For a probability measure of compact support μ on the set Pn of all positive definite matrices and t∈(0,1], let Pt(μ) be the unique positive solution of X=∫PnX♯tZdμ(Z). In this paper, we show that if μ and ν are probability measures on Pn and Pm, respectively, and Φ:Mn×Mm→Mk is a unital positive bilinear map, then Φ(Pt(μ),Pt(ν))≤Pt(λ) for all t∈[−1,1]∖{0}, where λ is the push-forward of the product measure μ×ν by the mapping Φ|Pn×Pm. In particular, Pt(μ)⊗Pt(ν)≤Pt(λ). Moreover, we show that Pt(μ)≤I⟹Ptp(ν)≤Pt(μ),for every p≥1, where ν(Z)=μ(Z1/p) and I denotes the identity matrix.

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