Asymptotics of compound means.

Given bivariate means m and M, we can form sequences an, bn defined recursively by an+1 = m(an, bn), bn+1 = M(an,bn) with a0, b0 > 0. These converge (under mild conditions) to a new mean, M(a0, b0), called a compound mean. For m and M homogeneous, M is also homogeneous and satisfies a functional equation. In this paper we study the asymptotic behaviour of M(1,x) as x → ∞ 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. [ABSTRACT FROM AUTHOR]