Transformations that minimize the Gini index of a random variable and applications

Let X be a continuous or discrete random variable with values in [0,M] and consider all functions (here called transformations) q:[0,M]→[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q:[0,M]\to [0,\infty )$\end{document} that are increasing and have given bounded rates B≤q(v)−q(u)v−u≤A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B \le \frac {q(v)-q(u)}{v-u} \le A$\end{document} for u < v. We prove that among such transformations, there is a transformation q that minimizes the Gini index of q(X), and such a q can be chosen as piecewise linear with only two rates, namely A and B. In the motivation for the study, X represents the incomes of a population. Our results imply that among all such tax policies with fixed allowable minimum and maximum tax rates, there is a tax policy that minimizes the Gini index of the disposable incomes of the population and such a tax policy has only two brackets with the given minimum and maximum rates.