Speaker
Description
For three homogeneous symmetric bivariate means $K$, $M$, $N$, let
$$
\mathcal{R}(K,M,N)(s,t)=K\bigl(M\bigl(s,N(s,t)\bigr),M\bigl(N(s,t),t\bigr) \bigr)
$$ be their resultant mean--map. Mean $M$ is said to be stable (or balanced), if $\mathcal{R}(M,M,M)=M$. Mean $M$ is called $(K,N)$-sub(super)-stabilizable, if $\mathcal{R}(K,M,N)\le(\ge)\, M$, and $M$ is between $K$ and $N$, where $K$ and $N$ are two non-trivial stable comparable means.
We present some 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
$$
R(x-t,x+t)= \mathcal{R}(K,M,N)(x-t,x+t)\sim\sum_{m=0}^\infty a_m^R t^{m} x^{-m+1},\quad x\to\infty.
$$
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.
