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An Induction Theorem for Rearrangements

Published online by Cambridge University Press:  20 November 2018

Kong-Ming Chong
Kong-Ming Chong*
Affiliation:
University of Malaya, Kuala Lumpur 22-11, Malaysia
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In this paper, an induction theorem for rearrangements involving w-tuples in Rn is proved, showing that a certain proposition regarding a pair of w-tuples related by the weak spectral order ≪ is true for any integer n ≧ 2 if and only if it is true for n = 2. This theorem contains as particular cases a well-known theorem of Hardy-Littlewood-Pólya [4, Lemma 2, p. 47], a theorem of Pólya [8], a theorem of Rado [9, pp. 1-2], two theorems of Mirsky [6, Theorem 2, p. 232; 7, Theorem 4, p. 90], a result given in [1, Corollary 2.4, p. 1333] and also [2, Proposition 2.1, p. 439].

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 1 , 01 February 1976 , pp. 154 - 160
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Chong, K. M., Some extensions of a theorem of Hardy, Littlew∞d and Pôlya and their applications, Can. J. Math. 26 (1974), 1321–1340.Google Scholar
2. Chong, K. M., Spectral order preserving matrices and Muirhead's theorem, Trans. Amer. Math. Soc. 200 (1974), 437–444.Google Scholar
3. Chong, K. M., A general induction theorem for rearrangements of n-tuples, J. Math. Anal. Appl. (to appear).Google Scholar
4. Hardy, G. H., Littlew∞d, J. E. and G., Pôlya, Inequalities (Cambridge, 1959).Google Scholar
5. Marshall, A. W. and Proschan, F., An inequality for convex functions involving majorization, J. Math. Anal. Appl. 12 (1965), 87–90.Google Scholar
6. Mirsky, L., Inequalities for certain classes of convex functions, Proc. Edin. Math. Soc. 11 (1959), 231–235.Google Scholar
7. Mirsky, L., On a convex set of matrices, Archiv der Math. 10 (1959), 88–92.Google Scholar
8. Pôlya, G., Remark on WeyVs note: Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci., 36 (1950), 49–51.Google Scholar
9. Rado, R., An inequality, J. London Math. Soc. 27 (1952), 1–6.Google Scholar