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 ∈ R } , i.e., we find necessary and sufficient conditions on the six parameters a, b, c, d, p, q such that the identity S p , q S a , b ( x , y ) , S c , d ( x , y ) = S p , q ( x , y ) ( x , y ∈ R + ) , be valid. We recall that, for pq( p − q) ≠ 0 and x ≠ y, the Stolarsky mean S p, q is defined by S p , q ( x , y ) ≔ q ( x p - y p ) p ( x q - y q ) 1 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.