  Since the pioneering work of Gerhard Gruss dating back to 1935, Gruss's
inequality and, more generally, Gruss-type bounds for covariances have
fascinated researchers and found numerous applications in areas such as
economics, insurance, reliability, and, more generally, decision making under
uncertainly. Gruss-type bounds for covariances have been established mainly
under most general dependence structures, meaning no restrictions on the
dependence structure between the two underlying random variables. Recent work
in the area has revealed a potential for improving Gruss-type bounds, including
the original Gruss's bound, assuming dependence structures such as quadrant
dependence (QD). In this paper we demonstrate that the relatively little
explored notion of `quadrant dependence in expectation' (QDE) is ideally suited
in the context of bounding covariances, especially those that appear in the
aforementioned areas of application. We explore this research avenue in detail,
establish general Gruss-type bounds, and illustrate them with newly constructed
examples of bivariate distributions, which are not QD but, nevertheless, are
QDE. The examples rely on specially devised copulas. We supplement the examples
with results concerning general copulas and their convex combinations. In the
process of deriving Gruss-type bounds, we also establish new bounds for central
moments, whose optimality is demonstrated.
