Gruss-Type Bounds for the Covariance of Transformed Random Variables

A number of problems in economics, finance, information theory, insurance, and generally in decision making under uncertainty rely on estimates of the covariance between (transformed) random variables, which can for example be losses, risks, incomes, financial returns, etc. Several avenues relying on inequalities for analyzing the covariance are available in the literature, bearing the names of Chebyshev, Gruss, Hoeffding, Kantorovich, and others. In the present paper we sharpen the upper bound of a Gruss-type covariance inequality by incorporating a notion of quadrant dependence between random variables and also utilizing the idea of constraining the means of the random variables