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 (Df,g,Dh,k), i.e. the equation G∘(Df,g,Dh,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.