By Dragomir S.S.
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Additional info for Inequalities for functions of selfadjoint operators in Hilbert spaces
4] J. Mi´ci´c, Y. -E. Takahasi and M. Tominaga, Inequalities of Furuta and Mond-Peµcari´c, Math. Ineq. , 2(1999), 83-111.  B. Mond and J. Peµcari´c, Convex inequalities in Hilbert space, Houston J. , 19(1993), 405-420.  B. Mond and J. Peµcari´c, On some operator inequalities, Indian J. , 35(1993), 221-232. 1 Introduction Jensen’s type inequalities in their various settings ranging from discrete to continuous case play an important role in di¤erent branches of Modern Mathematics. A simple search in the MathSciNet database of the American Mathematical Society with the key words "jensen" and "inequality" in the title reveals more than 300 items intimately devoted to this famous result.
The second inequality will provide the Mond-Peµcari´c inequality for convex functions whose derivatives are continuous. The following corollary is of interest: Corollary 71 Let I be an interval and f : I ! 77) in the operator order of B (H) : Proof. 78) any x 2 H with kxk = 1: µ We use the following Cebyš ev type inequality for functions of operators established by the author in : Let A be a selfadjoint operator with Sp (A) [m; M ] for some real numbers m < M: If h; g : [m; M ] ! 77). We provide now some particular inequalities of interest that can be derived from Theorem 69: Example 72 a.
43) hAf 0 (A) x; xi hAx; xi hf 0 (A) x; xi ; hf 0 (A) x; xi for any x 2 H with kxk = 1: Proof. 43). 49) for any x 2 H with kxk = 1: Indeed, we observe that if we consider the concave function f : (0; 1) ! 49). 48) for the convex (concave) function f (t) = tp ; p 2 ( 1; 0) [ [1; 1) (p 2 (0; 1)) by performing the required calculation. The details are omitted. 40 2. 3 Further Reverses The following results that provide perhaps more useful upper bounds for the nonnegative quantity hAf 0 (A) x; xi hf 0 (A) x; xi f hf (A) x; xi for x 2 H with kxk = 1; can be stated: Theorem 63 (Dragomir, 2008, ) Let I be an interval and f : I !