Further refinements of some inequalities involving unitarily invariant norm

Abstract

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:

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

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

Furthermore, we refine some inequalities as well.