An arithmetic-geometric mean inequality for products of three matrices

Consider the following noncommutative arithmetic geometric mean inequality: Given positive-semidefinite matrices A1,..., A(n), the following holds for each integer m <= n: 1/n(m) Sigma(j1,j2,..., jm=1) (n) vertical bar vertical bar vertical bar A(j1)A(j2) ... A(jm)vertical bar vertical bar vertical bar (n-m)!/n ! Sigma(j1,j2,..., jm=1) all distinct vertical bar vertical bar vertical bar A(j1)A(j2) ... A(jm)vertical bar vertical bar vertical bar, where vertical bar vertical bar vertical bar .vertical bar vertical bar vertical bar 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. (C) 2015 Elsevier Inc. All rights reserved.