Matrix Mean Inequalities for Sector Matrices

In this note, some inequalities involving matrix means of sectorial matrices are proved which are generalizations and refinements of previous known results. Among them, let A and B be two accretive matrices with \(A,B\in \mathcal {S}_{\theta }\), \(0 < mI \leqslant A, B \leqslant MI\) for positive real numbers M and m. If \(\sigma ,\sigma _1,\sigma _2\) are matrix means such that \(\sigma ^*\leqslant \sigma _1,\sigma _2\leqslant \sigma \), where \(\sigma ^*\) is the adjoint of \(\sigma \) and \(\Phi \) is a positive unital linear map, then for each \(p>0\),

$$\Phi ^{p}\Re (A \sigma _{1} B) \leqslant \sec ^{2p}\theta \alpha ^{p} \Phi ^{p}\Re (A \sigma _{2} B),$$

where

$$ \alpha = \max \left\{ K, 4^{1-\frac{2}{p}}K \right\} ,$$

and \( K= \frac{(M+m)^2}{4mM}\) is the Kantorovich constant.