Weak majorization inequalities for singular values (Q2869940)

Given two \(n\times n\) positive semi-definite matrices \(A, B\), it is proved that NEWLINE\[NEWLINEs(AB) \prec_w s\left(\int_0^1A^{1/2+t}B^{3/2-t} \, dt\right) \prec_w s\left( \Big ( \frac {A+B}2\Big)^2\right),NEWLINE\]NEWLINE where \(\prec_w\) denotes the weak majorization and \(s(X)\) denotes the vector of singular values of \(X\). This is a refinement of the inequality \(s(AB) \prec_w s\left( \Big ( \frac {A+B}2\Big)^2\right)\) of \textit{R. Bhatia} and \textit{F. Kittaneh} [Linear Algebra Appl. 308, No. 1--3, 203--211 (2000; Zbl 0974.15016)]. In the context of normal matrices, a generalization of the inequality \(s(A^m+B^m) \prec_w s((A+B)^m)\), where \(m\) is any positive integer, of \textit{R. Bhatia} and \textit{F. Kittaneh} [Lett. Math. Phys. 43, No. 3, 225--231 (1998; Zbl 0912.47005)] is given.