# Superadditivity of convex integral transform for positive operators in Hilbert spaces

## Abstract

For a continuous and positive function $$w\left( \lambda \right) ,$$ $$\lambda >0$$ and $$\mu$$ a positive measure on $$(0,\infty )$$ we consider the following convex integral transform

\begin{aligned} \mathcal {C}\left( w,\mu \right) \left( T\right) :=\int _{0}^{\infty }w\left( \lambda \right) T^{2}\left( \lambda +T\right) ^{-1}d\mu \left( \lambda \right) , \end{aligned}

where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other that, for all A$$B>0$$ with $$BA+AB\ge 0,$$

\begin{aligned} \mathcal {C}(w,\mu )\left( A+B\right) \ge \mathcal {C}(w,\mu )\left( A\right) +\mathcal {C}(w,\mu )\left( B\right) . \end{aligned}

In particular, we have for $$r\in (0,1],$$ the power inequality

\begin{aligned} \left( A+B\right) ^{r+1}\ge A^{r+1}+B^{r+1} \end{aligned}

and the logarithmic inequality

\begin{aligned} \left( A+B\right) \ln \left( A+B\right) \ge A\ln A+B\ln B. \end{aligned}

Some examples for operator monotone and operator convex functions and integral transforms $$\mathcal {C}\left( \cdot ,\cdot \right)$$ related to the exponential and logarithmic functions are also provided.

This is a preview of subscription content, access via your institution.

## References

1. Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics, vol. 169. Springer, New York (1997) (xii+347 pp. ISBN: 0-387-94846-5)

2. Dadkhah, A., Moslehian, M.S.: Non-linear positive maps between $$C^{\ast }$$-algebras. Linear Multilinear Algebra 68(8), 1501–1517 (2020)

3. Dinh, T.H., Tikhonov, O.E., Veselova, L.V.: Inequalities for the extended positive part of a von Neumann algebra related to operator-monotone and operator-convex functions. Ann. Funct. Anal. 10(3), 425–432 (2019)

4. Dragomir, S.S.: Operator monotonicity of an integral transform of positive operators in Hilbert spaces with applications. RGMIARes. Rep. Coll. 23, 15 (2020). https://rgmia.org/papers/v23/v23a65.pdf (Art. 65, online)

5. Fujii, J.I., Seo, Y.: On parametrized operator means dominated by power ones. Sci. Math. 1, 301–306 (1998)

6. Furuta, T.: Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation. Linear Algebra Appl. 429, 972–980 (2008)

7. Furuta, T.: Precise lower bound of $$f(A)-f(B)$$ for $$A>B>0$$ and non-constant operator monotone function $$f$$ on $$[0,\infty )$$. J. Math. Inequal. 9(1), 47–52 (2015)

8. Gustafson, K.: Interaction antieigenvalues. J. Math. Anal. Appl. 299(1), 174–185 (2004)

9. Heinz, E.: Beiträge zur Störungsteorie der Spektralzerlegung. Math. Ann. 123, 415–438 (1951)

10. Löwner, K.: Über monotone MatrixFunktionen. Math. Z. 38, 177–216 (1934)

11. Moslehian, M.S., Najafi, H.: Around operator monotone functions. Integr. Equations Oper. Theory 71, 575–582 (2011)

12. Moslehian, M.S., Najafi, H.: An extension of the Lö wner–Heinz inequality. Linear Algebra Appl. 437, 2359–2365 (2012)

13. Najafi, H., Moslehian, M.S., Fujii, M., Nakamoto, R.: Estimates of operator convex and operator monotone functions on bounded intervals. Hokkaido Math. J. 45(3), 325–336 (2016)

14. Zuo, H., Duan, G.: Some inequalities of operator monotone functions. J. Math. Inequal. 8(4), 777–781 (2014)

15. Incomplete Gamma and Related Functions, Definitions, Digital Library of Mathematical Functions, NIST (2020). https://dlmf.nist.gov/8.2 (Online)

16. Incomplete Gamma and Related Functions, Integral Representations, Digital library of mathematical functions, NIST (2020). https://dlmf.nist.gov/8.6 (online)

17. Generalized Exponential Integral, Digital Library of Mathematical Functions, NIST (2020). https://dlmf.nist.gov/8.19#E1 (online)

## Acknowledgements

The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper.

## Author information

Authors

### Corresponding author

Correspondence to Silvestru Sever Dragomir.

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Dragomir, S.S. Superadditivity of convex integral transform for positive operators in Hilbert spaces. RACSAM 115, 98 (2021). https://doi.org/10.1007/s13398-021-01037-z

• Accepted:

• Published:

• DOI: https://doi.org/10.1007/s13398-021-01037-z

### Keywords

• Operator monotone functions
• Operator convex functions
• Operator inequalities
• Löwner–Heinz inequality
• Logarithmic operator inequalities

• 47A63
• 47A60