A new convexity-based inequality, characterization of probability distributions and some free-of-distribution tests

A new inequality between some functional of probability distribution functions is given. The inequality is based on strict convexity of a function used in functional definition. Equality sign in the inequality gives a characteristic property of some probability distributions. This fact together with special character of functional is used to construct free-of-distribution two sample tests. Key words: convex functions; probability distances; characterization of distributions; Cram\'{e}r - von Mises distance; statistical tests.