Some results on matrix means (Q782514)

Let \(A\) and \(B\) be positive definite matrices with \(B\leq A\) where all eigenvalues lie in the range \([m,M]\) with \(0 < m < M\). It is shown that \(4mM(m+M)^{-2}B^{2}\leq mMB((m+M)I-B)^{-1}\leq mMA((m+M)I-A)^{-1}\leq A^{2}.\) This refines an inequality in [\textit{M. Fujii} et al., Nihonkai Math. J. 8, No. 2, 117--122 (1997; Zbl 0997.47505)]. The author also proves that the geometric mean \(A\#B:=A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{1/2}A^{1/2}\) satisfies \(A(A+B)^{\dag}B=(A\#B)(A+B)^{\dag}(A\#B),\) where \(\dag\) denotes the Moore-Penrose generalized inverse.