Abstract. Let $A$ and $B$ be positive operators. We remark that $A$ and $B$ are not necessarily invertible. Recently, Ito and Yamazaki showed relations between the two inequalities $$ (B^{r\over2}A^pB^{r\over2})^{r\over p+r} \ge B^r \mbox{ and } A^p \ge(A^{p\over2}B^rA^{p\over2})^{p\over p+r}, $$ for fixed positive numbers $p \ge0$ and $r \ge0$. In this paper, as extensions of these results, we shall show relations between the two inequalities $$ (B^{r\over2}A^pB^{r\over2})^{r-\delta\over p+r} \ge B^{r-\delta } \mbox{ and } A^{p\over2}B^{\delta }A^{p\over2} \ge(A^{p\over2}B^rA^{p\over2})^{\delta +p\over p+r}, $$ for fixed positive numbers $r \ge\delta \ge0$ and $p \ge0$. We shall also show a relation between the two inequalities $$ A^{p-\gamma } \ge(A^{p\over2}B^rA^{p\over2})^{p-\gamma\over p+r} \mbox{ and } (B^{r\over2}A^pB^{r\over2})^{\gamma +r\over p+r} \ge B^{r\over2}A^{\gamma }B^{r\over2}, $$ for fixed positive numbers $p \ge\gamma \ge0$ and $r \ge0$. Furthermore, we shall show a slight extension of a result on transitive properties of the first two inequalities by Yanagida as an application of these results.

AMS Subject Classification (1991): 47A63

Keyword(s): Positive operators, Furuta inequality

Received April 24, 2002, and in revised form October 24, 2002. (Registered under 2935/2009.)