Determination of Means by Invariance
If M is a mean on and M(f(x1),f(x2),…,f(xn))=f(M(x1,x2,…,xn)) then we say that M is invariant under f. The problem is to find a class of functions that by invariance determines a mean uniquely. We focus on the geometric mean, which can be transformed to obtain results for other means.
Ji, J., & Kicey, C. (2006). Determination of means by invariance. Journal of Mathematical Analysis and Applications, 322(1), 453-467. doi:DOI: 10.1016/j.jmaa.2005.09.035