Some Inequalities of Hermite-Hadamard Typefor Convex Functions of Commuting Selfadjoint Operators

Sever Silvestru Dragomir

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 3, page 491-511
  • ISSN: 0392-4041

Abstract

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Some operator inequalities for convex functions of commuting selfadjoint operators that are related to the Hermite-Hadamard inequality are given. Natural examples for some fundamental convex functions are presented as well.

How to cite

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Dragomir, Sever Silvestru. "Some Inequalities of Hermite-Hadamard Typefor Convex Functions of Commuting Selfadjoint Operators." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 491-511. <http://eudml.org/doc/294033>.

@article{Dragomir2013,
abstract = {Some operator inequalities for convex functions of commuting selfadjoint operators that are related to the Hermite-Hadamard inequality are given. Natural examples for some fundamental convex functions are presented as well.},
author = {Dragomir, Sever Silvestru},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {491-511},
publisher = {Unione Matematica Italiana},
title = {Some Inequalities of Hermite-Hadamard Typefor Convex Functions of Commuting Selfadjoint Operators},
url = {http://eudml.org/doc/294033},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Dragomir, Sever Silvestru
TI - Some Inequalities of Hermite-Hadamard Typefor Convex Functions of Commuting Selfadjoint Operators
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 491
EP - 511
AB - Some operator inequalities for convex functions of commuting selfadjoint operators that are related to the Hermite-Hadamard inequality are given. Natural examples for some fundamental convex functions are presented as well.
LA - eng
UR - http://eudml.org/doc/294033
ER -

