1. bookVolume 30 (2016): Issue 1 (September 2016)
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
access type Open Access

Reverse Jensen’s type Trace Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

Published Online: 23 Sep 2016
Volume & Issue: Volume 30 (2016) - Issue 1 (September 2016)
Page range: 39 - 62
Received: 22 Mar 2016
Accepted: 14 May 2016
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
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

MSC 2010

[1] Ando T., Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33–38.Search in Google Scholar

[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. 89–90.10.1007/978-3-0348-6324-7_8Search in Google Scholar

[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.Search in Google Scholar

[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. 73–140.10.1090/conm/529/10428Search in Google Scholar

[5] Chang D., A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl. 237 (1999), 721–725.10.1006/jmaa.1999.6433Search in Google Scholar

[6] Chen L., Wong C., Inequalities for singular values and traces, Linear Algebra Appl. 171 (1992), 109–120.10.1016/0024-3795(92)90253-7Search in Google Scholar

[7] Coop I.D., On matrix trace inequalities and related topics for products of Hermitian matrix, J. Math. Anal. Appl. 188 (1994), 999–1001.10.1006/jmaa.1994.1475Search in Google Scholar

[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.Search in Google Scholar

[9] Dragomir S.S., On a reverse of Jessen’s inequality for isotonic linear functionals, J. Ineqal. Pure Appl. Math. 2 (2001), No. 3, Art. 36.Search in Google Scholar

[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.10.1515/dema-2003-0308Search in Google Scholar

[11] Dragomir S.S., Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (2006), no. 3, 471–476.Search in Google Scholar

[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.Search in Google Scholar

[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].Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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. 7.10.1016/j.amc.2011.01.056Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[24] Dragomir S.S., Operator Inequalities of the Jensen, Čebyšev and Grüss Type, Springer Briefs in Mathematics, Springer, New York, 2012.10.1007/978-1-4614-1521-3Search in Google Scholar

[25] Dragomir S.S., Operator Inequalities of Ostrowski and Trapezoidal Type, Springer Briefs in Mathematics, Springer, New York, 2012.10.1007/978-1-4614-1779-8Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[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.Search in Google Scholar

[31] Helmberg G., Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, 1969.Search in Google Scholar

[32] Lee H.D., On some matrix inequalities, Korean J. Math. 16 (2008), no. 4, 565–571.Search in Google Scholar

[33] Liu L., A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), 1484–1486.10.1016/j.jmaa.2006.04.092Search in Google Scholar

[34] Manjegani S., Hölder and Young inequalities for the trace of operators, Positivity 11 (2007), 239–250.10.1007/s11117-006-2054-6Search in Google Scholar

[35] Neudecker H., A matrix trace inequality, J. Math. Anal. Appl. 166 (1992), 302–303.10.1016/0022-247X(92)90344-DSearch in Google Scholar

[36] Shebrawi K., Albadawi H., Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9 (2008), Art. 26, 1–10.Search in Google Scholar

[37] Shebrawi K., Albadawi H., Trace inequalities for matrices, Bull. Aust. Math. Soc. 87 (2013), 139–148.10.1017/S0004972712000627Search in Google Scholar

[38] Simon B., Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.Search in Google Scholar

[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.Search in Google Scholar

[40] McCarthy C.A., cp; Israel J. Math. 5 (1967), 249–271.10.1007/BF02771613Search in Google Scholar

[41] Mićić J., Seo Y., Takahasi S.-E., Tominaga M., Inequalities of Furuta and Mond-Pečarić, Math. Ineq. Appl. 2 (1999), 83–111.Search in Google Scholar

[42] Mond B., Pečarić J., Convex inequalities in Hilbert space, Houston J. Math. 19 (1993), 405–420.Search in Google Scholar

[43] Mond B., Pečarić J., On some operator inequalities, Indian J. Math. 35 (1993), 221–232.Search in Google Scholar

[44] B. Mond and J. Pečarić, Classical inequalities for matrix functions, Utilitas Math. 46 (1994), 155–166.Search in Google Scholar

[45] Riesz F., Sz-Nagy B., Functional Analysis, Dover Publications, New York, 1990.Search in Google Scholar

[46] Simić S., On a global upper bound for Jensen’s inequality, J. Math. Anal. Appl. 343 (2008), 414–419.10.1016/j.jmaa.2008.01.060Search in Google Scholar

[47] Ulukök Z., Türkmen R., On some matrix trace inequalities, J. Inequal. Appl. 2010, Art. ID 201486, 8 pp.10.1155/2010/201486Search in Google Scholar

[48] Yang X., A matrix trace inequality, J. Math. Anal. Appl. 250 (2000), 372–374.10.1006/jmaa.2000.7068Search in Google Scholar

[49] Yang X.M., Yang X.Q., Teo K.L., A matrix trace inequality, J. Math. Anal. Appl. 263 (2001), 327–331.10.1006/jmaa.2001.7613Search in Google Scholar

[50] Yang Y., A matrix trace inequality, J. Math. Anal. Appl. 133 (1988), 573–574.10.1016/0022-247X(88)90423-4Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo