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

摘要

We solve the so-called invariance equation in the class of two-variable Stolarsky means {Sp,q:p,q∈R}, i.e., we find necessary and sufficient conditions on the six parameters a, b, c, d, p, q such that the identitySp,qSa,b(x,y),Sc,d(x,y)=Sp,q(x,y)(x,y∈R+),be valid. We recall that, for pq(p − q) ≠ 0 and x ≠ y, the Stolarsky mean Sp, q is defined by≔Sp,q(x,y)≔q(xp-yp)p(xq-yq)1p-q.In the proof first we approximate the Stolarsky mean and we use the computer-algebra system Maple V Release 9 to compute the Taylor expansion of the approximation up to 12th order, which enables us to describe all the cases of the equality.