We derive a new inequality for uniform deviations of averages from their means. The inequality is a common generalization of previous results of Vapnik and Chervonenkis [1974, Theory of Pattern Recognition. Nauka, Moscow] and Pollard [1995, Uniform ratio limit theorems for empirical processes,...

We obtain Vapnik-Chervonenkis type upper bounds for the uniform deviation of probabilities from their expectations. The bounds sharpen previously known probability inequalities.