Notes on matrix inequalities ‎ ‎and‎ positive multilinear mappings

In this paper,  our aim is to prove some matrix inequalities involving arbitrary matrix means and positive multilinear mappings.  For example,   it is  shown  that for  Hermitian  matrices A_{i}, B_{i} such that ۰ < m \leq A_{i},B_{i} \leq M \,\,(i=۱,\cdots, k),\begin{align*}\Phi^{۲} (A_{۱} \sigma_{۱}  B_{۱}, \cdots ,A_{k} \sigma_{۱}  B_{k} ) \leq\left(K(u^{k})\right)^{۲}\Phi^{۲} (A_{۱} \sigma_{۲}  B_{۱}, \cdots ,A_{k} \sigma_{۲}  B_{k} ),\end{align*}where \sigma_{۱} and \sigma_{۲}  are two arbitrary matrix means between the arithmetic and harmonic means,  \Phi:\mathscr{M}_n^k(\mathbb{C}) \rightarrow  \mathscr{M}_l(\mathbb{C}) is a positive unital multilinear mapping, u=\frac{M}{m}  and K(u)=\frac{(۱+u)^{۲}}{۴ u}.  We also give the obtained results for the adjoint and the dual of an arbitrary matrix mean.

کلیدواژه ها:

Matrix means‎ ، ‎Kantorovich's constant‎ ، ‎Positive multilinear mapping‎ ، ‎Dual‎ ، ‎Adjoint‎