Efron's monotonicity property for measures on R2

In this paper, we prove some kernel representations for the covariance of two functions taken on the same random variable and we 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 R2. The new formulas are also used to derive some further quantitative estimates in Efron's monotonicity property.