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

TL;DR

This paper characterizes the conditions under which a specific invariance equation involving two-variable Stolarsky means holds, using computer algebra to analyze Taylor expansions and derive necessary and sufficient parameter conditions.

Contribution

It provides a complete solution to the invariance equation for Stolarsky means, identifying all parameter sets satisfying the identity.

Findings

Derived necessary and sufficient conditions for the invariance equation.

Used computer algebra to analyze Taylor expansions up to 12th order.

Established a comprehensive characterization of the invariance property.

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.