A quantitative Popoviciu type inequality for four positive semi-definite matrices (Q6618725)

In the author's previous paper [Math. Inequal. Appl. 27, No. 1, 149--158 (2024; Zbl 07926903)], it is proved that if \(A_1, A_2, \ldots, A_m \ (m \geqslant 3)\) are \(n \times n\) positive definite matrices, then \N\[\N\begin{aligned} & \operatorname{det}\left(\sum_{j=1}^m A_j\right)+(m-2) \sum_{j=1}^m \operatorname{det}\left(A_j\right) \\\N& \quad \geqslant \sum_{1 \leqslant i<j \leqslant m} \operatorname{det}\left(A_i+A_j\right)+\left[\left(m^n-m-\left(2^{n-1}-1\right)(m-1) m\right)\right]\left[\operatorname{det}\left(A_1 A_2 \cdots A_m\right)\right]^{\frac{1}{m}}. \end{aligned} \N\] \NThis refines a result of \textit{W. Berndt} and \textit{S. Sra} in [Linear Algebra Appl. 486, 317--327 (2015; Zbl 1327.15010)].\N\NIn the paper under review, the author further improves the previous inequality for \(m=4\). The main result is the following theorem.\N\NTheorem. Let \(A_1, A_2, A_3, A_4\) be positive definite \(n \times n\) matrices. Then\N\[\N\begin{aligned} & \operatorname{det}\left(\sum_{j=1}^4 A_j\right)+2 \sum_{j=1}^4 \operatorname{det}\left(A_j\right)-\sum_{1 \leqslant i<j \leqslant 4} \operatorname{det}\left(A_i+A_j\right) \\\N& \geqslant\left(3^n-3 \cdot 2^n+3\right)\left[\left(\operatorname{det}\left(A_1 A_2 A_3\right)\right)^{\frac{1}{3}}+\left(\operatorname{det}\left(A_1 A_2 A_4\right)\right)^{\frac{1}{3}}+\left(\operatorname{det}\left(A_1 A_3 A_4\right)\right)^{\frac{1}{3}}\right. \\\N& \left.\quad+\left(\operatorname{det}\left(A_2 A_3 A_4\right)\right)^{\frac{1}{3}}\right]+\left(4^n-4 \cdot 3^n+3 \cdot 2^{n+1}-4\right)\left(\operatorname{det}\left(A_1 A_2 A_3 A_4\right)\right)^{\frac{1}{4}}. \end{aligned} \N\]