Reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces
Abstract
Some reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest and reverses of Hölder and Schwarz trace inequalities are also given.
Keywords
trace class operators; Hilbert-Schmidt operators; trace; convex functions; Jensen’s inequality; trace inequalities for matrices
References
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6. Chen L., Wong C., Inequalities for singular values and traces, Linear Algebra Appl. 171 (1992), 109–120.
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8. Dragomir S.S., A converse result for Jensen’s discrete inequality via Gruss’ inequality and applications in information theory, An. Univ. Oradea Fasc. Mat. 7 (1999/2000), 178–189.
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11. Dragomir S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), no. 3, 471–476.
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27. Dragomir S.S., Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 116. Available at http://rgmia.org/papers/v17/v17a116.pdf.
28. Dragomir S.S., Ionescu N.M., Some converse of Jensen’s inequality and applications, Rev. Anal. Numér. Théor. Approx. 23 (1994), no. 1, 71–78.
29. Furuichi S., Lin M., Refinements of the trace inequality of Belmega, Lasaulce and Debbah, Aust. J. Math. Anal. Appl. 7 (2010), no. 2, Art. 23, 4 pp.
30. Furuta T., Mićić Hot J., Pečarić J., Seo Y., Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
31. Helmberg G., Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, 1969.
32. Lee H.D., On some matrix inequalities, Korean J. Math. 16 (2008), no. 4, 565–571.
33. Liu L., A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), 1484–1486.
34. Manjegani S., Hölder and Young inequalities for the trace of operators, Positivity 11 (2007), 239–250.
35. Neudecker H., A matrix trace inequality, J. Math. Anal. Appl. 166 (1992), 302–303.
36. Shebrawi K., Albadawi H., Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9 (2008), Art. 26, 1–10.
37. Shebrawi K., Albadawi H., Trace inequalities for matrices, Bull. Aust. Math. Soc. 87 (2013), 139–148.
38. Simon B., Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.
39. Matković A., Pečarić J., Perić I., A variant of Jensen’s inequality of Mercer’s type for operators with applications, Linear Algebra Appl. 418 (2006), no. 2-3, 551–564.
40. McCarthy C.A., c_p; Israel J. Math. 5 (1967), 249–271.
41. Mićić J., Seo Y., Takahasi S.-E., Tominaga M., Inequalities of Furuta and Mond-Pečarić, Math. Ineq. Appl. 2 (1999), 83–111.
42. Mond B., Pečarić J., Convex inequalities in Hilbert space, Houston J. Math. 19 (1993), 405–420.
43. Mond B., Pečarić J., On some operator inequalities, Indian J. Math. 35 (1993), 221–232.
44. Mond B., Pečarić J., Classical inequalities for matrix functions, Utilitas Math. 46 (1994), 155–166.
45. Riesz F., Sz-Nagy B., Functional Analysis, Dover Publications, New York, 1990.
46. Simić S., On a global upper bound for Jensen’s inequality, J. Math. Anal. Appl. 343 (2008), 414–419.
47. Ulukök Z., Türkmen R., On some matrix trace inequalities, J. Inequal. Appl. 2010, Art. ID 201486, 8 pp.
48. Yang X., A matrix trace inequality, J. Math. Anal. Appl. 250 (2000), 372–374.
49. Yang X.M., Yang X.Q., Teo K.L., A matrix trace inequality, J. Math. Anal. Appl. 263 (2001), 327–331.
50. Yang Y., A matrix trace inequality, J. Math. Anal. Appl. 133 (1988), 573–574.
2. Bellman R., Some inequalities for positive definite matrices, in: E.F. Beckenbach (Ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, Birkhäuser, Basel, 1980, pp. 89–90.
3. Belmega E.V., Jungers M., Lasaulce S., A generalization of a trace inequality for positive definite matrices, Aust. J. Math. Anal. Appl. 7 (2010), no. 2, Art. 26, 5 pp.
4. Carlen E.A., Trace inequalities and quantum entropy: an introductory course, in: Entropy and the quantum, Contemp. Math. 529, Amer. Math. Soc., Providence, RI, 2010, pp. 73–140.
5. Chang D., A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl. 237 (1999), 721–725.
6. Chen L., Wong C., Inequalities for singular values and traces, Linear Algebra Appl. 171 (1992), 109–120.
7. Coop I.D., On matrix trace inequalities and related topics for products of Hermitian matrix, J. Math. Anal. Appl. 188 (1994), 999–1001.
8. Dragomir S.S., A converse result for Jensen’s discrete inequality via Gruss’ inequality and applications in information theory, An. Univ. Oradea Fasc. Mat. 7 (1999/2000), 178–189.
9. Dragomir S.S., On a reverse of Jensen’s inequality for isotonic linear functionals, J. Ineqal. Pure Appl. Math. 2 (2001), No. 3, Art. 36.
10. Dragomir S.S., A Grüss type inequality for isotonic linear functionals and applications, Demonstratio Math. 36 (2003), no. 3, 551–562. Preprint RGMIA Res. Rep. Coll. 5 (2002), Suplement, Art. 12. Available at http://rgmia.org/v5(E).php.
11. Dragomir S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), no. 3, 471–476.
12. Dragomir S.S., Bounds for the deviation of a function from the chord generated by its extremities, Bull. Aust. Math. Soc. 78 (2008), no. 2, 225–248.
13. Dragomir S.S., Grüss’ type inequalities for functions of selfadjoint operators in Hilbert spaces. Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 11. Available at http://rgmia.org/v11(E).php.
14. Dragomir S.S., Some inequalities for convex functions of selfadjoint operators in Hilbert spaces, Filomat 23 (2009), no. 3, 81–92. Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 10.
