Comparison of Gini means with fixed number of variables

In this paper, we consider the global comparison problem of Gini means with fixed number of variables on a subinterval $I$ of ${\mathbb{R}}_{+}$, i.e., the following inequality \begin{align}\tag{$\star$}\label{ggcabs} G_{r,s}^{[n]}(x_1,\dots,x_n) \leq G_{p,q}^{[n]}(x_1,\dots,x_n), \end{align} where $n\in \mathbb{N},n\ge 2$ is fixed, $(p,q),(r,s)\in {\mathbb{R}}^2$ and $x_1,\dots ,x_n\in I$. Given a nonempty subinterval $I$ of ${\mathbb{R}}_{+}$ and $n\in \mathbb{N}$, we introduce the relations $${\mathrm{\Gamma }}_n(I):=\{((r,s),(p,q))\in {\mathbb{R}}^2\times {\mathbb{R}}^2\mid \text{eqref}ggcabsholdsforall{x}_1,\dots ,x_n\in I\},{\mathrm{\Gamma }}_{\mathrm{\infty }}(I):=\bigcap _{n=1}^{\mathrm{\infty }}{\mathrm{\Gamma }}_n(I)\mathrm{.}$$ In the paper, we investigate the properties of these sets and their dependence on $n$ and on the interval $I$ and we establish a characterizations of these sets via a constrained minimum problem by using a variant of the Lagrange multiplier rule. We also formulate two open problems at the end of the paper.