zeros of Reciprocal Polynomials of Odd Degree
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Some New Hardy Type Inequalities and their Limiting Inequalities
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Generalizations of the Triangle Inequality
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A Survey on Cauchy-Bunyakovsky-Schwarz Type Discrete Inequalities
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