Some inequalities for the matrix Heron mean

Original · EN

Let A, B be positive definite matrices, p=1, 2 and r≥ 0. It is shown that equation* ||A+ B + r(Aₜ B+A₁₋ₜ B)||ₚ ≤ ||A+ B + r(AᵗB¹⁻ᵗ + A¹⁻ᵗBᵗ)||ₚ. equation* We also prove that for positive definite matrices A and B equation*det (Pₜ(A, B)) ≤ (Qₜ(A, B)), equation* where Qₜ(A, B)= (Aᵗ+Bᵗ/2)¹/ᵗ and Pₜ(A, B) is the t-power mean of A and B. As a consequence, we obtain the determinant inequality for the matrix Heron mean: for any positive definite matrices A and B, (A+ B + 2(A B)) ≤ (A+ B + A¹/²B¹/² + A¹/²B¹/²)). These results complement those obtained by Bhatia, Lim and Yamazaki (LAA, 501 (2016) 112-122).