Generalized Cauchy means

Abstract

Given two means M and N, the operator \({\mathcal{M}_{M,N}}\) assigning to a given mean μ the mean $$\mathcal{M}_{M,N}(\mu )(x,y)=M(\mu (x,N(x,y)),\mu (N(x,y),y))$$was defined in Berrone and Moro (Aequationes Math 60:1–14, 2000) in connection with Cauchy means: the Cauchy mean generated by the pair f, g of continuous and strictly monotonic functions is the unique solution μ to the fixed point equation $$\mathcal{M}_{A_{(f)},A_{(g)}}(\mu )=\mu ,$$where A (f) and A (g) are the quasiarithmetic means respectively generated by f and g. In this article, the operator \({\mathcal{M}_{M,N}}\) is studied under less restrictive conditions and a general fixed point theorem is derived from an explicit formula for the iterates \({\mathcal{M} _{M,N}^{n}}\). The concept of class of generalized Cauchy means associated to a given family of mixing pairs of means is introduced and some distinguished families of pairs are presented. The question of equality in these classes of means remains a challenging open problem.