Asymptotics of compound means

Abstract:

Given bivariate means $m$ and $M$, we can form sequences $a_n$, $b_n$ defined recursively by $a_{n+1}=m(a_n,b_n)$, $b_{n+1}=M(a_n,b_n)$ with $a_0,b_0>0$. These converge (under mild conditions) to a new mean, $\mathcal {M}(a_0,b_0)$, called a compound mean. For $m$ and $M$ homogeneous, $\mathcal {M}$ is also homogeneous and satisfies a functional equation. In this paper we study the asymptotic behaviour of $\mathcal {M}(1,x)$ as $x\to \infty$ given that of $m$ and $M$, obtaining the main term up to a possible oscillatory function. We investigate when this oscillatory behaviour is in fact present, in particular for $m$ and $M$ coming from some well-known classes of means. We also present some numerics, which indicate the presence of oscillation is generic.