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

TL;DR

This paper uses computer algebra to analyze the invariance equation for two-variable Gini means, deriving necessary and sufficient conditions for the identity to hold, thus advancing the understanding of Gini mean properties.

Contribution

It provides a complete characterization of when the invariance equation holds for two-variable Gini means using computational algebra methods.

Findings

Derived necessary and sufficient conditions for invariance.

Used Taylor expansion up to 12th order for analysis.

Identified all cases where the invariance identity is valid.

Abstract

Our aim is to solve the so-called invariance equation in the class of two-variable Gini means ${G_{p,q}:p,q\in\R}$, i.e., to find necessary and sufficient conditions on the 6 parameters $a,b,c,d,p,q$ such that the identity [G_{p,q}\big(G_{a,b}(x,y),G_{c,d}(x,y)\big)=G_{p,q}(x,y) \qquad (x,y \in \R_+)] be valid. We recall that, for $p\neq q$, the Gini mean $G_{p,q}$ is defined by [G_{p,q}(x,y):=(\dfrac{x^p+y^p}{x^q+y^q})^{\frac1{p-q}}\qquad (x,y \in \R_+).] The proof uses the computer algebra system Maple V Release 9 to compute a Taylor expansion up to 12th order, which enables us to describe all the cases of the equality.