Quantile functions are equivalent alternatives to distribution functions in analysis and modelling of statistical data. Curves that measure inequality in incomes have been a topic of immense interest for more than a century ever since the work of Lorenz in 1905. A measure of income inequality is designed to provide an index that can abridge the variations prevailing in income among the individuals in a group. In the present paper, we study more aspects on the income inequality measures using quantile function approach. Aging concepts such as IFR, IFRA, NBU, HNBUE, ... have an important role in reliability. The inequality curves and indices and some links with reliability aging using quantile function approach are concentrated in this talk. We will focus on income inequality, quantile function and aging consepts and their links. We look into possible functional relationships between the income inequality measures and quantile function. We examine the possible relationships of the inequality measures as well as reliability concepts like mean residual life function and reversed mean residual life function. Then functional relationships enable us to establish characterization results for probability distributions.

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