Further refinements of some inequalities involving unitarily invariant norm

Yang Chaojun, Lu Fangyan


Let $A,B,X\in\mathbb{{M}}_n\mathbb{(C)}$ and $|||\cdot|||$ be an arbitrary unitarily invariant norm. We give a new log-convex function $f(t,s)$ such that $f(1/2,1/2)\le f(t,s)$ for any $t,s\in[0,1]$ which generalize the log-convex function defined in [4] and obtain the inequalities as follows:

&\le f(t,1-t)\\
&\quad\times ((1-t)|||A^*AX|||+t|||XB^*B|||-r(\sqrt{|||A^*AX|||}-\sqrt{|||XB^*B|||})^2),

where $t\in[0,1]$ and $r=\min\left\{ t,1-t \right\}.$

Furthermore, we refine some inequalities as well.


  • There are currently no refbacks.