Extensions of inequalities for unitarily invariant norms via log majorization (Q417482)

\textit{J. S. Matharu} and \textit{J. S. Aujla} [Linear Algebra Appl. 436, No. 6, 1623--1631 (2012; Zbl 1242.15018)] showed that for all \(n\times n\) positive definite matrices \(A,B\) and \(\alpha \in [0,1]\), NEWLINE\[NEWLINE\prod_{j=1}^{k}\lambda _{j}(A\#_{\alpha }B)\leq \prod\limits_{j=1}^{k}\lambda _{j}(A^{1-\alpha }B^{\alpha })\quad (1\leq k\leq n),NEWLINE\]NEWLINE where \(\#_{\alpha }\) denotes the operator mean and \(\lambda _{j}(X)\,\,(1\leq j\leq n)\) denote the eigenvalues of a Hermitian matrix \(X\) arranged in decreasing order. The author of the present paper provides further extensions of the result above via log majorization by using a variation of Corollary 1.2 of [the author, Invitation to linear operators. From matrices to bounded linear operators on a Hilbert space. London: Taylor and Francis (2001; Zbl 1029.47001)].