Norm inequalities for the difference between weighted and integral means of operator differentiable functions

in the operator order, for all 2 [0; 1] and for every selfadjoint operator A and B on a Hilbert space H whose spectra are contained in I: Notice that a function f is operator concave if f is operator convex. A real valued continuous function f on an interval I is said to be operator monotone if it is monotone with respect to the operator order, i.e., A B with Sp (A) ;Sp (B) I imply f (A) f (B) : For some fundamental results on operator convex (operator concave) and operator monotone functions, see [9] and the references therein. As examples of such functions, we note that f (t) = t is operator monotone on [0;1) if and only if 0 r 1: The function f (t) = t is operator convex on (0;1) if either 1 r 2 or 1 r 0 and is operator concave on (0;1) if 0 r 1: The logarithmic function f (t) = ln t is operator monotone and operator concave on (0;1): The entropy function f (t) = t ln t is operator concave on (0;1): The exponential function f (t) = e is neither operator convex nor operator monotone.