  We present analysis of some new means recently introduced by M. Ra\"{i}ssouli
and A. Rezgui. We establish comparison relations and results on
$(K,N)$-sub/super-stabilizability where $K$ and $N$ belong to the class of
power means, denoted by $B_p$, and $M$ is one of the classical or recently
studied new means. Assuming that means $K$, $M$ and $N$ have asymptotic
expansions, we present the complete asymptotic expansion of the resultant
mean-map. As an application of the obtained asymptotic expansions and the
asymptotic inequality between $M$ and $\mathcal{R}(B_p,M,B_q)$, we show how to
find the optimal parameters $p$ and $q$ for which $M$ is
$(B_p,B_q)$-sub/super-stabilizable.
