This paper rigorously demonstrates that for any unequal income distribution, the well-known Gini index of inequality is bounded above by the recently revived Bonferroni inequality index. The bound is exactly attained if and only if out of n incomes in the society, n − 1 poor incomes are identical. The boundedness theorem is shown to possess a duality-type inequality implication. These two inequality metrics, two popular members of a general class of inequality indices generated by Aaberge’s (J Econ Inequal 5:305–322, 2007) ‘scaled conditional mean curve’, may lead to different directional rankings of alternative income distributions because of some important differences between them. We then explicitly examine their sensitivity to Weymark’s (Math Soc Sci 1:409–430, 1981) ‘comonotonic additivity’ postulate.

本文严格证明，对于任何不平等的收入分配，著名的基尼不平等指数都以最近恢复的邦费罗尼不平等指数为界。当且仅当社会中n个收入中有n个− 1个贫困收入相同。有界定理显示具有对偶型不等式蕴涵。这两个不平等度量标准是Aaberge（J Econ Inequal 5：305–322，2007）“缩放的条件均值曲线”生成的一般不平等指数的两个流行成员，可能会导致替代收入分配的方向不同，这是因为它们之间的重要区别。然后，我们明确检查了它们对Weymark（Math Soc Sci 1：409-430，1981）“共音可加性”假设的敏感性。