Abstract
For a sequence of continuous, monotone functions \(f_1,\ldots ,f_n :I \rightarrow \mathbb {R}\) (I is an interval) we define the mapping \(M :I^n \rightarrow I^n\) as a Cartesian product of quasi-arithmetic means generated by \(f_j\)-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of \(I^n\). We will prove that whenever all \(f_j\)-s are \(\mathcal {C}^2\) with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit \(\frac{\text {Var}\, M^{k+1}(v)}{(\text {Var}\, M^{k}(v))^2}\) will be calculated in a nondegenerated case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arrow, K.J.: Aspects of the Theory of Risk-Bearing. Yrjö Jahnsson Foundation, Helsinki (1965)
Baják, Sz, Páles, Zs: Computer aided solution of the invariance equation for two-variable Gini means. Comput. Math. Appl. 58, 334–340 (2009)
Baják, Sz, Páles, Zs: Invariance equation for generalized quasi-arithmetic means. Aequ. Math. 77, 133–145 (2009)
Baják, Sz, Páles, Zs: Computer aided solution of the invariance equation for two-variable Stolarsky means. Appl. Math. Comput. 216(11), 3219–3227 (2010)
Baják, Sz, Páles, Zs: Solving invariance equations involving homogeneous means with the help of computer. Appl. Math. Comput. 219(11), 6297–6315 (2013)
Borwein, J.M., Borwein, P.B.: Pi and the AGM: A Study in the Analytic Number Theory and Computational Complexity. Wiley, New York (1987)
Burai, P.: A Matkowski–Sutô type equation. Publ. Math. Debr. 70, 233–247 (2007)
Daróczy, Z.: Functional equations involving means and Gauss compositions of means. Nonlinear Anal. 63(5–7), e417–e425 (2005)
Daróczy, Z., Páles, Zs.: A Matkowski–Sutô type problem for quasi-arithmetic means of order \(\alpha \). In: Daróczy, Z., Páles, Zs, editors. Functional Equations—Results and Advances, vol. 3 of Adv. Math. (Dordr.), pp. 189–200. Kluwer Acad. Publ., Dordrecht (2002)
Daróczy, Z., Páles, Zs: Gauss-composition of means and the solution of the Matkowski–Sutô problem. Publ. Math. Debr. 61(1–2), 157–218 (2002)
Daróczy, Z., Páles, Zs: The Matkowski–Sutô problem for weighted quasi-arithmetic means. Acta Math. Hungar. 100(3), 237–243 (2003)
de Finetti, B.: Sul concetto di media. Giornale dell’ Instituto, Italiano degli Attuarii 2, 369–396 (1931)
Foster, D.M.E., Phillips, G.M.: The arithmetic-harmonic mean. Math. Comput. 42(165), 183–191 (1984)
Gauss, C.F.: Nachlass: Aritmetisch-geometrisches Mittel. In: Werke 3 (Göttingem 1876), pp. 357–402. Königliche Gesellschaft der Wissenschaften (1818)
Głazowska, D.: A solution of an open problem concerning Lagrangian mean-type mappings. Cent. Eur. J. Math. 9(5), 1067–1073 (2011)
Głazowska, D.: Some Cauchy mean-type mappings for which the geometric mean is invariant. J. Math. Anal. Appl. 375(2), 418–430 (2011)
Jarczyk, J.: Invariance of weighted quasi-arithmetic means with continuous generators. Publ. Math. Debr. 71(3–4), 279–294 (2007)
Jarczyk, J., Matkowski, J.: Invariance in the class of weighted quasi-arithmetic means. Ann. Polon. Math. 88(1), 39–51 (2006)
Knopp, K.: Über Reihen mit positiven Gliedern. J. Lond. Math. Soc. 3, 205–211 (1928)
Kolmogorov, A.N.: Sur la notion de la moyenne. Rend. Accad. dei Lincei 6(12), 388–391 (1930)
Kuczma, M., Choczewski, B., Ger, R.: Iterative functional equations. In: Encyclopedia of Mathematics and its Applications, vol. 32. Cambridge University Press, Cambridge (1990)
Lehmer, D.H.: On the compounding of certain means. J. Math. Anal. Appl. 36, 183–200 (1971)
Matkowski, J.: Iterations of mean-type mappings and invariant means. Ann. Math. Sil. 13:211–226 (1999). European Conference on Iteration Theory (Muszyna-Złockie, 1998)
Matkowski, J.: On iteration semigroups of mean-type mappings and invariant means. Aequ. Math. 64(3), 297–303 (2002)
Matkowski, J.: Lagrangian mean-type mappings for which the arithmetic mean is invariant. J. Math. Anal. Appl. 309(1), 15–24 (2005)
Matkowski, J.: Iterations of the mean-type mappings and uniqueness of invariant means. Ann. Univ. Sci. Bp. Sect. Comput. 41, 145–158 (2013)
Matkowski, J., Páles, Zs: Characterization of generalized quasi-arithmetic means. Acta Sci. Math. (Szeged) 81(3—-4), 447–456 (2015)
Mikusiński, J.G.: Sur les moyennes de la forme \(\psi ^{-1}[\sum q\psi (x)]\). Studia Math. 10(1), 90–96 (1948)
Nagumo, M.: Über eine Klasse der Mittelwerte. Jpn. J. Math. 7, 71–79 (1930)
Pasteczka, P.: Iterated quasi-arithmetic mean type mappings. Colloq. Math. 144(2), 215–228 (2016)
Pratt, J.W.: Risk aversion in the small and in the large. Econometrica 32(1/2), 122–136 (1964)
Schoenberg, I.J.: Mathematical Time Exposures. Mathematical Association of America, Washington, DC (1982)
Toader, G., Toader, S.: Greek means and the arithmetic-geometric mean. In: RGMIA Monographs. Victoria University (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Pasteczka, P. On the quasi-arithmetic Gauss-type iteration. Aequat. Math. 92, 1119–1128 (2018). https://doi.org/10.1007/s00010-018-0568-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-018-0568-1