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

Abstract

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 \[ \lambda\bigl(a^2A+b^2B+c(A\natural B)\bigr) \prec_w \lambda\bigl(W_{a,b}(A,B)\bigr), \qquad 0\le c\le 2ab, \] 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 \[ \lambda\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w \lambda\bigl(W_{a,b}(A,B)\bigr). \] 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 \[ \lambda\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w \lambda\bigl((aA^{1/2}+bB^{1/2})^2\bigr), \] and consequently obtain the corresponding inequality for all unitarily invariant norms.