Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces
DOI:
https://doi.org/10.17398/2605-5686.37.2.261Keywords:
operator monotone functions, operator convex functions, operator inequalities, midpoint inequality, trapezoid inequalityAbstract
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.
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