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Integral Inequalities for Equimeasurable Rearrangements

Published online by Cambridge University Press:  20 November 2018

G. F. D. Duff
G. F. D. Duff*
Affiliation:
University of Toronto, Toronto, Ontario
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For a real-valued function f on the domain [0,b], the equimeasurable decreasing rearrangement f* of f is defined as a function μ–1 inverse to μ, where μ(y) is the measure of the set {x|f(x) > y}. Inequalities connected with rearrangements of sequences as well as functions play a considerable part in various branches of analysis, and, for example, the concluding chapter of Hardy, Littlewood, and Pólya [3] is devoted to rearrangement inequalities. Equimeasurable rearrangements of functions are also used by Zygmund [6, Vol. II, Chapters I and XII].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 408 - 430
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Banach, S., Sur les lignes rectifiables et les surfaces dont l'aire est fini, Fund. Math. 7 (1925), 224–236.Google Scholar
2. Duff, G. F. D., Differences, derivatives, and decreasing rearrangements, Can. J. Math. 19 (1967), 1153–1178.Google Scholar
3. Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities, 2nd ed. (Cambridge, at the University Press, 1952).Google Scholar
4. Pölya, G. and Szego, G., Isoperimetric inequalities in mathematical physics, Annals of Mathematics Studies, no. 27 (Princeton Univ. Press, Princeton, N.J., 1951).Google Scholar
5. Ryff, J. V., Measure preserving transformations and rearrangements (to appear).Google Scholar
6. Zygmund, A., Trigonometric series, 2nd éd., Vol. II (Cambridge Univ. Press, New York, 1959).Google Scholar