ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz

For independent X and Y in the inequality P(X.Y+.), we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).