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.

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中文翻译：

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复合均值的渐近线

摘要：给定双变量表示 $m$ 和 $M$，我们可以形成由 $a_{n+1}=m(a_n,b_n)$, $b_{n+1} 递归定义的序列 $a_n$, $b_n$ =M(a_n,b_n)$ 与 $a_0,b_0>0$。这些（在温和条件下）收敛到一个新的均值 $\mathcal {M}(a_0,b_0)$，称为复合均值。对于 $m$ 和 $M$ 齐次，$\mathcal {M}$ 也是齐次的并且满足函数方程。在本文中，我们研究了 $\mathcal {M}(1,x)$ 作为 $x\to \infty$ 的渐近行为，给定了 $m$ 和 $M$，获得了可能的振荡函数的主项. 我们调查这种振荡行为何时实际存在，特别是对于来自某些众所周知的均值类别的 $m$ 和 $M$。我们还提供了一些数字，表明振荡的存在是通用的。

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