In this paper, we shall show the following:

 If $A_1^{\delta} \ge A_2^{\delta}$ for some $\delta>0$

 and $\log A_2 \ge \log B$,

 then for each $\alpha$ such that $0 \le \alpha \le \delta$,

 \[

       (B^{\frac{r}{2}} A_1^t B^{\frac{r}{2}})^{\frac{\alpha+r}{t+r}}

   \ge (B^{\frac{r}{2}} A_2^p B^{\frac{r}{2}})^{\frac{\alpha+r}{p+r}}

 \]

 holds for all $p$, $t$ and $r$ such that

 $\delta \ge p \ge \frac{\alpha}{2}$, $t \ge \max\{\alpha, p\}$

 and $r \ge 0$.

And also we shall discuss a relation

 between $A_1^{\alpha} \ge A_2^{\alpha}$ and

 $(A_2^{\frac{-p}{2}} A_1^t A_2^{\frac{-p}{2}})^{\frac{p-\beta}{t-p}}

 \ge A_2^{p-\beta}$ for $0 \le \beta \le p<t$.

These results are extensions of Yang's recent ones.