M. Fujii and R. Nakamoto discuss the monotonity of the operator
function $F(r)= ((1-\mu) A^r + \mu B^r)^{\frac{1}{r}}$ ($r \in
{\mathbf{R}}$) for given $A,B>0$ and $\mu \in [0,1]$. They proved
it under the usual operator order: $F(r) \leq F(s)$ if $1 \leq r
\leq s$ or $1 \leq s \leq 2r$. Furthermore, they proved it under
the chaotic order: $F(r) \ll F(s)$ if $ r < s$ and consequently  $
\text{\bf s} - \lim_{r \rightarrow 0} F(r) = A \ \diamondsuit _\mu
B$, where $ \diamondsuit _\mu$ is the chaotic geometric mean
defined by $A \ \diamondsuit _\mu B := e^{(1- \mu) \log A + \mu
\log B}$.

The aim of this paper is to generalize the above mentioned as
follows: \\ \noindent Let $M_{k}^{[r]}({\mathbf{A}};w) := (
\sum_{j=1}^{k} \omega_j \ A_j^{r})^{1/r}$ ($r \in {\mathbf{R}}
\backslash \{0 \}$) be weighted power mean of posi\-ti\-ve
operators $A_j$, ${\mathsf{Sp}}(A_j) \subseteq [m,M]$
($j=1,\ldots, k$), $0<m<M$ and $\omega_j \in {\mathbf{R}}_+$,
$\sum_{j=1}^k \omega_j =1$. Let $M_{k}^{[0]}({\mathbf{A}};w)$ be
the corresponding chaotic geometric mean. If $r \leq s$ then real
constants $\alpha_1$ and $\alpha_1$ such that \(
  \alpha_2 M_{k}^{[s]}({\mathbf{A}};w) \leq
  M_{k}^{[r]}({\mathbf{A}};w) \leq
  \alpha_1 M_{k}^{[s]}({\mathbf{A}};w),
\) are determined, when $r \not \in \langle -1,1 \rangle$, $r \neq
0$ or $s \not \in \langle -1,1 \rangle$, $s \neq 0$. Furthermore,
if $r \leq s$ then real constant $\Delta$ such that \(
  \Delta M_{k}^{[s]}({\mathbf{A}};w) \ll
  M_{k}^{[r]}({\mathbf{A}};w) \ll
  M_{k}^{[s]}({\mathbf{A}};w),
\) is determined.