SINGULAR VALUE INEQUALITIES OF LEWENT TYPE

Abstract

Let $A_{i}$ be strictly contractive matrices and let $\lambda_{i}$ be nonnegative real numbers with $\displaystyle{\sum_{i=1}^{m}} \ \lambda_{i} =1$, $ \ i=1,\ldots,m.$ We prove that \begin{equation*} s \left(\frac{I+\displaystyle{\sum_{i=1}^{m}}\lambda_{i}A_{i}}{I- \displaystyle{\sum_{i=1}^{m}} \lambda_{i}A_{i}}\right) \prec_{\rm wlog} \displaystyle{\prod_{i=1}^{m}} \ s \left(\left(\frac{I+ |A_{i}|}{I- |A_{i}|}\right)^{\lambda_{i}}\right), \end{equation*} which generalizes a Lewent type determinantal inequality due to Lin [M. Lin, A Lewent type determinantal inequality, Taiwanese J. Math. 17(2013), 1303-1309]. On the other hand, we also prove \begin{equation*} s \left(\frac{I+\displaystyle{\sum_{i=1}^{m}}\lambda_{i}A_{i}}{I- \displaystyle{\sum_{i=1}^{m}} \lambda_{i}A_{i}} \right) \prec_{\rm wlog} \displaystyle{\sum_{i=1}^{m}} \ \lambda_{i} s \left( \frac{I+|A_{i}|}{I-|A_{i}|} \right). \end{equation*} Here ``$\prec_{\rm wlog}$" stands for weakly log-majorization. In addition, some other related inequalities are also obtained.