Efron's monotonicity property for measures on ${\mathbb{R}}^2$

First we prove some kernel representations for the covariance of two functions taken on the same random variable and deduce kernel representations for some functionals of a continuous one-dimensional measure. Then we apply these formulas to extend Efron’s monotonicity property, given in Efron [1965] and valid for independent log-concave measures, to the case of general measures on ${\mathbb{R}}^2$. The new formulas are also used to derive some further quantitative estimates in Efron’s monotonicity property.