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 A_i and B_i are positive definite matrices and Z = [Z_ij] is the block matrix such that Zij = B_i ^1/2 (m/Σ/k=1 A_k) B^1/2 _j for all i,j = 1, ...,m, then |||m/Σ/i=1 ( A²_i # B² _i )^r ||| ≤ ||| Z^r ||| ≤ ||| ((m/Σ/i=1 A_i)^rp/2 (m/Σ/i=1 B²)^rp (m/Σ/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. [ABSTRACT FROM AUTHOR]