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

Open Mathematics (2011)

- Volume: 9, Issue: 5, page 1067-1073
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topDorota Głazowska. "A solution of an open problem concerning Lagrangian mean-type mappings." Open Mathematics 9.5 (2011): 1067-1073. <http://eudml.org/doc/269793>.

@article{DorotaGłazowska2011,

abstract = {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.},

author = {Dorota Głazowska},

journal = {Open Mathematics},

keywords = {Mean; Lagrangian mean; Invariant mean; Functional equation; mean; invariant mean; functional equation},

language = {eng},

number = {5},

pages = {1067-1073},

title = {A solution of an open problem concerning Lagrangian mean-type mappings},

url = {http://eudml.org/doc/269793},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Dorota Głazowska

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

JO - Open Mathematics

PY - 2011

VL - 9

IS - 5

SP - 1067

EP - 1073

AB - 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.

LA - eng

KW - Mean; Lagrangian mean; Invariant mean; Functional equation; mean; invariant mean; functional equation

UR - http://eudml.org/doc/269793

ER -

## References

top- [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 Zbl1205.26042
- [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 Zbl1119.26029
- [6] Matkowski J., Invariant and complementary quasi-arithmetic means, Aequationes Math., 1999, 57(1), 87–107 http://dx.doi.org/10.1007/s000100050072 Zbl0930.26014
- [7] Matkowski J., Iterations of mean-type mappings and invariant means, Ann. Math. Sil., 1999, 13, 211–226 Zbl0954.26015
- [8] Matkowski J., On invariant generalized Beckenbach-Gini means, In: Functional Equations - Results and Advances, Adv. Math. (Dordr.), 3, Kluwer, Dordrecht, 2002, 219–230 Zbl0996.39019
- [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 Zbl1084.39019