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, infinity) that are increasing and have - given bounded rates B <= q(v)-q(u)/v-u <= A 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.