The Furuta inequality says that it characterizes the operator order:

 For $A\geq B\geq 0$, $A \ge C \ge B$ if and only if

 \begin{equation*} 

(A^{\frac{r}{2}}C^pA^{\frac{r}{2}})^{\frac{1}{q}}

\geq  (C^{\frac{r}{2}}C^pC^{\frac{r}{2}})^{\frac{1}{q}}

\geq   (B^{\frac{r}{2}}C^pB^{\frac{r}{2}})^{\frac{1}{q}}

\end{equation*}

holds for all $p, r \geq 0$ and $q \ge 1$ with $(1+r)q \ge p+r$.

 

 From the viepoint of this, we give some characterizations of $p$-hyponormal operators and simplify the privious characterization by one of the authors. Furthermore we characterize log-hyponormal operators.