In this paper, firstly we shall show the equivalence relation between Ando-Hiai inequality ``For $A,B>0$, $A\ \sharp_{\alpha}\ B \le I$ ensures $A^r\ \sharp_{\alpha}\ B^r \le I$ for $r \ge 1$" and the inequalty ``$A \ge B \ge 0$ ensures $A^{-r}\ \sharp_{\frac{r}{p+r}}\ B^p \le I$ for $p \ge 0$ and $r \ge 0$." Next we shall show a complementary result of Ando-Hiai inequality: If $A\ \sharp_{\alpha}\ B \le I$, then $A^r\ \sharp_{\alpha}\ B^r \le A\ \sharp_{\alpha}\ B$ for $0 \le r \le 1$.