## Abstract

Let \(\sigma\) be a non-trivial operator mean in the sense of Kubo and Ando, and let \(OM_+^1\) be the set of normalized positive operator monotone functions on \((0, \infty )\). In this paper, we study the class of \(\sigma\)-subpreserving functions \(f\in OM_+^1\) satisfying

for all invertible positive operators *A* and *B*. We provide some criteria for *f* to be trivial, i.e., \(f(t)=1\) or \(f(t)=t\). We also establish characterizations of \(\sigma\)-preserving functions \(f\in OM_+^1\) satisfying

for all invertible positive operators *A* and *B*. In particular, when \(\lim _{t\rightarrow 0} (1\sigma t) =0\), the function \(f\in OM_+^1\backslash \{1,t\}\) preserves \(\sigma\) if and only if *f* and \(1\sigma t\) are representing functions for a weighted harmonic mean.

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## Acknowledgements

The research of the second author is partially supported by JSPS KAKENHI Grant number JP17K05285.

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Communicated by Takeaki Yamazaki.

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Dinh, T.H., Osaka, H. & Wada, S. Functions preserving operator means.
*Ann. Funct. Anal.* **11**, 1203–1219 (2020). https://doi.org/10.1007/s43034-020-00080-y

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DOI: https://doi.org/10.1007/s43034-020-00080-y

### Keywords

- Operator means
- Operator monotone functions
- Positive matrices
- Operator convexity
- Maps preserving operator means