A solution of an open problem concerning Lagrangian mean-type mappings

• Autorzy: Dorota Głazowska
• Języki publikacji: EN

Abstrakty

EN

The problem of invariance of the geometric mean in the class of Lagrangian means was considered in [Głazowska D., Matkowski J., An invariance of geometric mean with respect to Lagrangian means, J. Math. Anal. Appl., 2007, 331(2), 1187–1199], where some necessary conditions for the generators of Lagrangian means have been established. The question if all necessary conditions are also sufficient remained open. In this paper we solve this problem.

Słowa kluczowe

EN

Mean; Lagrangian mean; Invariant mean; Functional equation

Kategorie tematyczne

• 26A18: Iteration
• 39B12: Iteration theory, iterative and composite equations
• 26E60: Means

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 5
• Strony: 1067-1073

Daty

wydano

2011-10-01

online

2011-07-26

Twórcy

autor

Dorota Głazowska

• University of Zielona Góra

Bibliografia

• [1] Borwein J.M., Borwein P.B., Pi and the AGM, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York-Chichester-Brisbane-Toronto-Singapore, 1987
• [2] Bullen P.S., MitrinoviĆ D.S., VasiĆ P.M., Means and Their Inequalities, Math. Appl. (East European Ser.), 31, D. Reidel, Dordrecht-Boston-Lancaster-Tokyo, 1988
• [3] Daróczy Z., Páles Zs., Gauss-composition of means and the solution of the Matkowski-Sutô problem, Publ. Math. Debrecen, 2002, 61(1–2), 157–218
• [4] Głazowska D., Some Cauchy mean-type mappings for which the geometric mean is invariant, J. Math. Anal. Appl., 2011, 375(2), 418–430 http://dx.doi.org/10.1016/j.jmaa.2010.09.036
• [5] Głazowska D., Matkowski J., An invariance of geometric mean with respect to Lagrangian means, J. Math. Anal. Appl., 2007, 331(2), 1187–1199 http://dx.doi.org/10.1016/j.jmaa.2006.09.005
• [6] Matkowski J., Invariant and complementary quasi-arithmetic means, Aequationes Math., 1999, 57(1), 87–107 http://dx.doi.org/10.1007/s000100050072
• [7] Matkowski J., Iterations of mean-type mappings and invariant means, Ann. Math. Sil., 1999, 13, 211–226
• [8] Matkowski J., On invariant generalized Beckenbach-Gini means, In: Functional Equations - Results and Advances, Adv. Math. (Dordr.), 3, Kluwer, Dordrecht, 2002, 219–230
• [9] Matkowski J., Lagrangian mean-type mappings for which the arithmetic mean is invariant, J. Math. Anal. Appl., 2005, 309(1), 15–24 http://dx.doi.org/10.1016/j.jmaa.2004.10.033

Identyfikatory

DOI

10.2478/s11533-011-0059-2