References

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  1. ALLASIA, G. - GIORDANO, C. - PEČARIĆ, J., Hadamard-type inequalities for (2r)-convex functions with applications, Atti Acad. Sci. Torino-Cl. Sc. Fis., 133 (1999), 1-14. MR1799453
  2. AZPEITIA, A. G., Convex functions and the Hadamard inequality, Rev.-Colombiana-Mat., 28 (1) (1994), 7-12. Zbl0832.26015MR1304041
  3. BECKENBACH, E. F. - BELLMAN, R., Inequalities, 4th Edition, Springer-Verlag, Berlin, 1983. MR158038
  4. DRAGOMIR, S. S., An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math.3, no. 2 (2002), 8. Zbl0994.26018MR1906400
  5. DRAGOMIR, S. S., An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math.3, no. 3 (2002), 8. Zbl0995.26009MR1917794
  6. DRAGOMIR, S. S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc.74 (3) (2006), 471-476. Zbl1113.26021MR2273755DOI10.1017/S000497270004051X
  7. DRAGOMIR, S. S., Hermite-Hadamard's type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Linear Algebra and its Applications, 436, Issue 5, Pages 1503-1515. Zbl1248.47017MR2890934DOI10.1016/j.laa.2011.08.050
  8. DRAGOMIR, S. S., Hermite-Hadamard's type inequalities for operator convex functions, Applied Mathematics and Computation, 218, Issue 3, Pages 766-772. Zbl1239.47009MR2831305DOI10.1016/j.amc.2011.01.056
  9. DRAGOMIR, S. S. - CERONE, P. - SOFO, A., Some remarks on the midpoint rule in numerical integration, Studia Univ. Babes - Bolyai Math.45, no. 1 (2000), 63-74. Preprint RGMIA Res. Rep. Coll., 1, no. 2 (1998), 25-33. Zbl1027.26021MR2062533
  10. DRAGOMIR, S. S. - CERONE, P. - SOFO, A., Some remarks on the trapezoid rule in numerical integration. Indian J. Pure Appl. Math.31, no. 5 (2000), 475-494. Preprint RGMIA Res. Rep. Coll. , 2, no. 5 (1998), Art. 1. Zbl0967.41019MR1757670
  11. DRAGOMIR, S. S. - PEARCE, C. E. M., Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. [http://www.staff.vu.edu.au/RGMIA/monographs/hermite_hadamard.html]. 
  12. DRAGOMIR, S. S., Some inequalities of Čebyšev type for functions of operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 15 (2012), Article 41, 12 pp. MR3280660DOI10.5644/SJM.10.2.08
  13. FINK, A. M., Toward a theory of best possible inequalities, Nieuw Archief von Wiskunde, 12 (1994), 19-29. Zbl0827.26018MR1284677
  14. FURUTA, T. - MIĆIĆHOT, J. - PEČARIĆ, J. - SEO, Y., Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005. Zbl1135.47012MR3026316
  15. GANTMACHER, F. R., The Theory of Matrices, Chelsea, 1959. Zbl0085.01001MR107649
  16. GAVREA, B., On Hadamard's inequality for the convex mappings defined on a convex domain in the space, Journal of Ineq. in Pure1, no. 1(2000), Article 9, http://jipam.vu.edu.au/. Zbl0949.26008MR1756660
  17. GOHBERG, I. - LANCASTER, P. - RODMAN, L., Invariant Subspaces of Matrices with Application, Wiley-Interscience, 1986. MR873503
  18. GILL, P. M. - PEARCE, C. E. M. - PEČARIĆ, J., Hadamard's inequality for r-convex functions, J. of Math. Anal. and Appl., 215 (1997), 461-470. Zbl0891.26013MR1490762DOI10.1006/jmaa.1997.5645
  19. LEE, K.-C. - TSENG, K.-L., On a weighted generalisation of Hadamard's inequality for G-convex functions, Tamsui Oxford Journal of Math. Sci., 16 (1) (2000), 91-104. MR1772077
  20. LUPAS, A., A generalisation of Hadamard's inequality for convex functions, Univ. Beograd. Publ. Elek. Fak. Ser. Mat. Fiz., 544-576, (1976), 115-121. MR444865
  21. MAKSIMOVIĆ, D. M., A short proof of generalized Hadamard's inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., no. 634-677 (1979), 126-128. MR579274
  22. MATKOVIĆ, A. - PEČARIĆ, J. - PERIĆ, I., A variant of Jensen's inequality of Mercer's type for operators with applications. Linear Algebra Appl., 418, no. 2-3 (2006), 551- 564. Zbl1105.47017MR2260210DOI10.1016/j.laa.2006.02.030
  23. MCCARTHY, C. A., c p ; Israel J. Math., 5 (1967), 249-271. MR225140DOI10.1007/BF02771613
  24. MIĆIĆ, J. - SEO, Y. - TAKAHASI, S.-E. - TOMINAGA, M., Inequalities of Furuta and Mond-Pečarić, Math. Ineq. Appl., 2 (1999), 83-111. MR1667794DOI10.7153/mia-02-08
  25. MITRINOVIĆ, D. S. - LACKOVIĆ, I., Hermite and convexity, Aequat. Math., 28 (1985), 229-232. MR791622DOI10.1007/BF02189414
  26. MITRINOVIĆ, D. S. - PEČARIĆ, J. - FINK, A. M., Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993. Zbl0771.26009MR1220224DOI10.1007/978-94-017-1043-5
  27. MOND, B. - PEČARIĆ, J., Convex inequalities in Hilbert space, Houston J. Math., 19 (1993), 405-420. Zbl0813.46016MR1242427
  28. MOND, B. - PEČARIĆ, J., On some operator inequalities, Indian J. Math., 35 (1993), 221-232. MR1291724
  29. MOND, B. - PEČARIĆ, J., Classical inequalities for matrix functions, Utilitas Math., 46 (1994), 155-166. Zbl0823.15018MR1301304
  30. RIESZ, F. - SZ-NAGY, B., Functional Analysis, New York, Dover Publications, 1990. MR1068530
  31. SUPRUNENKO, D. A. - TYSHKEVICH, R. I., Commutative Matrices, Academic, 1968. MR201472

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