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Iterated quasi-arithmetic mean-type mappings

Volume 144 / 2016

Paweł Pasteczka Colloquium Mathematicum 144 (2016), 215-228 MSC: Primary 26E60; Secondary 26A18, 26B15, 39B12. DOI: 10.4064/cm6479-2-2016 Published online: 16 March 2016


We work with a fixed $N$-tuple of quasi-arithmetic means $M_1,\dots ,M_N$ generated by an $N$-tuple of continuous monotone functions $f_1,\dots ,f_N: I \to \mathbb {R}$ ($I$ an interval) satisfying certain regularity conditions. It is known [initially Gauss, later Gustin, Borwein, Toader, Lehmer, Schoenberg, Foster, Philips et al.] that the iterations of the mapping $I^N \ni b \mapsto (M_1(b),\dots ,M_N(b))$ tend pointwise to a mapping having values on the diagonal of $I^N$. Each of [all equal] coordinates of the limit is a new mean, called the Gaussian product of the means $M_1,\dots ,M_N$ taken on $b$. We effectively measure the speed of convergence to that Gaussian product by producing an effective—doubly exponential with fractional base—majorization of the error.


  • Paweł PasteczkaInstitute of Mathematics
    Pedagogical University of Cracow
    Podchorążych 2
    30-084 Kraków, Poland

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