Upper bounding the distance covariance of bounded random vectors

A classical statistical inequality is used to show that the distance covariance of two bounded random vectors is bounded from above by a simple function of the dimensionality and the bounds of the random vectors. Two special cases that further simplify the result are considered: one in which both random vectors have the same number of components, each component taking values in an interval of unit length, and the other in which both random vectors have one component.