Bounds for Gini’s mean difference based on first four moments, with some applications

In this paper, we obtain lower and upper bounds for the Gini mean difference for the case of independent and identically distributed random variables based on the information about mean, variance, skewness, and kurtosis of the distribution. We also obtain some relationships between the three dispersion measures in the general case. The established results improve some well-known bounds and inequalities. These results are then used to sharpen some inequalities concerning Gini’s index, order statistics and premium principles. Examples demonstrate that the proposed bounds perform much better than the existing ones.