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

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.

Invariance of means

“…For this reason it was hard to decide if the geometric mean G is, in fact, invariant with respect to the pair L f , L g . This problem remained unsolved until 2011 when G lazowska [68] showed that this is not the case. This was done by calculating partial derivatives of order 7 of L f and L g satisfying (5.5).…”

Characterization of the Hardy property of means and the best Hardy constants

“…J. M. Borwein and P. B. Borwein [9] extended some earlier ideas [19,32,63] and generalized this iteration to a vector of continuous, strict means of an arbitrary length. For several recent results about Gaussian product of means see the papers by Baják and Páles [4,5,6,7], by Daróczy and Páles [13,17,18], by Głazowska [21,22], by Matkowski [39,40,41,42], and by Matkowski and Páles [43]. Given N ∈ N and a vector (M 1 , .…”

Quasiarithmetic-type invariant means on probability space

“…Inviant means in a family of quasi-arithmetic means were studied by many authors, for example Burai [7], Daróczy-Páles [11], J. Jarczyk [18], J. Jarczyk and Matkowski [20]. In fact invariant means were extensively studied during recent years, see for example the papers by Baják-Páles [1,2,3,4], by Daróczy-Páles [10,12,13], by Głazowska [15,16], by Matkowski [23,24,25], by Matkowski-Páles [27], by Pasteczka [29,32,30] and Matkowski-Pasteczka [26]. For details we refer the reader to the recent paper of J. Jarczyk and W. Jarczyk [19].…”