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Let A, B be positive operators on a Hilbert space with 0 < m ≤ A, B ≤ M. Then for every unital positive linear map Φ, Φ²((A + B)/2) ≤ K²(h)Φ²(A ♯ B), and Φ²((A+B)/2) ≤ K²(h)(Φ(A) ♯ Φ(B))², where A ♯ B is the geometric mean and K(h) = (h+1)²/(4h) with h = M/m.

Fiedler and Markham (1994) proved $${\left(\frac{\det \widehat{H}}{k}\right)}^k\ge \det H,$$ where $H={\left(H_{ij}\right)}_{i,j=1}^n$ is a positive semidefinite matrix partitioned into $n\times n$ blocks with each block $k\times k$ and $\widehat{H}={\left(\mathrm{tr}{H}_{ij}\right)}_{i,j=1}^n$. We revisit this inequality mainly using some terminology from quantum information theory. Analogous results are included. For example, under the same condition, we prove $$\det (I_n+\widehat{H})\ge \det {(I_{nk}+kH)}^{1/k}.$$

Companion matrices of the second type are characterized by properties that involve bilinear maps.