Abstract. Let $A$ and $B$ be positive operators on a Hilbert space, and $p$, $q$ and $\lambda $ be positive numbers. Ito and Yamazaki showed relations between the two operator inequalities $(B^{r\over2}A^pB^{r\over2})^{r\over p+r} \ge B^r$ and $A^p \ge(A^{p\over2}B^rA^{p\over2})^{p\over p+r}$. We shall show parallel relations between the two weaker inequalities $$ {rB^{r\over2}A^pB^{r\over2}+p\lambda ^{p+r}I\over(p+r)\lambda ^p}\ge B^r \hbox{ and } A^p \ge{(p+r)\lambda ^pA^{p\over2}B^rA^{p\over2}\over rA^{p\over2}B^rA^{p\over2}+p\lambda ^{p+r}I}, $$ which can be obtained by applying the arithmetic--geometric--harmonic mean inequality to the inequalities studied by Ito and Yamazaki. As an application of these relations, we shall show {\it ``an operator $T$ is normal if $T$ and $T^*$ are paranormal,''} which is an extension of Ando's result, that is, he showed the same result with the additional kernel condition $N(T)=N(T^*)$. We shall also show an extension of a result on normality of $w$-hyponormal operators via Aluthge transformation by Chō, Huruya and Kim.

AMS Subject Classification (1991): 47A63, 47B20

Received July 20, 2001. (Registered under 2902/2009.)