An upper bound and characterization of Gini's mean difference for correlated random variables

In this paper, we obtain an upper bound for the Gini mean difference for the case of jointly distributed random variables based on the knowledge about mean, variance and correlation of the distribution. We also derive some closed formulas for the Gini mean difference when the random variables involved have an absolutely continuous joint distribution. As applications of our results, we examine some particular examples based on elliptically contoured distributions. Specifically, multivariate normal and Student-$t$ are analyzed to illustrate the obtained results.