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References [1] V. Bacak and R. Türkmen, Refinements of Hermite-Hadamard type inequalities for operator convex functions. J. Inequal. Appl. 2013 , 2013:262, 10 pp. [2] B. Li, Refinements of Hermite-Hadamard’s type inequalities for operator convex functions. Int. J. Contemp. Math. Sci. 8 2013, no. 9-12, 463–467. [3] M. D. Choi, Positive linear maps on C*-algebras. Canad. J. Math. 24 1972, 520–529. [4] S. S. Dragomir, Hermite–Hadamard’s type inequalities for operator convex functions, Applied Mathematics and Computation , 218 2011, Issue 3, pp. 766

Georgian Math. J. 20 (2013), 625–640 DOI 10.1515/gmj-2013-0035 © de Gruyter 2013 Operator-valued Bochner integrable functions and Jensen’s inequality Farid Bahrami, Ali Bayati Eshkaftaki and Seyed Mahmoud Manjegani Abstract. In this paper we obtain a Jensen’s type inequality for operator-valued integrable functions, which generalizes some of the previous results in this regard. More precisely, if .;†;/ is a probability measure space and if is an operator convex function, then, under suitable conditions, we show that . R  gfg d/ R  g ı fg d, where f W  ! B

References [1] T. Antczak, Mean value in invexity analysis, Nonlinear Anal ., 60 (2005), 1473–1484 [2] E. F. Beckenbach, Convex functions, Bull. Amer. Math. Soc ., 54 (1948), 439–460. [3] R. Bhatia, Matrix Analysis , GTM 169, Springer-Verlag, New York, 1997. [4] S. S. Dragomir, Hermite-Hadamards type inequalities for operator convex functions, Applied Mathematics and Computation ., 218 (2011), 766–772 [5] A. G. Ghazanfari, M. Shakoori, A. Barani, S. S. Dragomir, Hermite-Hadamard type inequality for operator preinvex functions, arXiv:1306.0730v1 [6] R. A

References [1] CONDE, C.: A version of the Hermite-Hadamard inequality in a nonpositive curvature space, Banach J. Math. Anal. 6 (2012), 159-167. [2] DAHMANI, Z.: On Minkowski and Hermite-Haamard integral inequalities via fractional integration, Ann. Funct. Anal. 1 (2010), 51-58. [3] DRAGOMIR, S. S.: Hermite-Hadamard’s type inequalities for operator convex functions, Appl. Math. Comput. 218 (2011), 766-772. [4] DRAGOMIR, S. S.-PEČARIĆ, J.-PERSSON, L.-E.: Some inequalities of Hadamard type, Soochow J. Math. 21 (1995), 335-341. [5] FUJII, J. I.-KIAN, M

logarithmic mean, Resonance 13 (2008), no. 6, 583-594. [5] Carlson B.C., The logarithmic mean, Amer. Math. Monthly 79 (1972), no. 6, 615-618. [6] Cerone P., Dragomir S.S., Mathematical inequalities. A perspective, CRC Press, Boca Raton, 2011. [7] Conde C., A version of the Hermite-Hadamard inequality in a nonpositive curvature space, Banach J. Math. Anal. 6 (2012), no. 2, 159-167. [8] Dragomir S.S., Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc. 74 (2006), no. 3, 471-478. [9] Dragomir S.S., Hermite-Hadamard’s type inequalities for operator convex

. Set, On some inequalities of Hermite-Hadamard type via m- convexity, Appl. Math. Lett., 23(2010), 1065-1070. [14] M. E. Ödemir, M. Avci, H. Kavurmaci, Hermite-Hadamard-type inequalities via (fi;m)- convexity, Comput. Math. Appl., 61(2011), 2614-2620. [15] S. S. Dragomir, Hermite-Hadamard's type inequalities for operator convex functions, Appl. Math. Comput., 218(2011), 766-772. [16] S. S. Dragomir, Hermite-Hadamard's type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Linear Algebra Appl., 436(2012), 1503-1515. [17] M. Z. Sarikaya, N

Hilbert Spaces, Second Edition, Springer, (2017). [6] I. D. Berg and I. G. Nikolaev, Quasilinearization and curvature of Aleksanderov spaces , Geom. Dedicata. 133 (2008), 195–218. [7] J.M.Borwein, Fifty years of maximal monotonicity , Optim. Lett. 4 (2010), 473–490. [8] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundelhern Math. Wiss., Springer, (1999). [9] R. S. Burachik and B. F. Svaiter, Maximal monotone operators, convex functions and a special family of enlargements , Set-Valued Anal. 10 (2002), 297–316. [10] P.Chaipunya and P

-Hadamard’s type inequalities for operator convex functions, Appl. Math. Comput. 218 (2011), 766–772. http://dx.doi.org/10.1016/j.amc.2011.01.056 [8] DRAGOMIR, S. S.: Hermite-Hadamard’s type inequalities for convex functions of self-adjoint operators in Hilbert spaces, Linear Algebra Appl. 436 (2012), 1503–1515. http://dx.doi.org/10.1016/j.laa.2011.08.050 [9] DUTKIEWICZ, A.: On the Aronszajn property for a differential equation of fractional order in Banach spaces, Math. Slovaca 61 (2011), 571–578. http://dx.doi.org/10.2478/s12175-011-0029-y [10] KILBAS, A. A.— SRIVASTAVA, H. M

. 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

that if Φ : 𝔹 ⁢ ( ℋ ) → 𝔹 ⁢ ( 𝒦 ) {\Phi\colon\mathbb{B}({\mathscr{H}})\to\mathbb{B}({\mathscr{K}})} is a unital positive linear map and φ is an operator convex function on an interval J , then φ ⁢ ( Φ ⁢ ( A ) ) ≤ Φ ⁢ ( φ ⁢ ( A ) ) \varphi(\Phi(A))\leq\Phi(\varphi(A)) for every self-adjoint operator A on ℋ {\mathscr{H}} whose spectrum is contained in J . In [ 16 ], Toader defined the m -convexity. Let J = [ 0 , b ] {J=[0,b]} for some b ∈ ℝ {b\in\mathbb{R}} or J = [ 0 , ∞ ) {J=[0,\infty)} . A function φ : J → ℝ {\varphi\colon J