OP-00266 Open

Bourin's crossed-Heinz inequality

Let $|||\cdot|||$ be a unitarily invariant norm. The inequality below is immediate for $t = 0, 1$; Kittaneh and Ricard proved it for $t \in [1/4, 3/4]$, and Sababheh and Bourin for $n = 2$ and every $t$. For the remaining $t$ it is known with the constant $2^{2|t - 1/2| - 1/2}$ on the right.

Problem. Is $$|||A^t B^{1-t} + B^t A^{1-t}||| \le |||A + B|||$$ for every $n \ge 1$, all positive definite $A, B \in \mathbb{C}^{n \times n}$, every $t \in [0, 1]$ and every unitarily invariant norm $|||\cdot|||$?

Source: Bourin (2009), Question 4 in arXiv:0805.1954.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: FA — Functional Analysis
Secondary: NA
Keywords: norm inequalities, OpenProblemsInNLA, unitarily invariant norms
Literature checked: Oct. 8, 2026 by clemens-admin

References

@misc{sababheh2026,
  author        = {Sababheh, Mohammad and Bourin, Jean-Christophe},
  title         = {Crossed-Heinz Inequalities: The Two-Dimensional Threshold and Higher-Dimensional Stability},
  year          = {2026},
  eprint        = {2610.04567},
  archivePrefix = {arXiv},
}
@article{bourin2009,
  author        = {Bourin, Jean-Christophe},
  title         = {Matrix subadditivity inequalities and block-matrices},
  journal       = {Internat. J. Math.},
  volume        = {20},
  number        = {6},
  year          = {2009},
  pages         = {679--691},
  doi           = {10.1142/S0129167X09005509},
  eprint        = {0805.1954},
  archivePrefix = {arXiv},
}
@article{kittaneh2025,
  author  = {Kittaneh, Fuad and Ricard, {\'E}ric},
  title   = {On a question of {B}ourin},
  journal = {Linear Algebra Appl.},
  volume  = {710},
  year    = {2025},
  pages   = {356--362},
  doi     = {10.1016/j.laa.2025.02.004},
}
@misc{zhang2026,
  author        = {Zhang, Teng},
  title         = {Bourin-type inequalities for $\tau$-measurable operators in fully symmetric spaces},
  year          = {2026},
  eprint        = {2602.00358},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi02,
  author       = {Townsend, Alex},
  title        = {{MI-02}: Bourin's crossed-Heinz inequality},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-02/README.md},
  note         = {Read at commit e375a6f, 2026-09-30},
}

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00266, "Bourin's crossed-Heinz inequality", clopen.solutions.