15. Dragomir S.S., Some Jensen’s type inequalities for twice differentiable functions of selfadjoint operators in Hilbert spaces, Filomat 23 (2009), no. 3, 211–222. Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 13.
16. Dragomir S.S., Some new Grüss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Sarajevo J. Math. 6(18) (2010), no. 1, 89–107. Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 12. Available at http://rgmia.org/v11(E).php.
17. Dragomir S.S., New bounds for the Čebyšev functional of two functions of selfadjoint operators in Hilbert spaces, Filomat 24 (2010), no. 2, 27–39.
18. Dragomir S.S., Some Jensen’s type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Bull. Malays. Math. Sci. Soc. 34 (2011), no. 3, 445–454. Preprint RGMIA Res. Rep. Coll. 13 (2010), Suplement, Art. 2.
19. Dragomir S.S., Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces, J. Ineq. & Appl. (2010), Art. ID 496821. Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 15. Available at http://rgmia.org/v11(E).php.
20. Dragomir S.S., Some Slater’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Rev. Un. Mat. Argentina 52 (2011), no. 1, 109–120. Preprint RGMIA Res. Rep. Coll. 11(e) (2008), Art. 7.
21. Dragomir S.S., Hermite-Hadamard’s type inequalities for operator convex functions, Appl. Math. Comp. 218 (2011), 766–772. Preprint RGMIA Res. Rep. Coll. 13 (2010), no. 1, Art. 7.
22. Dragomir S.S., Hermite-Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces. Preprint RGMIA Res. Rep. Coll. 13 (2010), no. 2, Art 1.
23. Dragomir S.S., New Jensen’s type inequalities for differentiable log-convex functions of selfadjoint operators in Hilbert spaces, Sarajevo J. Math. 19 (2011), no. 1, 67–80. Preprint RGMIA Res. Rep. Coll. 13 (2010), Suplement, Art. 2.
24. Dragomir S.S., Operator Inequalities of the Jensen, Čebyšev and Grüss Type, Springer Briefs in Mathematics, Springer, New York, 2012.
25. Dragomir S.S., Operator Inequalities of Ostrowski and Trapezoidal Type, Springer Briefs in Mathematics, Springer, New York, 2012.
26. Dragomir S.S., Some trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 115. Available at http://rgmia.org/papers/v17/v17a115.pdf.
27. Dragomir S.S., Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Preprint RGMIA Res. Rep. Coll. 17 (2014), Art. 116. Available at http://rgmia.org/papers/v17/v17a116.pdf.
28. Dragomir S.S., Ionescu N.M., Some converse of Jensen’s inequality and applications, Rev. Anal. Numér. Théor. Approx. 23 (1994), no. 1, 71–78.
29. Furuichi S., Lin M., Refinements of the trace inequality of Belmega, Lasaulce and Debbah, Aust. J. Math. Anal. Appl. 7 (2010), no. 2, Art. 23, 4 pp.
30. Furuta T., Mićić Hot J., Pečarić J., Seo Y., Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
31. Helmberg G., Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, 1969.
32. Lee H.D., On some matrix inequalities, Korean J. Math. 16 (2008), no. 4, 565–571.
33. Liu L., A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), 1484–1486.
34. Manjegani S., Hölder and Young inequalities for the trace of operators, Positivity 11 (2007), 239–250.
35. Neudecker H., A matrix trace inequality, J. Math. Anal. Appl. 166 (1992), 302–303.
36. Shebrawi K., Albadawi H., Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9 (2008), Art. 26, 1–10.
37. Shebrawi K., Albadawi H., Trace inequalities for matrices, Bull. Aust. Math. Soc. 87 (2013), 139–148.
38. Simon B., Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.
39. Matković A., Pečarić J., Perić I., A variant of Jensen’s inequality of Mercer’s type for operators with applications, Linear Algebra Appl. 418 (2006), no. 2-3, 551–564.
40. McCarthy C.A., c_p; Israel J. Math. 5 (1967), 249–271.
41. Mićić J., Seo Y., Takahasi S.-E., Tominaga M., Inequalities of Furuta and Mond-Pečarić, Math. Ineq. Appl. 2 (1999), 83–111.
42. Mond B., Pečarić J., Convex inequalities in Hilbert space, Houston J. Math. 19 (1993), 405–420.
43. Mond B., Pečarić J., On some operator inequalities, Indian J. Math. 35 (1993), 221–232.
44. Mond B., Pečarić J., Classical inequalities for matrix functions, Utilitas Math. 46 (1994), 155–166.
45. Riesz F., Sz-Nagy B., Functional Analysis, Dover Publications, New York, 1990.
46. Simić S., On a global upper bound for Jensen’s inequality, J. Math. Anal. Appl. 343 (2008), 414–419.
47. Ulukök Z., Türkmen R., On some matrix trace inequalities, J. Inequal. Appl. 2010, Art. ID 201486, 8 pp.
48. Yang X., A matrix trace inequality, J. Math. Anal. Appl. 250 (2000), 372–374.
49. Yang X.M., Yang X.Q., Teo K.L., A matrix trace inequality, J. Math. Anal. Appl. 263 (2001), 327–331.
50. Yang Y., A matrix trace inequality, J. Math. Anal. Appl. 133 (1988), 573–574.
DragomirS. S. (2016). Reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces. Annales Mathematicae Silesianae, 30, 39-62. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13955
Sever S. Dragomir
sever.dragomir@vu.edu.au
Mathematics, College of Engineering & Science, Victoria University, Australia Australia
Mathematics, College of Engineering & Science, Victoria University, Australia Australia
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