Some Cauchy mean-type mappings for which the geometric mean is invariant

The invariance of the geometric mean G with respect to the Cauchy mean-type mapping ( D f , g , D h , k ) , i.e. the equation G ∘ ( D f , g , D h , k ) = G , is considered. We give some necessary, and necessary and sufficient conditions under assumption that one of the generators of each Cauchy means is a power function.