JENSEN’S TYPE TRACE INEQUALITIES FOR CONVEX FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES
Abstract
Some Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces are provided. Applications for some convex functions of interest are also given.
Keywords
Keywords
Full Text:
PDFReferences
T. Ando: Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33–38.
R. Bellman: 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.
E. V. Belmega, M. Jungers and S. Lasulce : A generalization of a trace inequality for positive definite matrices. Aust. J. Math. Anal. Appl. 7 (2010), no. 2, Art. 26, 5 pp.
E. A. Carlen: Trace inequalities and quantum entropy: an introductory course, [Online http://www.mathphys.org/AZschool/material/AZ09-carlen.pdf].
D. Chang: A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl. 237 (1999) 721–725.
L. Chen and C. Wong: Inequalities for singular values and traces, Linear Algebra Appl. 171 (1992), 109–120.
I. D. Coop: On matrix trace inequalities and related topics for products of Hermitian matrix, J. Math. Anal. Appl. 188 (1994) 999–1001.
S. S. Dragomir: 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.
S. S. Dragomir: On a reverse of Jessen’s inequality for isotonic linear functionals, J. Ineq. Pure & Appl. Math., 2(2001), No. 3, Article 36.
S. S. Dragomir: 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. [ONLINE:http://rgmia.org/v5(E).php].
S. S. Dragomir: Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (3) (2006), 471-476.
S. S. Dragomir: Bounds for the deviation of a function from the chord generated by its extremities. Bull. Aust. Math. Soc. 78 (2008), no. 2, 225–248.
S. S. Dragomir: Grüss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 11. [ONLINE: http://rgmia.org/v11(E).php].
S. S. Dragomir: 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.
S. S. Dragomir: 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.
S. S. Dragomir: 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. [ONLINE: http://rgmia.org/v11(E).php].
S. S. Dragomir: New bounds for the Čebyšev functional of two functions of selfadjoint operators in Hilbert spaces, Filomat 24 (2010), No. 2, 27-39.
S. S. Dragomir: Some Jensen’s type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Bull. Malays. Math. Sci. Soc. 34 (2011), No. 3. Preprint, RGMIA Res. Rep. Coll., 13 (2010), Sup. Art. 2.
S. S. Dragomir: Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces, J. Ineq. & Appl. Vol. 2010, Article ID 496821. Preprint, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 15. [ONLINE: http://rgmia.org/v11(E).php].
S. S. Dragomir: 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.
S. S. Dragomir: 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.
S. S. Dragomir: Hermite-Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Linear Algebra Appl. 436 (2012), no. 5, 1503–1515. Preprint, RGMIA Res. Rep. Coll. 13(2010), No. 2, Art 1.
S. S. Dragomir: 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), Sup. Art. 2.
S. S. Dragomir: Operator Inequalities of the Jensen, Čebyšev and Grüss Type. Springer Briefs in Mathematics. Springer, New York, 2012. xii+121 pp. ISBN: 978-1-4614-1520-6.
S. S. Dragomir: Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1
S. S. Dragomir: Some trace inequalities for convex functions of selfadjoint operators in Hilbert spaces, Korean J. Math. 24 (2016), No. 2, pp. 273-296, Preprint, RGMIA Res. Rep. Coll. 17 (2014), Art 115.
S. S. Dragomir and N. M. Ionescu: Some converse of Jensen’s inequality and applications. Rev. Anal. Numér. Théor. Approx. 23 (1994), no. 1, 71–78. MR:1325895 (96c:26012).
S. Furuichi and M. Lin: Refinements of the trace inequality of Belmega, Lasaulce and Debbah. Aust. J. Math. Anal. Appl. 7 (2010), no. 2, Art. 23, 4 pp.
T. Furuta, J. Mićić Hot, J. Pečarić and Y. Seo: Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
G. Helmberg: Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, 1969.
H. D. Lee: On some matrix inequalities, Korean J. Math. 16 (2008), No. 4, pp. 565-571.
L. Liu: A trace class operator inequality, J. Math. Anal. Appl. 328 (2007) 1484–1486.
S. Manjegani: Hölder and Young inequalities for the trace of operators, Positivity 11 (2007), 239–250.
H. Neudecker: A matrix trace inequality, J. Math. Anal. Appl. 166 (1992) 302–303.
K. Shebrawi and H. Albadawi: Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9(1) (2008), 1–10, article 26.
K. Shebrawi and H. Albadawi: Trace inequalities for matrices, Bull. Aust. Math. Soc. 87 (2013), 139–148.
B. Simon: Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.
A. Matković, J. Pečarić and I. Perić: A variant of Jensen’s inequality of Mercer’s type for operators with applications. Linear Algebra Appl. 418 (2006), no. 2-3, 551–564.
C. A. McCarthy: cp, Israel J. Math., 5(1967), 249-271.
J. Mićić, Y. Seo, S.-E. Takahasi and M. Tominaga: Inequalities of Furuta and Mond-Pečarić, Math. Ineq. Appl. 2 (1999), 83-111.
B. Mond and J. Pečarić: Convex inequalities in Hilbert space, Houston J. Math. 19 (1993), 405-420.
B. Mond and J. Pečarič: On some operator inequalities, Indian J. Math. 35 (1993), 221-232.
B. Mond and J. Pečarić: Classical inequalities for matrix functions, Utilitas Math. 46 (1994), 155-166.
S. Simić: On a global upper bound for Jensen’s inequality, J. Math. Anal. Appl. 343 (2008), 414-419.
Z. Ulukök and R. Türkmen: On some matrix trace inequalities. J. Inequal. Appl. 2010, Art. ID 201486, 8 pp.
X. Yang: A matrix trace inequality, J. Math. Anal. Appl. 250 (2000) 372–374.
X. M. Yang, X. Q. Yang and K. L. Teo: A matrix trace inequality, J. Math. Anal. Appl. 263 (2001), 327–331.
Y. Yang: A matrix trace inequality, J. Math. Anal. Appl. 133 (1988) 573–574.
DOI: https://doi.org/10.22190/FUMI1605981D
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)