### Abstract

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.