Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean $A{\mathrm{♯}}_{1-\alpha }B=A^{1/2}(A^{-1/2}B{A}^{-1/2}{)}^{1-\alpha }{A}^{1/2},0\le \alpha \le 1$ and its ‘naïve’ extension given by $Q_{\alpha ,z}(A,B)=(B^{\frac{1-\alpha }{2z}}{A}^{\frac{\alpha }{z}}{B}^{\frac{1-\alpha }{2z}}{)}^z$, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality $\parallel A+B+2rA{\mathrm{♯}}_{1-\alpha }B{\parallel }_p\le \parallel A+B+2r{Q}_{\alpha ,z}(A,B){\parallel }_p,$ holds for p = 1, 2, $\alpha =1,z$, and $r\ge 0$, among other related inequalities.

Bhatia、Lim 和 Yamazaki 研究了正半定矩阵的几个 Kubo-Ando 均值的范数极小性。最近，Hiai 证明了一个涉及加权几何平均值的范数极小结果$\mathrm{一种}{\mathrm{♯}}_{1-\alpha }乙={\mathrm{一种}}^{1/2}({\mathrm{一种}}^{-1/2}乙{\mathrm{一种}}^{-1/2}{)}^{1-\alpha }{\mathrm{一种}}^{1/2},0\le \alpha \le 1$及其“天真的”扩展由${问}_{\alpha ,z}(\mathrm{一种},乙)=({乙}^{\frac{1-\alpha }{2z}}{\mathrm{一种}}^{\frac{\alpha }{z}}{乙}^{\frac{1-\alpha }{2z}}{)}^z$，它是量子α - z -Rényi散度定义中的矩阵函数。结合这些结果，对于半正定矩阵，我们证明了不等式$\parallel \mathrm{一种}+乙+2r\mathrm{一种}{\mathrm{♯}}_{1-\alpha }乙{\parallel }_p\le \parallel \mathrm{一种}+乙+2r{问}_{\alpha ,z}(\mathrm{一种},乙){\parallel }_p,$p =  1, 2,$\alpha =1,z$， 和$r\ge 0$，以及其他相关的不平等。