An arithmetic-geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic-geometric mean inequality: Given positive-semidefinite matrices ^A1,...,^An, the ≤ m≤n:1;bsupesup ;bsupesup bsup esuphellip bsupesup=1n(Equation presented)A;bsupbsub Aj2...A^jm≥(n-m)!n!∑^j1,^j2,...,^jm=1all distinctn(Equation presented)A^j1A^j2...^Ajm(Equation presented), where (Equation presented) 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≤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.