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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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An induction principle for spectral and rearrangement inequalitities
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by Kong Ming Chong
Trans. Amer. Math. Soc. 196 (1974), 371-383
DOI: https://doi.org/10.1090/S0002-9947-1974-0344396-7

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.
References
    K. M. Chong, Equimeasurable rearrangements of functions with applications to analysis, Thesis, Queen’s University, Kingston, Ontario, Canada, 1972.
  • Kong Ming Chong, Spectral orders, uniform integrability and Lebesgue’s dominated convergence theorem, Trans. Amer. Math. Soc. 191 (1974), 395–404. MR 369646, DOI 10.1090/S0002-9947-1974-0369646-2
  • —, Some extensions of a theorem of Hardy, Littlewood and Póly a and their applications, Canad. J. Math. —, Spectral inequalities involving the sums and products of functions (submitted for publication). K. M. Chong, Spectral inequalities involving the infima and suprema of functions (submitted for publication).
  • K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Applied Mathematics, No. 28, Queen’s University, Kingston, Ont., 1971. MR 0372140
  • G. H. Hardy, J. E. Littlewood and G. Pólya, Some simple inequalities satisfied by convex functions, Mess, of Math. 58 (1929), 145-152. —, Inequalities, Cambridge Univ. Press, New York, 1934.
  • G. G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953), 176–179. MR 52476, DOI 10.2307/2307574
  • W. A. J. Luxemburg, Rearrangement invariant Banach function spaces, Queen’s Papers in Pure and Appl. Math., no. 10, Queen’s University, Kingston, Ont., 1967, pp. 83-144.
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • 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