Geodesic
Lower Bounds for Matrices, II
Canadian journal of mathematics, Tome 44 (1991) no. 1, pp. 54-74

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1992-003-9
Mots-clés : 26D15, 40G05, 47A30, 47B37 
Bennett, Grahame. Lower Bounds for Matrices, II. Canadian journal of mathematics, Tome 44 (1991) no. 1, pp. 54-74. doi: 10.4153/CJM-1992-003-9
@article{10_4153_CJM_1992_003_9,
     author = {Bennett, Grahame},
     title = {Lower {Bounds} for {Matrices,} {II}},
     journal = {Canadian journal of mathematics},
     pages = {54--74},
     year = {1991},
     volume = {44},
     number = {1},
     doi = {10.4153/CJM-1992-003-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1992-003-9/}
}
TY  - JOUR
AU  - Bennett, Grahame
TI  - Lower Bounds for Matrices, II
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 54
EP  - 74
VL  - 44
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1992-003-9/
DO  - 10.4153/CJM-1992-003-9
ID  - 10_4153_CJM_1992_003_9
ER  - 
%0 Journal Article
%A Bennett, Grahame
%T Lower Bounds for Matrices, II
%J Canadian journal of mathematics
%D 1991
%P 54-74
%V 44
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1992-003-9/
%R 10.4153/CJM-1992-003-9
%F 10_4153_CJM_1992_003_9

[1] 1. Bennett, G., Lower bounds for matrices, Linear Algebra and Appl. 82(1986), 81–98. Google Scholar

[2] 2. Bennett, G., Some elementary inequalities, Quart. Jour. Math. Oxford (2) 38(1987), 401–42. Google Scholar

[3] 3. Bennett, G., Some elementary inequalities, II, Quart. Jour. Math. Oxford (2) 39(1988), 385–400. Google Scholar

[4] 4. Bennett, G., Coin tossing and moment sequences, Discrete Math., 84(1990), 111–118. Google Scholar

[5] 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] 6. Ghatage, P., On the spectrum of the Bergman Hilbert matrix, Linear Algebra and App. 97(1987), 57–63. Google Scholar

[7] 7. Hardy, G.H., AM inequality for Hausdorff means, Jour. London Math. Soc. 18(1943), 46–50. Google Scholar

[8] 8. Hardy, G.H., Divergent series, Oxford University Press, 1949. Google Scholar

[9] 9. Hardy, G.H., Littlewood, J.E. and Polya, G., Inequalities, 2nd edition, Cambridge University Press, 1967. Google Scholar

[10] 10. Hausdorff, F., Summationsmethoden und Momentfolgen, I, Math. Z. 9(1921), 74–109. Google Scholar

[11] 11. Knopp, K., Uber Reihen mit positiven Gliedern (Zweite Mitteilung), Jour. London Math. Soc. 5(1930), 13–21. Google Scholar

[12] 12. Lenard, A., Personal communication. Google Scholar

[13] 13. Lyons, R., A lower bound on the Cesàro operator, Proc. Amer. Math. Soc. 86(1982), 694. Google Scholar

[14] 14. Marshall, A.W. and Olkin, W., Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. Google Scholar

[15] 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] 16. Rassias, J.M., On the generalized Cesàro operators, Math, analysis, 32–34. Teubner Texte Math. 79, Teubner, Leipzig, (1985). Google Scholar

[17] 17. Renaud, P.F.,A reversed Hardy inequality, Bull. Australian Math. Soc. 34(1986), 225–232. Google Scholar

[18] 18. Rhoades, B.E., A sufficient condition for total monotonicity, Trans. Amer. Math. Soc. 107( 1963), 309–319. Google Scholar

[19] 19. Rhoades, B.E., A sufficient condition for total monotonicity, II, Jour. Indian Math. Soc. 41(1977), 221–232. Google Scholar

[20] 20. Rhoades, B.E., Lower bounds for some matrices, Linear and Multilinear Algebra 20(1987), 347–352. Google Scholar

[21] 21. Rhoades, B.E., Square roots for Hausdorff operators, Integral Equations and Operator Theory 11(1988), 292–296. Google Scholar

[22] 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] 23. Wall, H.S., Continuedfractions and totally monotone sequences, Trans. Amer. Math. Soc. 48(1940), 165–184. Google Scholar

[24] 24. Widder, D.V., The Laplace Transform, Princeton University Press, Princeton, N.J. 1946. Google Scholar

[25] 25. Zeller, K. and Beekman, W., Théorie der Limitierungsverfahren, Ergebnisse der Math., 15, Springer-Verlag, Berlin, 1970. Google Scholar

Cité par Sources :