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

Abstract

We solve the so-called invariance equation in the class of two-variable Stolarsky means ${S_{p,q}:p,q\in\R}$, i.e., we find necessary and sufficient conditions on the 6 parameters $a,b,c,d,p,q$ such that the identity [S_{p,q}\big(S_{a,b}(x,y),S_{c,d}(x,y)\big)=S_{p,q}(x,y) \qquad (x,y \in \R_+)] be valid. We recall that, for $pq(p-q)\neq 0$ and $x\neq y$, the Stolarsky mean $S_{p,q}$ is defined by [S_{p,q}(x,y):=(\dfrac{q(x^p-y^p)}{p(x^q-y^q)})^{\frac1{p-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.