Comparison of Gini means with fixed number of variables

Abstract

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 \[ Γ_n(I):=\{((r,s),(p,q))\in\mathbb{R}^2\times\mathbb{R}^2\mid \eqref{ggcabs}\mbox{ holds for all } x_1,\dots,x_n\in I\},\qquad Γ_\infty(I):=\bigcap_{n=1}^\inftyΓ_n(I). \] 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.