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Hermite-Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces. (English) Zbl 1248.47017

In the continuation of his paper [Appl. Math. Comput. 218, No. 3, 766–772 (2011; Zbl 1239.47009)], the author uses functional calculus to present some Hermite-Hadamard type inequalities for convex functions of self-adjoint operators in Hilbert spaces under appropriate assumptions. He also improves the Hölder-McCarthy inequality for positive operators and applies his results to Ky Fan’s inequality for real numbers.

MSC:

47A63 Linear operator inequalities
47A60 Functional calculus for linear operators

Citations:

Zbl 1239.47009
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References:

[1] Allasia, G.; Giordano, C.; Pečarić, J., Hadamard-type inequalities for (2r)-convex functions with applications, Atti accad. sci. Torino cl. sci. fis. mat. natur., 133, 1-14, (1999)
[2] Azpeitia, A.G., Convex functions and the Hadamard inequality, Rev. colombiana mat., 28, 1, 7-12, (1994) · Zbl 0832.26015
[3] Beckenbach, E.F.; Bellman, R., Inequalities, (1983), Springer-Verlag Berlin · Zbl 0513.26003
[4] Dragomir, S.S., Some refinements of Ky fan’s inequality, J. math. anal. appl., 163, 2, 317-321, (1992) · Zbl 0768.26006
[5] Dragomir, S.S., Some refinements of jensen’s inequality, J. math. anal. appl., 168, 2, 518-522, (1992) · Zbl 0765.26007
[6] S.S. Dragomir, Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces, J. Inequal. Appl. (2010) (Article ID 496821, Preprint RGMIA Res. Rep. Coll. 11(e) (2008) (Art. 15)).
[7] S.S. Dragomir, Some Jensen’s type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Bull. Malaysian Math. Sci. Soc. 34 (3) (2011) (Preprint RGMIA Res. Rep. Coll. 13 (2010) (Sup. Art. 2)). · Zbl 1239.47014
[8] S.S. Dragomir, C.E.M. Pearce, Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. <http://www.rgmia.org/monographs.php>.
[9] Fink, A.M., Toward a theory of best possible inequalities, Nieuw arch. wisk., 12, 19-29, (1994) · Zbl 0827.26018
[10] T. Furuta, J. Mićić Hot, J. Pečarić, Y. Seo, Mond-Pečarić Method in Operator Inequalities, Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
[11] B. Gavrea, On Hadamard’s inequality for the convex mappings defined on a convex domain in the space, J. Inequal. Pure Appl. Math. 1 (1) (2000) (Article 9) <http://www.jipam.vu.edu.au/>. · Zbl 0949.26008
[12] Gill, P.M.; Pearce, C.E.M.; Pečarić, J., Hadamard’s inequality for \(r\)-convex functions, J. math. anal. appl., 215, 461-470, (1997) · Zbl 0891.26013
[13] Lee, K.C.; Tseng, K.L., On a weighted generalisation of hadamard’s inequality for G-convex functions, Tamsui oxf. J. math. sci., 16, 1, 91-104, (2000)
[14] Lupaş, A., A generalisation of hadamard’s inequality for convex functions, Univ. beograd. publ. elektrotehn. fak. ser. mat., 544-576, 115-121, (1976)
[15] Maksimović, D.M., A short proof of generalized hadamard’s inequalities, Univ. beograd. publ. elektrotehn. fak. ser. mat., 634-677, 126-128, (1979) · Zbl 0452.26006
[16] Matković, A.; Pečarić, J.; Perić, I., A variant of jensen’s inequality of mercer’s type for operators with applications, Linear algebra appl., 418, 2-3, 551-564, (2006) · Zbl 1105.47017
[17] Mitrinović, D.S.; Lacković, I., Hermite and convexity, Aequationes math., 28, 229-232, (1985) · Zbl 0572.26004
[18] McCarthy, C.A., \(c_p\), Israel J. math., 5, 249-271, (1967)
[19] Mićić, J.; Seo, Y.; Takahasi, S.E.; Tominaga, M., Inequalities of Furuta and mond – pečarić, Math. inequal. appl., 2, 83-111, (1999) · Zbl 0924.47013
[20] Mitrinović, D.S.; Pečarić, J.; Fink, A.M., Classical and new inequalities in analysis, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0771.26009
[21] Mond, B.; Pečarić, J., Convex inequalities in Hilbert space, Houston J. math, 19, 405-420, (1993) · Zbl 0813.46016
[22] Mond, B.; Pečarić, J., On some operator inequalities, Indian J. math., 35, 221-232, (1993) · Zbl 1052.47505
[23] Mond, B.; Pečarić, J., Classical inequalities for matrix functions, Util. math., 46, 155-166, (1994) · Zbl 0823.15018
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