Operator Monotone Functions and Convexity of Its Derivatives Norms

Authors
Zahra Rahimi Chegeni
amir ghasem ghazanfari
Kamal Falahi
10.52547/mmr.7.1.43
Abstract
Introduction

Given the important role convex and quasi-convex functions play in many areas of mathematics and especially in optimization, one of the inequalities that has attracted the attention of many mathematicians in recent decades is Hermit-Hadamard’s famous inequality. Significant generalizations and refinements have been obtained for this inequality in a diverse variety of convexity, including convex operator functions of self adjoint operators on Hilbert spaces, matrix functions, quasi-convex, s-convex and log-convex functions.

In this paper, we generalize this inequality to differentiable functions whose norm of their derivatives are convex functions.

Results and discussion

In this paper, we consider differentiable mappings which norm of the induced maps by them on the set of self adjoint operators is convex, quasi convex or s-convex. We show that if is an operator monotone function on , is a strictly positive operator and a unitarily invariant norm, then for all positive integers . We also prove that

is a quasi-convex function on the set of all strictly positive operators in B(H). Examples and applications for particular cases of interest are also illustrated. Finally, an error estimate for the Simpson formula is addressed.

Conclusion

The following conclusions were drawn from this research.

As an important application of the results in this paper, we find bounds for

in terms of , which is one of the central problems in perturbation theory.

We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-convex or s- convex../files/site1/files/71/5.pdf
Keywords
Hermite-Hadamard inequality
Differentiable functions
Unitarily invariant norms
Operator monotone functions
1. Barnet N. S., Cerone P., Dragomir S. S., "Some new inequalities for Hermite-Hadamard divergence in information theory", Stochastic analysis and applications. Vol. 3, Nova Sci. Publ., Hauppauge, NY (2003) 7-19.## 2. Dragomir S. S., Pearce C. E. M., "Selected Topics on Hermite-Hadamard Inequalities and Applications", (RGMIA Monographs http:// rgmia.Vu.Edu.au/ monographs/ Hermite hadamard. Html), Victoria University (2000). ## 3. Wu S., "On the weighted generalization of the Hermite-Hadamard inequality and its applications", Rocky Mountain J. Math. 39 (5) (2009) 1741-1749. ## 4. Hudzik H., Maligranda L., "Some remarks on s-convex functions", Aequationes Math., 48 (1994) 100-111. ## 5. Bhatia R., "Matrix analysis", Springer-verlag, New York (1997). ## 6. Furuta T., Micic Hot J., Pecaric J., Seo Y., "Mond-Pecaric Method in Operator Inequalities (Inequalities for bounded self adjoint operators on a Hilbert space)", Monographs in Inequalities, Vol. 1. Element, Zagreb (2005). ## 7. Bhatia R., "Perturbation bounds for the operator absolute value", Linear Algebra Appl. 226 (1995) 539-545. ## 8. Bhatia R., "First and second order perturbation bounds for the operator absolute value", Linear Algebra Appl. 208 (1994) 367-376. ## 9. Bhatia R., Sinha K. B., "Variation of real powers of positive operators", Indiana Univ. Math. J. 43 (1994) 913-925. ## 10. Dragomir S. S., Agarwal R. P., "Tow inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula", Appl. Math. Lett. 11(5) (1998) 91-95. ## 11. Ghazanfari A. G., "Hermite-Hadamard type inequalities for functions whose derivatives are operator convex", Complex Anal. Oper. Theory 10 (8) (2016) 1695-1703. ##