In this paper, we shall discuss generalizations of the results on Ando-Hiai inequality and a generalized Furuta-type operator function. Firstly we shall obtain a generalization of our recent result on generalized Ando-Hiai inequality, that is, if $A^{-r}\ \sharp_{\frac{r}{p+r}}\ B^p \le I$ for $A,B>0$ and $p,r>0$, then \[ A^{-r} \ \sharp_{\frac{\delta+r}{p+r}}\ B^{p} \le A^{-t}\ \sharp_{\frac{\delta+t}{s+t}}\ B^{s} \] for $0 \le s \le p$, $0 \le t \le r$ and $-t \le \delta \le s$, Secondly, as a related result to Furuta's and our recent ones, we shall show the following: Let $A,B>0$. If $A^t \ge B^t \ge 0$ for some $t \in (0,1]$ and $p \ge 1$, then \[ F(\lambda,\mu) =A^{-\lambda}\ \sharp_{\frac{1-t+\lambda}{(p-t)\mu+\lambda}}\ (A^{\frac{-t}{2}} B^p A^{\frac{-t}{2}})^{\mu}. \] satisfies \( F(q,w) \ge F(r,s) \) for any $s \ge 1$, $r \ge t$, $\frac{1-t}{p-t} \le w \le s$ and $0 \le q \le r$, and also these two theorems lead Grand Furuta inequality. Moreover we discuss further extensions of the results on these two topics.