ON A CONJECTURE RELATED TO THE GEOMETRIC MEAN AND NORM INEQUALITIES

A conjecture of Dinh, Ahsani, and Tam, was recently settled in [7]. In this note, we give a refinement to that result, namely if Ai and Bi are positive definite matrices and Z = [Z(ij)] is the block matrix such that Z(ij) = B-i(1/2) (Sigma(m)(k=1)A(k)) B-j(1/2) for all i, j = 1, center dot center dot center dot, m, then |||Sigma(m)(i=1) (A(i)(2) (sic) B-i(2))(r) ||| <= |||Z(r) ||| <= ||| ((Sigma(m)(i=1) A(i) )(rp/2) (Sigma(m)(i=1) B-i )(rp) (Sigma(m)(i=1) A(i) )(rp/2) )1/p ||| for all unitarily invariant norms, for all p > 0 and r >= 1 such that rp >= 1. Our approach provides us with an alternative proof without using the method of majorization that was used in [7]. As a byproduct, we get a refinement to a result of Audenaert in 2015.