Some inequalities for the matrix Heron mean

2016/05/11 by Hoa, Dinh Trung
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1605.03516

Abstract

Let A, B be positive definite matrices, p=1, 2 and r≥ 0. It is shown that ||A+ B + r(A\sharp_t B+A\sharp_1-t B)||_p ≤ ||A+ B + r(A^tB^1-t + A^1-tB^t)||_p. We also prove that for positive definite matrices A and B \Dt (P_t(A, B)) ≤ \Dt (Q_t(A, B)), where Q_t(A, B)= ((A^t+B^t)/(2))^1/t and P_t(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, \Dt(A+ B + 2(A\sharp B)) ≤ \Dt(A+ B + A^1/2B^1/2 + A^1/2B^1/2)). These results complement those obtained by Bhatia, Lim and Yamazaki (LAA, \bf 501 (2016) 112-122).

Related

• On matrix inequalities between the power means: counterexamples
• New Improvements to Heron and Heinz Inequality Using Matrix Techniques
• On norm inequalities related to the geometric mean
• Inequalities related to Bourin and Heinz means with a complex parameter
• A Determinantal Inequality for the Geometric Mean with an Application in\n Diffusion Tensor Imaging