Further norm inequalities for positive semidefinite matrices

Let A and B be \(n\times n\) $n\times n$ positive semidefinite matrices, and let \( ||\cdot ||_{2}\) $||·{||}_2$ be the Hilbert-Schmidt norm. Bhatia and then Hayajneh and Kittaneh, using different techniques, proved that \(\begin{aligned} ||A^{v}B^{1- v}+B^{v}A^{1-v}||{2}\le ||A+B||{2} \end{aligned}\) $$\begin{array}{c}\hfill ||A^v{B}^{1-v}+B^v{A}^{1-v}||2\le ||A+B||2\end{array}$$ for \(v\in \left[ \frac{1}{4},\frac{3}{4}\right] \) $v\in \left[\frac{1}{4},,,\frac{3}{4}\right]$ , which gives an affirmative answer to an open problem posed by Bourin for the special case of the Hilbert–Schmidt norm. In this paper, we prove a general unitarily invariant norm inequality from which we obtain a new proof of the above Hilbert–Schmidt norm inequality. We also prove that if \(r\ge 1,\) $r\ge 1,$ then \(\begin{aligned} |||A^{v}B^{1-v}+B^{v}A^{1-v}|||\le |||(A^{1/r}+B^{1/r})^{r}||| \end{aligned}\) $$\begin{array}{c}\hfill |||A^v{B}^{1-v}+B^v{A}^{1-v}|||\le |||{(A^{1/r}+B^{1/r})}^r|||\end{array}$$ for \(\frac{1}{2r}\le v\le \frac{2r-1}{2r}\) $\frac{1}{2r}\le v\le \frac{2r-1}{2r}$ , where \(|||\cdot |||\) $|||·|||$ denotes any unitarily invariant norm.