The invariance of the geometric mean G with respect to the Cauchy mean-type mappingWe give some necessary, and necessary and sufficient conditions under assumption that one of the generators of each Cauchy means is a power function.
The invariance of the geometric mean G with respect to the Lagrangian mean-type mapping (L f , L g ), i.e. the equation G • (L f , L g ) = G, is considered. We show that the functions f and g must be of high class regularity. This fact allows to reduce the problem to a differential equation and determine the second derivatives of the generators f and g.
Under some regularity assumptions imposed on the generators f , g, we determine all the quasi-arithmetic means M [f ] , M [g] and all real numbers λ and µ such that λM [f ] + µM [g] = A, where A is the arithmetic mean.
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.
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