Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means

We prove sharp Heron-type majorization inequalities for two quadratic matrix expressions associated with the spectral and Kubo-Ando geometric means. For the spectral geometric mean cross term, we show that $$\lambda (a^{2}A+b^{2}B+c(A\mathrm{♮}B)){\prec }_w\lambda (W_{a,b}(A,B)),0\le c\le 2ab,$$ where $W_{a,b}(A,B)$ is the weighted Bures-Wasserstein expression. The coefficient $2ab$ is sharp, and at this endpoint the weak majorization becomes majorization. For the Kubo-Ando geometric mean, we prove the direct comparison $$\lambda (a^{2}A+b^{2}B+2ab(A\mathrm{\#}B)){\prec }_w\lambda (W_{a,b}(A,B))\mathrm{.}$$ This settles, in the two-variable setting, Bhatia’s question of whether the Heron-type norm inequality of Bhatia-Lim-Yamazaki admits a weak-majorization refinement. More precisely, we prove $$\lambda (a^{2}A+b^{2}B+2ab(A\mathrm{\#}B)){\prec }_w\lambda ((a{A}^{1\mathrm{/}2}+b{B}^{1\mathrm{/}2}{)}^2),$$ and consequently obtain the corresponding inequality for all unitarily invariant norms.