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Inequalities for certain Classes of Convex Functions

Published online by Cambridge University Press:  20 January 2009

L. Mirsky
L. Mirsky
Affiliation:
The University, Sheffield
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Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 11 , Issue 4 , November 1959 , pp. 231 - 235
Copyright
Copyright © Edinburgh Mathematical Society 1959

References

REFERENCES

Fan, K., Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci., 37 (1951), 760–766.Google Scholar
Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities (Cambridge, 1934).Google Scholar
Karamata, J., Sur une inégalité relative aux fonctions convexes, Publ. math. Univ. Belgrade, 1 (1932), 145–148.Google Scholar
Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. (Oxford). To appear.Google Scholar
Mirsky, L., On a convex set of matrices, Archiv der Math., 10 (1959), 88–92.Google Scholar
von Neumann, J., Some matrix-inequalities and metrization of matricspace, Tomsk Univ. Rev. 1 (1937), 286–299.Google Scholar
Pólya, G., Remark on Weyl´s note : Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci., 36 (1950), 49–51.Google Scholar
Rado, R., An inequality, J. London Math. Soc, 27 (1952), 1–6.CrossRefGoogle Scholar
Schatten, R., A Theory of Cross-spaces (Princeton, 1950).Google Scholar