Comparison of Gini means with fixed number of variables

TL;DR

This paper studies the inequalities between Gini means with fixed variables over subintervals of positive reals, characterizing the conditions for their comparison and exploring the properties of the related relation sets.

Contribution

It introduces the sets of parameters where Gini mean inequalities hold for fixed variables and characterizes these sets using constrained optimization techniques.

Findings

Characterization of the sets mma_n(I) and mma_(I) for Gini mean inequalities.

Dependence of these sets on the number of variables n and the interval I.

Formulation of open problems related to Gini mean inequalities.

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 $n\in\mathbb{N},n\geq2$ 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 \[ \Gamma_n(I):=\{((r,s),(p,q))\in\mathbb{R}^2\times\mathbb{R}^2\mid \eqref{ggcabs}\mbox{ holds for all } x_1,\dots,x_n\in I\},\qquad \Gamma_\infty(I):=\bigcap_{n=1}^\infty\Gamma_n(I). \] 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.