An Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Abstract

Let H be a Hilbert space. Assume that f is continuously differentiable on I with ‖f′‖_{I,∞}:=sup_{t∈I}|f′(t)|<∞ and A, B are selfadjoint operators with Sp(A), Sp(B)⊂I, then ‖f((1-λ)A⊗1+λ1⊗B)-∫₀¹f((1-u)A⊗1+u1⊗B)du‖ ≤‖f′‖_{I,∞}[(1/4)+(λ-(1/2))²]‖1⊗B-A⊗1‖ for λ∈[0,1]. In particular, we have the midpoint inequality ‖f(((A⊗1+1⊗B)/2))-∫₀¹f((1-u)A⊗1+u1⊗B)du‖ ≤(1/4)‖f′‖_{I,∞}‖1⊗B-A⊗1‖.

Keywords

• Tensorial product
• Selfadjoint operators
• Operator norm
• Ostrowski’s inequality

References

1. T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Lin. Alg. & Appl. 26 (1979), 203-241.
2. H. Araki and F. Hansen, Jensen's operator inequality for functions of several variables, Proc. Amer. Math. Soc. 128 (2000), No. 7, 2075-2084.
3. J. S. Aujila and H. L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon. 42 (1995), 265-272.
4. N. S. Barnett, P. Cerone and S. S. Dragomir, Some new inequalities for Hermite-Hadamard divergence in information theory. in Stochastic Analysis and Applications. Vol. 3, 7-19, Nova Sci. Publ., Hauppauge, NY, 2003. Preprint RGMIA Res. Rep. Coll. 5 (2002), No. 4, Art. 8, 11 pp. [Online https://rgmia.org/papers/v5n4/NIHHDIT.pdf]
5. S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74(3)(2006), 417-478.
6. S. S. Dragomir, Some tensorial Hermite-Hadamard type inequalities for convex functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll. 25 (2022), Art. 90, 14 pp. [Online https://rgmia.org/papers/v25/v25a90.pdf]
7. A. Koranyi. On some classes of analytic functions of several variables. Trans. Amer. Math. Soc., 101 (1961), 520ñ554.
8. A. Ebadian, I. Nikoufar and M. E. Gordji, Perspectives of matrix convex functions, Proc. Natl. Acad. Sci. USA, 108 (2011), no. 18, 7313-7314.

9. J. I. Fujii, The Marcus-Khan theorem for Hilbert space operators. Math. Jpn. 41 (1995), 531-535.
10. T. Furuta, J. Micic Hot, J. Pecaric and Y. Seo, Mond-Pecaric Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
11. K. Kitamura and Y. Seo, Operator inequalities on Hadamard product associated with Kadison's Schwarz inequalities, Scient. Math. 1 (1998), No. 2, 237-241.
12. I. Nikoufar and M. Shamohammadi, The converse of the Loewner-Heinz inequality via perspective, Lin. & Multilin. Alg., 66 (2018), N0. 2, 243-249.
13. A. Ostrowski, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.
14. S. Wada, On some refinement of the Cauchy-Schwarz Inequality, Lin. Alg. & Appl. 420 (2007), 433-440.