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Asymptotics of compound means. (English) Zbl 1483.26026

In this paper, the author studies bivariate means \(m\) and \(M\) which can be formed from 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 \(a_0, b_0 > 0\). He investigates under mild conditions when these means will converge to a new mean \(\mathcal{M}(a_0,b_0)\), called a compound mean. The special case \(m\) and \(M\) are homogeneous, it is shown that \(\mathcal{M}\) is also homogeneous and satisfies a functional equation. Further, the author studies the asymptotic behaviour of \(\mathcal{M}(1, x)\) as \(x \to \infty\) given that of \(m\) and \(M\), and obtains the main term up to a possible oscillatory function. The oscillatory behaviour is also investigated if \(m\) and \(M\) are coming from some well-known classes of means. Some numerical computations are also reported which show that the oscillations are generic.

MSC:

26E60 Means
26A18 Iteration of real functions in one variable
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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