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 other...
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| Format: | Article |
| Language: | English |
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University of Extremadura
2022-10-01
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| Series: | Extracta Mathematicae |
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| Online Access: | https://publicaciones.unex.es/index.php/EM/article/view/1781 |
| _version_ | 1797989888541327360 |
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| author | S.S. Dragomir |
| author_facet | S.S. Dragomir |
| author_sort | S.S. Dragomir |
| collection | DOAJ |
| description | 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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| first_indexed | 2024-04-11T08:27:51Z |
| format | Article |
| id | doaj.art-48984c3a1cdf4f93acdc29b1a850e824 |
| institution | Directory Open Access Journal |
| issn | 0213-8743 2605-5686 |
| language | English |
| last_indexed | 2024-04-11T08:27:51Z |
| publishDate | 2022-10-01 |
| publisher | University of Extremadura |
| record_format | Article |
| series | Extracta Mathematicae |
| spelling | doaj.art-48984c3a1cdf4f93acdc29b1a850e8242022-12-22T04:34:41ZengUniversity of ExtremaduraExtracta Mathematicae0213-87432605-56862022-10-01Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spacesS.S. Dragomir0Mathematics, College of Engineering & Science Victoria University, PO Box 14428, Melbourne City 8001, AustraliaFor 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. https://publicaciones.unex.es/index.php/EM/article/view/1781operator monotone functionsoperator convex functionsoperator inequalitiesmidpoint inequalitytrapezoid inequality |
| spellingShingle | S.S. Dragomir Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces Extracta Mathematicae operator monotone functions operator convex functions operator inequalities midpoint inequality trapezoid inequality |
| title | Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces |
| title_full | Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces |
| title_fullStr | Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces |
| title_full_unstemmed | Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces |
| title_short | Second derivative Lipschitz type inequalities for an integral transform of positive operators in Hilbert spaces |
| title_sort | second derivative lipschitz type inequalities for an integral transform of positive operators in hilbert spaces |
| topic | operator monotone functions operator convex functions operator inequalities midpoint inequality trapezoid inequality |
| url | https://publicaciones.unex.es/index.php/EM/article/view/1781 |
| work_keys_str_mv | AT ssdragomir secondderivativelipschitztypeinequalitiesforanintegraltransformofpositiveoperatorsinhilbertspaces |