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Remarks on a Question of Bourin for Positive Semidefinite Matrices

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Abstract

Let A and B be positive semidefinite matrices. For \(t\in \left[ \frac{3}{4},1\right] \) and for every unitarily invariant norm, it is shown that

$$\begin{aligned} {\left| \left| \left| A^{t}B^{1-t}+B^{t}A^{1-t} \right| \right| \right| }\le 2^{2\left( t-\frac{3}{4}\right) }{\left| \left| \left| A+B \right| \right| \right| } \end{aligned}$$

and for \(t\in \left[ 0,\frac{1}{4}\right] \),

$$\begin{aligned} {\left| \left| \left| A^{t}B^{1-t}+B^{t}A^{1-t} \right| \right| \right| }\le 2^{2\left( \frac{1}{4}-t\right) }{\left| \left| \left| A+B \right| \right| \right| }. \end{aligned}$$

These norm inequalities are sharper than an earlier norm inequality due to Alakhrass and closely related to an open question of Bourin. In fact, they lead to an affirmative solution of Bourin’s question for \(t=\frac{1}{4}\) and \(\frac{3}{4}\), which is a result due to Hayajneh and Kittaneh (Int J Math 32 (2150043):7, 2021).

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Authors and Affiliations

  1. Department of Mathematics, Yarmouk University, Irbid, Jordan

    Mostafa Hayajneh

  2. Department of Mathematics, The University of Jordan, Amman, Jordan

    Saja Hayajneh & Fuad Kittaneh

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  1. Mostafa Hayajneh
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  2. Saja Hayajneh
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  3. Fuad Kittaneh
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All authors contributed to each part of this work equally, and they all read and approved the final manuscript.

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Correspondence to Fuad Kittaneh.

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Hayajneh, M., Hayajneh, S. & Kittaneh, F. Remarks on a Question of Bourin for Positive Semidefinite Matrices. Results Math 78, 88 (2023). https://doi.org/10.1007/s00025-023-01870-1

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