Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces

For a continuous and positive function w (λ), λ > 0 µ measure on (0, ∞) we consider the following integral transform
 D (w, µ) (T ) := ∫0∞w (λ) (λ + T −1 dµ ,
 where is assumed to exist for operator complex Hilbert space H. We show among others that, if A ≥ m 1 0, B 2 then
 ||D (B) − (A) (D µ)) (B A)||
 ≤|B A|2×[D(w,µ)(m2)−D(w,µ)(m1)−(m2- m1)D’(w,µ)(m1)]/(m2−m1)2 m1≠m2,
 ≤ D’’(w, µ)(m)/2 m1=m2=m,
 Fréchet derivative of as second real function.
 also prove norm inequalities power r ∈ 1] A, 0,
 ||∫01((1−t)A+tB)r−1dt−((A+B)/2)r−1|| (1−r) (2−r) mr−3||B−A||2/24
 and
 ||((Ar−1+Br−1 )/2) ∫01((1−t) A+tB)r−1dt|| mr−3||B A||2/12.