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