We develop a unified, nonparametric framework for sharp partial identification and inference on inequality indices when the econometrician only has coarse observations in the dimension of interest — for example via grouped tables or individual interval reports, possibly with additional linear restrictions such as income ratios. First, for a broad class of Schur-convex inequality measures, we characterize extremal allocations and show that sharp bounds are attained by distributions with simple, finite support, reducing the underlying infinite-dimensional problem to finite-dimensional optimization. Second, for indices that admit linear-fractional representations after suitable ordering of the data (including the Gini coefficient, quantile ratios, and the Hoover index), we recast the bound problems as linear or quadratic programs, yielding fast computation of numerically sharp bounds. Third, we establish $\sqrt{n}$ inference for bound endpoints using a uniform directional delta method and a bootstrap procedure for standard errors. In our empirical examples we compute sharp Gini bounds from household wealth data with mixed point and interval observations, and use historical U.S.\ income grouping tables to provide bounds on the time-series for the Gini, quantile ratios, and Hoover index.