Explicit solutions of the invariance equation for means

TL;DR

This paper generalizes invariance identities for bivariate means, providing explicit formulas for M-complementary pairs and demonstrating limitations in extending these results to higher dimensions.

Contribution

It introduces a generalized invariance identity for means and derives explicit formulas for M-complementary pairs, expanding the theoretical framework of mean functions.

Findings

Explicit formulas for M-complementary means

Limitations of the method in higher dimensions

Examples illustrating the theory

Abstract

Extending the notion of projective means we first generalize an invariance identity related to the Carlson log given in a recent paper of P. Kahlig and J. Matkowski, and then, more generally, given a bivariate symmetric, homogeneous and monotone mean M, we give explicit formula for a rich family of pairs of M-complementary means. We prove that this method cannot be extended for higher dimension. Some examples are given and two open questions are proposed.