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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

An induction principle for spectral and rearrangement inequalitities


Author: Kong Ming Chong
Journal: Trans. Amer. Math. Soc. 196 (1974), 371-383
MSC: Primary 26A87
DOI: https://doi.org/10.1090/S0002-9947-1974-0344396-7
MathSciNet review: 0344396
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Abstract: In this paper, expressions of the form $ f \prec g$ or $ f \prec \prec g$ (where $ \prec $ and $ \prec \prec $ denote the Hardy-Littlewood-Pólya spectral order relations) are called spectral inequalities. Here a general induction principle for spectral and rearrangement inequalities involving a pair of n-tuples in $ {R^n}$ as well as their decreasing and increasing rearrangements is developed. This induction principle proves that such spectral or rearrangement inequalities hold iff they hold for the case when $ n = 2$, and that, under some mild conditions, this discrete result can be generalized to include measurable functions with integrable positive parts. A similar induction principle for spectral and rearrangement inequalities involving more than two measurable functions is also established. With this induction principle, some well-known spectral or rearrangement inequalities are obtained as particular cases and additional new results given.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0344396-7
Keywords: Equimeasurability, decreasing and increasing rearrangements, spectral orders, spectral inequalities, rearrangement inequalities
Article copyright: © Copyright 1974 American Mathematical Society