Let
$M_{k}^{[r]}({\mathbf{A}},{\mathbf{\Phi}},{\mathbf{\omega}}):=(
\sum_{j=1}^{k} \omega_j \ \Phi_j ( A_j^{r} ))^{1/r}$ ($r \in
{\mathbf{R}} \backslash \{0 \}$) be a weighted power mean of
positive operators $A_j$ with ${\mathsf{Sp}}(A_j) \subseteq [m,M]$
for some scalars $0<m<M$, where $\Phi_j$  are normalized positive
linear maps from ${\mathcal{B}}(H)$ to ${\mathcal{B}}(K)$ and
$\omega_j \in {\mathbf{R}}_+$ such that $\sum_{j=1}^k \omega_j =1$
($j=1,\ldots, k$). As a continuation of our paper [Sci. Math.
Japon., {\bf 57} (2003), 139--148] we determined real constants
$\alpha_1$, $\alpha_2$, $\beta_1$ and $\beta_2$ such that $$
\alpha_2
M_{k}^{[s]}({\mathbf{A}},{\mathbf{\Phi}},{\mathbf{\omega}}) \leq
  M_{k}^{[r]}({\mathbf{A}},{\mathbf{\Phi}},{\mathbf{\omega}}) \leq
  \alpha_1 M_{k}^{[s]}({\mathbf{A}},{\mathbf{\Phi}},{\mathbf{\omega}})$$ and
$$  \beta_2 I \leq
M_{k}^{[s]}({\mathbf{A}},{\mathbf{\Phi}},{\mathbf{\omega}}) -
  M_{k}^{[r]}({\mathbf{A}},{\mathbf{\Phi}},{\mathbf{\omega}}) \leq
  \beta_1 I$$
hold if $r \leq s$, $r,s \neq 0$. As applications, these
inequalities for some special maps were given.