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

In this article we will define new inequalities, connecting some functionals of probability distribution functions. These inequalities are based on the strict convexity of functions that are used in functional definition.
The starting point of of this article is the work “Cramér–von Mises distance: probabilistic interpretation, confidence intervals and neighbourhood of model validation” by Ludwig Baringhaus and Norbert Henze. In our article we provide a generalization of inequality obtained in probabilistic interpretation of the Cramér–von Mises distance. If equality holds there appears a chance to give characterization of some probability distribution functions. Considering this fact and a special character of functional, it is possible to create a class of free-of-distribution two sample tests.