Geodesic
The inverse problem for generalized contraharmonic means
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 3-9.

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In this paper we introduce the generalized contraharmanic mean associated to a Kubo-Ando mean $\sigma$ as $$ C_\sigma(X, Y) = X\sigma Y - X\sigma^\perp Y, $$ where $\sigma^\perp$ is the dual mean of $\sigma$ and $X, Y$ are positive definite matrices. We show that for a symmetric Kubo-Ando mean $\sigma$ such as $\sigma \ge \sharp$ and for any positive definite matrices $A \ge B$ the inverse problem \begin{equation*} A=C_\sigma(X, Y), \ \ B=X^{1/2}(X^{-1/2}YX^{-1/2})^{1/2}X^{1/2} \end{equation*} has a positive solution $(X, Y)$.
Keywords: Kubo-Ando means, geometric mean, generalized contraharmonic mean, inverse problem, Brouwer's fixed point theorem, non-linear matrix equations. 
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T. H. Dinh; C. T. Le; B. K. Vo. The inverse problem for generalized contraharmonic means. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2022), pp. 3-9. http://geodesic.mathdoc.fr/item/IVM_2022_7_a0/

[1] Kubo F., Ando T., “Means of positive linear operators”, Math. Ann., 246 (1980), 205–224 | DOI | MR | Zbl

[2] Anderson W. N., Mays M. E., Morley T. D., Trapp G. E., “The contraharmonic mean of HSD matrices”, SIAM J. Algebra Discr. Meth., 8 (1987), 674–682 | DOI | MR | Zbl

[3] Lim Y. D., “The inverse mean problem of geometric mean and contraharmonic means”, Linear Algebra Appl., 408 (2005), 221–229 | DOI | MR | Zbl

[4] Dinh T. H., Dumitru R., Franco J., “New characterizations of operator monotone functions”, Linear Algebra Appl., 546 (2018), 169–186 | DOI | MR | Zbl