Computer aided solution of the invariance equation for two-variable Gini means

Our aim is to solve the so-called invariance equation in the class of two-variable Gini means {Gp,q:p,q∈R}{Gp,q:p,q∈R}, i.e., to find necessary and sufficient conditions on the 6 parameters a,b,c,d,p,qa,b,c,d,p,q such that the identity Gp,q(Ga,b(x,y),Gc,d(x,y))=Gp,q(x,y)(x,y∈R+) be valid. We recall that, for p≠qp≠q, the Gini mean Gp,qGp,q is defined by Gp,q(x,y):=(xp+ypxq+yq)1p−q(x,y∈R+). The proof uses the computer algebra system Maple V Release 9 to compute a Taylor expansion up to 12th order, which enables us to describe all the cases of the equality.