Some inequalities for matrix means and positive linear maps

We investigate comparison inequalities for several matrix means arising beyond the Kubo–Ando framework, with emphasis on the weighted F-mean and the two-variable Bures–Wasserstein mean. We establish quantitative weak log-majorization relations between these means, yielding sharp spectral-radius comparisons and showing that the Loewner inequality Ft(A,B)⪯A⋄tB does not hold in general. The compound-matrix argument is strengthened to retain the first k eigenvalues of the reference matrix; we also record the determinant endpoint and the resulting constant-one weak log-majorization in dimension two. We further study comparison inequalities for the Bures–Wasserstein mean under unital positive linear maps and derive explicit two-sided bounds under uniform spectral constraints. Finally, we introduce a class of alternative quasi-means generated by two-sided functional calculus and obtain corresponding inequalities under positive linear maps. The positive-map estimates are formulated for arbitrary positive continuous generating functions, so the operator-monotone and decreasing cases follow as special cases.