Recently Uchiyama gave a nice comment on the implication of 

the Furuta inequality to the chaotic Furuta inequality, i.e., 

$\log A \ge \log B$ for positive invertible operators $A$ and $B$ 

if and only if 

$A^r \ge (A^{\frac{r}{2}}B^pA^{\frac{r}{2}})^{\frac{r}{p+r}}$ 

for all $p,\ r \ge 0$. 

The purpose of this note is to show the converse implication. 

That is, the chaotic Furuta inequality and the Furuta inequality 

are equivalent.