Relations between the inequality indices Gini, Pietra, and Kolkata: Theory and data analysis.

We study relations between three inequality indices, namely the Gini (g), Pietra (p) and Kolkata (k) introduced in 1912, 1915, and 2014 respectively, and all are derived from the Lorenz function L(x) introduced in 1905. The Kolkata index [which corresponds to a fixed point of the complementary Lorenz function L_{c}(x)≡1-L(x)] gives the fraction k of wealth possessed by the richest 1-k fraction of people (k = 0.8 corresponds to Pareto's 80-20 law from 1896). We show rigorously that while the Pietra index value p should be greater than or equal to 2k-1, the Robin Hood index should strictly be equal to the excess wealth fraction 2k-1 possessed by the richest 1-k fraction of people. Our numerical data analysis for U.S. IRS income data (1983-2022), Bollywood (India) movie income data (1999-2024), and the citation inequalities across the publications by forty Nobel laureates (2020-2025) in economics, physics, chemistry, and medicine clearly show that p/(2k-1) is always greater than unity but the deviation is never more than 5%. Assuming some simple analytic form for the Lorenz function, we also derived the relations k=(1/2)+(3/8)g for small g values and p/g=3/4. However, by considering the Lorenz function appropriate for the generalized Pareto power law distribution, we obtain an extended range (1_{+}<p/g≤e/2) that captures most of the empirical data.