Dragomir, Sever S ORCID: 0000-0003-2902-6805 (2022) Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces. Extracta Mathematicae, 37 (2). pp. 261-282. ISSN 0213-8743

Abstract

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 ≥ m1 > 0, B ≥ m2 > 0, then ∥D (w, µ) (B) − D (w, µ) (A) − D (D (w, µ)) (A) (B − A) ∥ ≤ ∥B − A∥2 × D(w,µ)(m)−D(w,µ)(m)−(m−m)D′(w,µ)(m1) (m2−m1)2 if m1 ≠ m2, 1/2 D′′ (w, µ) (m) if m1 = m2 = m, where D (D (w, µ)) is the Fréchet derivative of D (w, µ) as a function of operator and D00 (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−1 dt − (A + B/2)r−1∥ ≤ 1/24 (1 − r) (2 − r) mr−3 ∥B − A∥2 and ∥ Ar−1 + Br−1/2 − ∫ 01 ((1 − t) A + tB)r−1 dt ∥ ≤ 1/12 (1 − r) (2 − r) mr−3 ∥B − A∥2

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