Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910256859185152 |
|---|---|
| author | Vuong, Trung Dung Nguyen, Anh Thi Dinh, Trung Hoa |
| author_facet | Vuong, Trung Dung Nguyen, Anh Thi Dinh, Trung Hoa |
| contents | 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 \[ λ\bigl(a^2A+b^2B+c(A\natural B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr), \qquad 0\le c\le 2ab, \] 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 \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr). \] 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 \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl((aA^{1/2}+bB^{1/2})^2\bigr), \] and consequently obtain the corresponding inequality for all unitarily invariant norms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26141 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means Vuong, Trung Dung Nguyen, Anh Thi Dinh, Trung Hoa Functional Analysis 15A42, 15A45, 15A60, 47A64 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 \[ λ\bigl(a^2A+b^2B+c(A\natural B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr), \qquad 0\le c\le 2ab, \] 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 \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl(W_{a,b}(A,B)\bigr). \] 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 \[ λ\bigl(a^2A+b^2B+2ab(A\#B)\bigr) \prec_w λ\bigl((aA^{1/2}+bB^{1/2})^2\bigr), \] and consequently obtain the corresponding inequality for all unitarily invariant norms. |
| title | Heron-Wasserstein majorization inequalities for spectral and Kubo-Ando geometric means |
| topic | Functional Analysis 15A42, 15A45, 15A60, 47A64 |
| url | https://arxiv.org/abs/2605.26141 |