Solving invariance equations involving homogeneous means with the help of computer

摘要

Given three strict means M,N,K:R+2→R+, we say that the triple (M,N,K) satisfies the invariance equation ifKM(x,y),N(x,y)=K(x,y)(x,y∈R+)holds. It is well known that K is uniquely determined by M and N, and it is called the Gauss composition M⊗N of M and N.Our aim is to solve the invariance equation when each of the means M,N,K is either a Gini or a Stolarsky mean with possibly different parameters. This implies that we have to consider six different invariance equations. With the help of the computer algebra system Maple V Release 9, which enables us to perform the tedious computations, we completely describe the general solutions of these six equations.