An arithmetic–geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic–geometric mean inequality: Given positive-semidefinite matrices A1,…,AnA1,…,An, the following holds for each integer m≤nm≤n:1nm∑j1,j2,…,jm=1n⦀Aj1Aj2…Ajm⦀≥(n−m)!n!∑j1,j2,…,jm=1all distinctn⦀Aj1Aj2…Ajm⦀, where ⦀⋅⦀⦀⋅⦀ 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≤3m≤3. The proofs for m=1,2m=1,2 are straightforward; to derive the proof for m=3m=3, we appeal to a variant of the classic Araki–Lieb–Thirring inequality for permutations of matrix products.