Skip to main content

Superadditivity of convex integral transform for positive operators in Hilbert spaces


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.


  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Dragomir, S.S.: Operator monotonicity of an integral transform of positive operators in Hilbert spaces with applications. RGMIARes. Rep. Coll. 23, 15 (2020). (Art. 65, online)

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

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Incomplete Gamma and Related Functions, Definitions, Digital Library of Mathematical Functions, NIST (2020). (Online)

  16. Incomplete Gamma and Related Functions, Integral Representations, Digital library of mathematical functions, NIST (2020). (online)

  17. Generalized Exponential Integral, Digital Library of Mathematical Functions, NIST (2020). (online)

Download references


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 and Affiliations


Corresponding author

Correspondence to Silvestru Sever Dragomir.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dragomir, S.S. Superadditivity of convex integral transform for positive operators in Hilbert spaces. RACSAM 115, 98 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


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

Mathematics Subject Classification

  • 47A63
  • 47A60