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.