Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T11:29:14.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower Bounds for Matrices, II

Published online by Cambridge University Press:  20 November 2018

Grahame Bennett
Grahame Bennett*
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our main result is the following monotonicity property for moment sequences μ. Let p be fixed, 1 ≤ p < ∞: then

is an increasing function of r(r = 1,2,…). From this we derive a sharp lower bound for an arbitrary Hausdorff matrix acting on ℓp.The corresponding upper bound problem was solved by Hardy.

Keywords

26D1540G0547A3047B37
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 1 , 01 December 1991 , pp. 54 - 74
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bennett, G., Lower bounds for matrices, Linear Algebra and Appl. 82(1986), 81–98.Google Scholar
2. Bennett, G., Some elementary inequalities, Quart. Jour. Math. Oxford (2) 38(1987), 401–42.Google Scholar
3. Bennett, G., Some elementary inequalities, II, Quart. Jour. Math. Oxford (2) 39(1988), 385–400.Google Scholar
4. Bennett, G., Coin tossing and moment sequences, Discrete Math., 84(1990), 111–118.Google Scholar
5. Garabedian, H.L., Hille, E. and Wall, H.S., Formulations of the Hausdorff inclusion problem Trans. Amer. Math. Soc. 8(1941), 193–213.Google Scholar
6. Ghatage, P., On the spectrum of the Bergman Hilbert matrix, Linear Algebra and App. 97(1987), 57–63.Google Scholar
7. Hardy, G.H., AM inequality for Hausdorff means, Jour. London Math. Soc. 18(1943), 46–50.Google Scholar
8. Hardy, G.H., Divergent series, Oxford University Press, 1949.Google Scholar
9. Hardy, G.H., Littlewood, J.E. and Polya, G., Inequalities, 2nd edition, Cambridge University Press, 1967.Google Scholar
10. Hausdorff, F., Summationsmethoden und Momentfolgen, I, Math. Z. 9(1921), 74–109.Google Scholar
11. Knopp, K., Uber Reihen mit positiven Gliedern (Zweite Mitteilung), Jour. London Math. Soc. 5(1930), 13–21.Google Scholar
12. Lenard, A., Personal communication.Google Scholar
13. Lyons, R., A lower bound on the Cesàro operator, Proc. Amer. Math. Soc. 86(1982), 694.Google Scholar
14. Marshall, A.W. and Olkin, W., Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979.Google Scholar
15. Polya, G., Remark on Weyl's note “Inequalities between the two kinds of eigenvalues of a linear transformation”, Proc. Nat. Acad. Sci. U.S.A. 36(1950), 49–51.Google Scholar
16. Rassias, J.M., On the generalized Cesàro operators, Math, analysis, 32–34. Teubner Texte Math. 79, Teubner, Leipzig, (1985).Google Scholar
17. Renaud, P.F.,A reversed Hardy inequality, Bull. Australian Math. Soc. 34(1986), 225–232.Google Scholar
18. Rhoades, B.E., A sufficient condition for total monotonicity, Trans. Amer. Math. Soc. 107( 1963), 309–319.Google Scholar
19. Rhoades, B.E., A sufficient condition for total monotonicity, II, Jour. Indian Math. Soc. 41(1977), 221–232.Google Scholar
20. Rhoades, B.E., Lower bounds for some matrices, Linear and Multilinear Algebra 20(1987), 347–352.Google Scholar
21. Rhoades, B.E., Square roots for Hausdorff operators, Integral Equations and Operator Theory 11(1988), 292–296.Google Scholar
22. Tomic, M., Théorème de Gauss relatif au centre de gravité et son application, Bull. Soc. Math. Phys. Serbie 1(1949), 31–40.Google Scholar
23. Wall, H.S., Continuedfractions and totally monotone sequences, Trans. Amer. Math. Soc. 48(1940), 165–184.Google Scholar
24. Widder, D.V., The Laplace Transform, Princeton University Press, Princeton, N.J. 1946.Google Scholar
25. Zeller, K. and Beekman, W., Théorie der Limitierungsverfahren, Ergebnisse der Math., 15, Springer-Verlag, Berlin, 1970.Google Scholar