A note on the non-commutative arithmetic-geometric mean inequality

Abstract: This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$, \begin{equation} \frac{1}{n^3}\Big|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big| \geq \frac{(n-3)!}{n!} \Big|\sum_{\substack{j_1,j_2,j_3=1,\\text{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}A_{j_1}A_{j_2}A_{j_3}\Big|, \end{equation} where $|\cdot|$ represents the operator norm. This inequality is a special case of a recent conjecture by Recht and R\'e.