Let $B^n_p=\{x\in \mathbb R^n|\|x\|_p\geq1\}$ be the unit ball in $\ell^n_p$. We prove the inequalities for the volume of the $B^n_p$: \[ V^{\frac1{n+1}}_{B^{n+1}_p}<V^{\frac1n}_{B^n_p} \] \[ 2\Gamma\Big(\frac1p+1\Big)\sqrt[p]{\frac p{n+p}V_{B^{n+1}_p}}\leq V_{B^{n+1}_p} \] for all $n\geq1$ and $p\geq1$, where $V_{B^n_p}$ denotes the volumes of $B^n_p$. Furthermore, we obtain the upper and lower bounds of $V^{\frac n{n+1}}_{B_p^{n+1}}/V_{B^n_p}$ and $V_{B^{n+1}_p}/V_{B^n_p}$. Our results are generalizations for inequalities in $R^n$ proved and refined by G. D. Anderson et al., K. H. Borgwardt, D. A.Klain and G. -C. Rota and H. Alzer.