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 ${G_{p,q}:p,q\in\R}$, i.e., to find necessary and sufficient conditions on the 6 parameters $a,b,c,d,p,q$ such that the identity [G_{p,q}\big(G_{a,b}(x,y),G_{c,d}(x,y)\big)=G_{p,q}(x,y) \qquad (x,y \in \R_+)] be valid. We recall that, for $p\neq q$, the Gini mean $G_{p,q}$ is defined by [G_{p,q}(x,y):=(\dfrac{x^p+y^p}{x^q+y^q})^{\frac1{p-q}}\qquad (x,y \in \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.