An arithmetic-geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices $\mathbf{A}_1, \dots, \mathbf{A}_n$, the following holds for each integer $m \leq n$: $\frac{1}{n^m}\sum_{j_1, j_2, \dots, j_m = 1}^{n} ||| \mathbf{A}_{j_1} \mathbf{A}_{j_2} \dots \mathbf{A}_{j_m} ||| \geq \frac{(n-m)!}{n!} \sum_{\substack{j_1, j_2, \dots, j_m = 1 \\ \text{all distinct}}}^{n} ||| \mathbf{A}_{j_1} \mathbf{A}_{j_2} \dots \mathbf{A}_{j_m} |||,$ where $||| \cdot |||$ denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, $m \leq 3$. The proofs for $m = 1,2$ are straightforward; to derive the proof for $m=3$, we appeal to a variant of the classic Araki-Lieb-Thirring inequality for permutations of matrix products.

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