Expansion of a bivariate symmetric mean in the neighbourhood of the first bisector

Abstract

In this paper, we investigate the behavior of a bivariate mean $M$ near the first bisector by
establishing, in several significant cases, an important expansion of $M$ derived from the
Taylor expansion of a single-variable function. These expansions are made explicit for a
number of classical means. This motivates the introduction of the concept of the
\emph{characteristic function} $Q_M$ of a mean $M$, defined as the second partial
derivative of $M$ with respect to its first variable, evaluated along the diagonal.
The function $Q_M$ measures the proximity of $M$ to the arithmetic mean near the first
bisector and provides a univariate analytic framework for comparing and classifying means.
We prove that inequalities between characteristic functions yield local inequalities between
the corresponding means, and that in the case of homogeneous means, such inequalities
hold globally.  We also examine several important classes of means, both classical and novel, including:
\emph{normal means}, \emph{additive means}, \emph{integral means of the first kind},
\emph{integral means of the second kind}, \emph{weighted integral means of the first kind},
and \emph{weighted integral means of the second kind}. For each class, we determine the
specific form taken by the characteristic functions $Q_M$ of the means $M$ it contains,
and we then study the injectivity and the surjectivity of the mapping $M \mapsto Q_M$
within the class. We also use characteristic functions to investigate intersections between
certain classes of means, highlighting one of the key strengths of this concept. Finally, we
introduce and study, for a given mean $M$, the class of \emph{$M$-means}, and show, in
particular, that the arithmetic-geometric mean $\AGM$ is an $M$-mean for a specific
weighted integral mean of the first kind $M$.